probability

Anonymous
timer Asked: Dec 25th, 2018
account_balance_wallet $30

Question description


In a small grocery store, an average of 2 out of 5 customers pay with a credit card. A random sample of 12 customers is selected.

(1) Find the probability that exactly 8 customers pay with a credit card. Show calculations.

(2) Find the probability that less than 2 customers pay with a credit card. Show calculations.

(3) Find the probability that more than 2 customer pay with a credit card. Show calculations.

Statistics for Business and Economics (13e) Statistics for Business and Economics (13e) Anderson, Sweeney, Williams, Camm, Cochran © 2017 Cengage Learning Slides by John Loucks St. Edwards University © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 1 Statistics for Business and Economics (13e) Chapter 1 - Data and Statistics • • • • • • • • • • Statistics Applications in Business and Economics Data Data Sources Descriptive Statistics Statistical Inference Analytics Big Data and Data Mining Computers and Statistical Analysis Ethical Guidelines for Statistical Practice © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 2 Statistics for Business and Economics (13e) What is Statistics? • The term statistics can refer to numerical facts such as averages, medians, percentages, and maximums that help us understand a variety of business and economic situations. • Statistics can also refer to the art and science of collecting, analyzing, presenting, and interpreting data. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 3 Statistics for Business and Economics (13e) Applications in Business and Economics Accounting • Public accounting firms use statistical sampling procedures when conducting audits for their clients. Economics • Economists use statistical information in making forecasts about the future of the economy or some aspect of it. Finance • Financial advisors use price-earnings ratios and dividend yields to guide their investment advice. p/e ratio: the price an investor is paying for $1 of a company's earnings or profit. In other words, if a company is reporting basic earnings per share of $2 and the stock is selling for $20 per share, the p/e ratio is 10 ($20 per share divided by $2 earnings per share = 10 p/e) © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 4 Statistics for Business and Economics (13e) Applications in Business and Economics Marketing • Electronic point-of-sale scanners at retail checkout counters are used to collect data for a variety of marketing research applications. Production • A variety of statistical quality control charts are used to monitor the output of a production process. Information Systems • A variety of statistical information helps administrators assess the performance of computer networks. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 5 Statistics for Business and Economics (13e) Data and Data Sets • Data are the facts and figures collected, analyzed, and summarized for presentation and interpretation. • All the data collected in a particular study are referred to as the data set for the study. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 6 Statistics for Business and Economics (13e) Elements, Variables, and Observations • Elements are the entities on which data are collected. • A variable is a characteristic of interest for the elements. • The set of measurements obtained for a particular element is called an observation. • A data set with n elements contains n observations. • The total number of data values in a complete data set is the number of elements multiplied by the number of variables. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 7 Statistics for Business and Economics (13e) Data, Data Sets, Elements, Variables, and Observations Variables Element Names Company Stock Exchange Annual Sales ($M) Earnings per share ($) Dataram NQ 73.10 0.86 EnergySouth N 74.00 1.67 Keystone N 365.70 0.86 LandCare NQ 111.40 0.33 N 17.60 0.13 Psychemedics Observation Data Set © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 8 Statistics for Business and Economics (13e) Scales of Measurement • Scales of measurement include • • • • Nominal Ordinal Interval Ratio • The scale determines the amount of information contained in the data. • The scale indicates the data summarization and statistical analyses that are most appropriate. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 9 Statistics for Business and Economics (13e) Scales of Measurement Nominal scale • Data are labels or names used to identify an attribute of the element. • A nonnumeric label or numeric code may be used. Example Students of a university are classified by the school in which they are enrolled using a nonnumeric label such as Business, Humanities, Education, and so on. Alternatively, a numeric code could be used for the school variable (e.g. 1 denotes Business, 2 denotes Humanities, 3 denotes Education, and so on). © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 10 Statistics for Business and Economics (13e) Scales of Measurement Ordinal scale • The data have the properties of nominal data and the order or rank of the data is meaningful. • A nonnumeric label or numeric code may be used. Example Students of a university are classified by their class standing using a nonnumeric label such as Freshman, Sophomore, Junior, or Senior. Alternatively, a numeric code could be used for the class standing variable (e.g. 1 denotes Freshman, 2 denotes Sophomore, and so on). © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 11 Statistics for Business and Economics (13e) Scales of Measurement Interval scale • The data have the properties of ordinal data, and the interval between observations is expressed in terms of a fixed unit of measure. • Interval data are always numeric. Example Melissa has an SAT score of 1985, while Kevin has an SAT score of 1880. Melissa scored 105 points more than Kevin. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 12 Statistics for Business and Economics (13e) Scales of Measurement Ratio scale • Data have all the properties of interval data and the ratio of two values is meaningful. • Ratio data are always numerical. Example: Price of a book at a retail store is $ 200, while the price of the same book sold online is $100. The ratio property shows that retail stores charge twice the online price. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 13 Statistics for Business and Economics (13e) Categorical and Quantitative Data • Data can be further classified as being categorical or quantitative. • The statistical analysis that is appropriate depends on whether the data for the variable are categorical or quantitative. • In general, there are more alternatives for statistical analysis when the data are quantitative. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 14 Statistics for Business and Economics (13e) Categorical Data • • • • • Labels or names are used to identify an attribute of each element Often referred to as qualitative data Use either the nominal or ordinal scale of measurement Can be either numeric or nonnumeric Appropriate statistical analyses are rather limited © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 15 Statistics for Business and Economics (13e) Quantitative Data • Quantitative data indicate how many or how much. • Quantitative data are always numeric. • Ordinary arithmetic operations are meaningful for quantitative data. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 16 Statistics for Business and Economics (13e) Cross-Sectional Data Cross-sectional data are collected at the same or approximately the same point in time. Example Data detailing the number of building permits issued in November 2013 in each of the counties of Ohio. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 17 Statistics for Business and Economics (13e) Time Series Data Time series data are collected over several time periods. Example Data detailing the number of building permits issued in Lucas County, Ohio in each of the last 36 months. Graphs of time series data help analysts understand • what happened in the past • identify any trends over time, and • project future levels for the time series © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 18 Statistics for Business and Economics (13e) Time Series Data Graph of Time Series Data © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 19 Statistics for Business and Economics (13e) Data Sources Existing Sources • Internal company records – almost any department • Business database services – Dow Jones & Co. • Government agencies - U.S. Department of Labor • Industry associations – Travel Industry Association of America • Special-interest organizations – Graduate Management Admission Council (GMAT) • Internet – more and more firms © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 20 Statistics for Business and Economics (13e) Data Sources Data Available From Internal Company Records Record Some of the Data Available Employee records Name, address, social security number Production records Part number, quantity produced, direct labor cost, material cost Inventory records Part number, quantity in stock, reorder level, economic order quantity Sales records Product number, sales volume, sales volume by region Credit records Customer name, credit limit, accounts receivable balance Customer profile Age, gender, income, household size © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 21 Statistics for Business and Economics (13e) Data Sources Data Available From Selected Government Agencies Government Agency Web address Some of the Data Available Census Bureau www.census.gov Population data, number of households, household income Federal Reserve Board www.federalreserve.gov Data on money supply, exchange rates, discount rates Office of Mgmt. & Budget www.whitehouse.gov/omb Data on revenue, expenditures, debt of federal government Department of Commerce www.doc.gov Data on business activity, value of shipments, profit by industry Bureau of Labor Statistics Customer spending, unemployment rate, hourly earnings, safety record www.bls.gov © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 22 Statistics for Business and Economics (13e) Data Sources Statistical Studies – Observational • In observational (nonexperimental) studies no attempt is made to control or influence the variables of interest. • Example - Survey • Studies of smokers and nonsmokers are observational studies because researchers do not determine or control who will smoke and who will not smoke. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 23 Statistics for Business and Economics (13e) Data Sources Statistical Studies – Experimental • In experimental studies the variable of interest is first identified. Then one or more other variables are identified and controlled so that data can be obtained about how they influence the variable of interest. • The largest experimental study ever conducted is believed to be the 1954 Public Health Service experiment for the Salk polio vaccine. Nearly two million U.S. children (grades 1- 3) were selected. • Across the US, 623,972 schoolchildren were injected with vaccine or placebo, and more than a million others participated as “observed” controls. The results, announced in 1955, showed good statistical evidence that Salk polio vaccine was 8090% effective in preventing paralytic poliomyelitis. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 24 Statistics for Business and Economics (13e) Data Acquisition Considerations Time Requirement • Searching for information can be time consuming. • Information may no longer be useful by the time it is available. Cost of Acquisition • Organizations often charge for information even when it is not their primary business activity. Data Errors • Using any data that happen to be available or were acquired with little care can lead to misleading information. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 25 Statistics for Business and Economics (13e) Descriptive Statistics • Most of the statistical information in newspapers, magazines, company reports, and other publications consists of data that are summarized and presented in a form that is easy to understand. • Such summaries of data, which may be tabular, graphical, or numerical, are referred to as descriptive statistics. Example The manager of Hudson Auto would like to have a better understanding of the cost of parts used in the engine tune-ups performed in her shop. She examines 50 customer invoices for tune-ups. The costs of parts, rounded to the nearest dollar, are listed on the next slide. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 26 Statistics for Business and Economics (13e) Example: Hudson Auto Repair Sample of Parts Cost ($) for 50 Tune-ups 91 78 93 57 75 52 99 80 97 62 71 69 72 89 66 75 79 75 72 76 104 74 62 68 97 105 77 65 80 109 85 97 88 68 83 68 71 69 67 74 62 82 98 101 79 105 79 69 62 73 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 27 Statistics for Business and Economics (13e) Tabular Summary: Frequency and Percent Frequency Parts Cost ($) Frequency Percent Frequency 50-59 2 4% 60-69 13 26% 70-79 16 32% 80-89 7 14% 90-99 7 14% 100-109 5 10% 50 100% TOTAL © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 28 Statistics for Business and Economics (13e) Graphical Summary: Histogram Example: Hudson Auto Tune-up Parts Cost 18 16 14 Frequency 12 10 8 6 4 2 0 50-59 60-69 70-79 80-89 90-99 Parts Cost ($) © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 29 Statistics for Business and Economics (13e) Numerical Descriptive Statistics • The most common numerical descriptive statistic is the mean (or average). • The mean demonstrates a measure of the central tendency, or central location of the data for a variable. • Hudson’s mean cost of parts, based on the 50 tune-ups studied is $79 (found by summing up the 50 cost values and then dividing by 50). © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 30 Statistics for Business and Economics (13e) Statistical Inference Population: The set of all elements of interest in a particular study. Sample: A subset of the population. Statistical inference: The process of using data obtained from a sample to make estimates and test hypotheses about the characteristics of a population. Census: Collecting data for the entire population. Sample survey: Collecting data for a sample. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 31 Statistics for Business and Economics (13e) Process of Statistical Inference Example: Hudson Auto Step 1 Step 2 Step 3 Step 4 • Population consists of all tune ups. Average cost of parts is unknown. • A sample of 50 engine tune-ups is examined. • The sample data provides a sample average parts cost of $79 per tune-up. • The sample average is used to estimate the population average. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 32 Statistics for Business and Economics (13e) Analytics Analytics is the scientific process of transforming data into insight for making better decisions. Techniques: • Descriptive analytics: This describes what has happened in the past. • Predictive analytics: Use models constructed from past data to predict the future or to assess the impact of one variable on another. • Prescriptive analytics: The set of analytical techniques that yield a best course of action. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 33 Statistics for Business and Economics (13e) Big data and Data Mining: Big data: Large and complex data set. Three V’s of Big data: Volume : Amount of available data Velocity: Speed at which data is collected and processed Variety: Different data types © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 34 Statistics for Business and Economics (13e) Data warehousing Data warehousing is the process of capturing, storing, and maintaining the data. • Organizations obtain large amounts of data on a daily basis by means of magnetic card readers, bar code scanners, point of sale terminals, and touch screen monitors. • Wal-Mart captures data on 20-30 million transactions per day. • Visa processes 6,800 payment transactions per second. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 35 Statistics for Business and Economics (13e) Data Mining • Methods for developing useful decision-making information from large databases. • Using a combination of procedures from statistics, mathematics, and computer science, analysts “mine the data” to convert it into useful information. • The most effective data mining systems use automated procedures to discover relationships in the data and predict future outcomes prompted by general and even vague queries by the user. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 36 Statistics for Business and Economics (13e) Data Mining Applications • The major applications of data mining have been made by companies with a strong consumer focus such as retail, financial, and communication firms. • Data mining is used to identify related products that customers who have already purchased a specific product are also likely to purchase (and then pop-ups are used to draw attention to those related products). • Data mining is also used to identify customers who should receive special discount offers based on their past purchasing volumes. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 37 Statistics for Business and Economics (13e) Data Mining Requirements • Statistical methodology such as multiple regression, logistic regression, and correlation are heavily used. • Also needed are computer science technologies involving artificial intelligence and machine learning. • A significant investment in time and money is required as well. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 38 Statistics for Business and Economics (13e) Data Mining Model Reliability • Finding a statistical model that works well for a particular sample of data does not necessarily mean that it can be reliably applied to other data. • With the enormous amount of data available, the data set can be partitioned into a training set (for model development) and a test set (for validating the model). • There is, however, a danger of overfitting the model to the point that misleading associations and conclusions appear to exist. • Careful interpretation of results and extensive testing is important. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 39 Statistics for Business and Economics (13e) Ethical Guidelines for Statistical Practice • In a statistical study, unethical behavior can take a variety of forms including: • • • • • Improper sampling Inappropriate analysis of the data Development of misleading graphs Use of inappropriate summary statistics Biased interpretation of the statistical results • One should strive to be fair, thorough, objective, and neutral as you collect, analyze, and present data. • As a consumer of statistics, one should also be aware of the possibility of unethical behavior by others. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 40 Statistics for Business and Economics (13e) Ethical Guidelines for Statistical Practice • The American Statistical Association developed the report “Ethical Guidelines for Statistical Practice”. • It contains 67 guidelines organized into 8 topic areas: • • • • • • • • Professionalism Responsibilities to Funders, Clients, Employers Responsibilities in Publications and Testimony Responsibilities to Research Subjects Responsibilities to Research Team Colleagues Responsibilities to Other Statisticians/Practitioners Responsibilities Regarding Allegations of Misconduct Responsibilities of Employers Including Organizations, Individuals, Attorneys, or Other Clients © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 41 Statistics for Business and Economics (13e) End of Chapter 1 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 42
Statistics for Business and Economics (13e) Statistics for Business and Economics (13e) Anderson, Sweeney, Williams, Camm, Cochran © 2017 Cengage Learning Slides by John Loucks St. Edwards University © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 1 Statistics for Business and Economics (13e) Chapter 2 Part A - Descriptive Statistics: Tabular and Graphical Displays • Summarizing Data for a Categorical Variable • Categorical data use labels or names to identify categories of like items. • Summarizing Data for a Quantitative Variable • Quantitative data are numerical values that indicate how much or how many. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 2 Statistics for Business and Economics (13e) Summarizing Categorical Data • • • • • Frequency Distribution Relative Frequency Distribution Percent Frequency Distribution Bar Chart Pie Chart © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 3 Statistics for Business and Economics (13e) Frequency Distribution • A frequency distribution is a tabular summary of data showing the number (frequency) of observations in each of several non-overlapping categories or classes. • The objective is to provide insights about the data that cannot be quickly obtained by looking only at the original data. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 4 Statistics for Business and Economics (13e) Frequency Distribution Example: Marada Inn • Guests staying at Marada Inn were asked to rate the quality of their accommodations as being excellent, above average, average, below average, or poor. • The ratings provided by a sample of 20 guests are: Below Average Average Above Average Above Average Above Average Above Average Above Average Below Average Below Average Average Poor Poor Above Average Excellent Above Average Average Above Average Average Above Average Average © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 5 Statistics for Business and Economics (13e) Frequency Distribution • Example: Marada Inn Rating Frequency Poor 2 Below Average 3 Average 5 Above Average 9 Excellent 1 Total 20 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 6 Statistics for Business and Economics (13e) Relative Frequency Distribution • The relative frequency of a class is the fraction or proportion of the total number of data items belonging to the class. Relative frequency of a class = Frequency of the class 𝑛 • A relative frequency distribution is a tabular summary of data showing the relative frequency for each class. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 7 Statistics for Business and Economics (13e) Percent Frequency Distribution • The percent frequency of a class is the relative frequency multiplied by 100. • A percent frequency distribution is a tabular summary of a set of data showing the percent frequency for each class. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 8 Statistics for Business and Economics (13e) Relative Frequency and Percent Frequency Distributions • Example: Marada Inn Rating Relative Frequency Percent Frequency Poor .10 10 Below Average .15 15 Average .25 25 Above Average .45 45 Excellent .05 5 1.00 100 Total © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 9 Statistics for Business and Economics (13e) Bar Chart • A bar chart is a graphical display for depicting qualitative data. • On one axis (usually the horizontal axis), we specify the labels that are used for each of the classes. • A frequency, relative frequency, or percent frequency scale can be used for the other axis (usually the vertical axis). • Using a bar of fixed width drawn above each class label, we extend the height appropriately. • The bars are separated to emphasize the fact that each class is a separate category. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 10 Statistics for Business and Economics (13e) Bar Chart Marada Inn Quality Ratings 10 9 Frequency 8 7 6 5 4 3 2 1 Poor Below Average Average Above Average Excellent Quality Rating © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 11 Statistics for Business and Economics (13e) Pareto Diagram • In quality control, bar charts are used to identify the most important causes of problems. • When the bars are arranged in descending order of height from left to right (with the most frequently occurring cause appearing first) the bar chart is called a Pareto diagram. • This diagram is named for its founder, Vilfredo Pareto, an Italian economist. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 12 Statistics for Business and Economics (13e) Pie Chart • The pie chart is a commonly used graphical display for presenting relative frequency and percent frequency distributions for categorical data. • First draw a circle; then use the relative frequencies to subdivide the circle into sectors that correspond to the relative frequency for each class. • Since there are 360 degrees in a circle, a class with a relative frequency of .25 would consume .25(360) = 90 degrees of the circle. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 13 Statistics for Business and Economics (13e) Pie Chart Marada Inn Quality Ratings Excellent 5% Poor 10% Above Average 45% Below Average 15% Average 25% Average 25% © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 14 Statistics for Business and Economics (13e) Example: Marada Inn from thesurveyed Preceding Pie Chart a quality rating of “above • Insights One-half Gained of the customers gave Marada average” or “excellent” (looking at the left side of the pie). This might please the manager. • For each customer who gave an “excellent” rating, there were two customers who gave a “poor” rating (looking at the top of the pie). This should displease the manager. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 15 Statistics for Business and Economics (13e) Summarizing Quantitative Data • Frequency Distribution • Relative Frequency and Percent Frequency Distributions • Dot Plot • Histogram • Cumulative Distributions • Stem-and-Leaf Display © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 16 Statistics for Business and Economics (13e) Frequency Distribution • Example: Hudson Auto Repair The manager of Hudson Auto would like to gain a better understanding of the cost of parts used in the engine tune-ups performed in the shop. She examines 50 customer invoices for tune-ups. The costs of parts, rounded to the nearest dollar, are listed on the next slide. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 17 Statistics for Business and Economics (13e) Frequency Distribution • Example: Hudson Auto Repair Sample of Parts Cost($) for 50 Tune-ups 91 71 104 85 62 78 69 74 97 82 93 72 62 88 98 57 89 68 68 101 75 66 97 83 79 52 75 105 68 105 99 79 77 71 79 80 75 65 69 69 97 72 80 67 62 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 62 76 109 74 73 18 Statistics for Business and Economics (13e) Frequency Distribution The three steps necessary to define the classes for a frequency distribution with quantitative data are: 1. Determine the number of non-overlapping classes. 2. Determine the width of each class. 3. Determine the class limits. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 19 Statistics for Business and Economics (13e) Frequency Distribution • Guidelines for Determining the Number of Classes • Use between 5 and 20 classes. • Data sets with a larger number of elements usually require a larger number of classes. • Smaller data sets usually require fewer classes. • The goal is to use enough classes to show the variation in the data, but not so many classes that some contain only a few data items. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 20 Statistics for Business and Economics (13e) Frequency Distribution • Guidelines for Determining the Width of Each Class • Use classes of equal width. • Approximate Class Width = Largest data value − Smallest data value Number of classes • Making the classes the same width reduces the chance of inappropriate interpretations. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 21 Statistics for Business and Economics (13e) Frequency Distribution • Note on Number of Classes and Class Width • In practice, the number of classes and the appropriate class width are determined by trial and error. • Once a possible number of classes is chosen, the appropriate class width is found. • The process can be repeated for a different number of classes. • Ultimately, the analyst uses judgment to determine the combination of the number of classes and class width that provides the best frequency distribution for summarizing the data. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 22 Statistics for Business and Economics (13e) Frequency Distribution • Guidelines for Determining the Class Limits • Class limits must be chosen so that each data item belongs to one and only one class. • The lower class limit identifies the smallest possible data value assigned to the class. • The upper class limit identifies the largest possible data value assigned to the class. • An open-end class requires only a lower class limit or an upper class limit. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 23 Statistics for Business and Economics (13e) Frequency Distribution • Class Midpoint • In some cases, we want to know the midpoints of the classes in a frequency distribution for quantitative data. • The class midpoint is the value halfway between the lower and upper class limits. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 24 Statistics for Business and Economics (13e) Frequency Distribution • Example: Hudson Auto Repair If we choose six classes: Approximate Class Width = (109 - 52)/6 = 9.5 10 Parts Cost ($) 50-59 60-69 70-79 80-89 90-99 100-109 Frequency 2 13 16 7 7 5 Total 50 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 25 Statistics for Business and Economics (13e) Relative Frequency and Percent Frequency Distributions • Example: Hudson Auto Repair Parts Cost ($) 50-59 60-69 70-79 80-89 90-99 100-109 Relative Frequency .04 = 2/50 .26 .32 .14 .14 .10 Total 1.00 Percent Frequency 4 = .04(100) 26 32 14 14 10 100 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 26 Statistics for Business and Economics (13e) Relative Frequency and Percent Frequency Distributions • Example: Hudson Auto Repair Insights Gained from the Percent Frequency Distribution: • Only 4% of the parts costs are in the $50-59 class. • 30% of the parts costs are under $70. • The greatest percentage (32% or almost one-third) of the parts costs are in the $70-79 class. • 10% of the parts costs are $100 or more. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 27 Statistics for Business and Economics (13e) Dot Plot • One of the simplest graphical summaries of data is a dot plot. • A horizontal axis shows the range of data values. • Then each data value is represented by a dot placed above the axis. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 28 Statistics for Business and Economics (13e) Dot Plot • Example: Hudson Auto Repair 50 60 70 80 90 100 110 Tune-up Parts Cost ($) © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 29 Statistics for Business and Economics (13e) Histogram • Another common graphical display of quantitative data is a histogram. • The variable of interest is placed on the horizontal axis. • A rectangle is drawn above each class interval with its height corresponding to the interval’s frequency, relative frequency, or percent frequency. • Unlike a bar graph, a histogram has no natural separation between rectangles of adjacent classes. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 30 Statistics for Business and Economics (13e) Histogram • Example: Hudson Auto Repair 18 Tune-up Parts Cost 16 Frequency 14 12 10 8 6 4 2 50-59 60-69 70-79 80-89 90-99 100-110 Parts Cost ($) © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 31 Statistics for Business and Economics (13e) Histograms Showing Skewness • Symmetric • Left tail is the mirror image of the right tail • Example: Heights of People Relative Frequency .35 .30 .25 .20 .15 .10 .05 0 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 32 Statistics for Business and Economics (13e) Histograms Showing Skewness • Moderately Skewed Left • A longer tail to the left • Example: Exam Scores Relative Frequency .35 .30 .25 .20 .15 .10 .05 0 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 33 Statistics for Business and Economics (13e) Histograms Showing Skewness • Moderately Right Skewed • A Longer tail to the right • Example: Housing Values Relative Frequency .35 .30 .25 .20 .15 .10 .05 0 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 34 Statistics for Business and Economics (13e) Histograms Showing Skewness • Highly Skewed Right • A very long tail to the right • Example: Executive Salaries Relative Frequency .35 .30 .25 .20 .15 .10 .05 0 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 35 Statistics for Business and Economics (13e) Skewness – Left and Right © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 36 Statistics for Business and Economics (13e) Cumulative Distributions • Cumulative frequency distribution - shows the number of items with values less than or equal to the upper limit of each class. • Cumulative relative frequency distribution – shows the proportion of items with values less than or equal to the upper limit of each class. • Cumulative percent frequency distribution – shows the percentage of items with values less than or equal to the upper limit of each class. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 37 Statistics for Business and Economics (13e) Cumulative Distributions • The last entry in a cumulative frequency distribution always equals the total number of observations. • The last entry in a cumulative relative frequency distribution always equals 1.00. • The last entry in a cumulative percent frequency distribution always equals 100. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 38 Statistics for Business and Economics (13e) Frequency Distribution • Example: Hudson Auto Repair If we choose six classes: Approximate Class Width = (109 - 52)/6 = 9.5 10 Parts Cost ($) 50-59 60-69 70-79 80-89 90-99 100-109 Frequency 2 13 16 7 7 5 Total 50 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 39 Statistics for Business and Economics (13e) Relative Frequency and Percent Frequency Distributions • Example: Hudson Auto Repair Parts Cost ($) 50-59 60-69 70-79 80-89 90-99 100-109 Relative Frequency .04 = 2/50 .26 .32 .14 .14 .10 Total 1.00 Percent Frequency 4 = .04(100) 26 32 14 14 10 100 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 40 Statistics for Business and Economics (13e) Cumulative Distributions • Hudson Auto Repair Cost ($) < 59 < 69 < 79 < 89 < 99 < 109 Cumulative Frequency 2 15 = 2+13 31 38 45 50 Cumulative Relative Frequency .04 .30 = 15/50 .62 .76 .90 1.00 Cumulative Percent Frequency 4 30 = .30(100) 62 76 90 100 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 41 Statistics for Business and Economics (13e) Stem-and-Leaf Display • A stem-and-leaf display shows both the rank order and shape of a distribution of data. • It is similar to a histogram on its side, but it has the advantage of showing the actual data values. • The leading digits of each data item are arranged to the left of a vertical line. • To the right of the vertical line we record the last digit for each item in rank order. • Each line (row) in the display is referred to as a stem. • Each digit on a stem is a leaf. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 42 Statistics for Business and Economics (13e) Example: Hudson Auto Repair The manager of Hudson Auto would like to gain a better understanding of the cost of parts used in the engine tune-ups performed in the shop. She examines 50 customer invoices for tune-ups. The costs of parts, rounded to the nearest dollar, are listed on the next slide. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 43 Statistics for Business and Economics (13e) Stem-and-Leaf Display • Example: Hudson Auto Repair Sample of Parts Cost ($) for 50 Tune-ups 91 71 104 85 62 78 69 74 97 82 93 72 62 88 98 57 89 68 68 101 75 66 97 83 79 52 75 105 68 105 99 79 77 71 79 80 75 65 69 69 97 72 80 67 62 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 62 76 109 74 73 44 Statistics for Business and Economics (13e) Stem-and-Leaf Display • Example: Hudson Auto Repair 5 6 7 8 9 10 Stems 2 2 1 0 1 1 7 2 1 0 3 4 2 2 2 7 5 2 2 3 7 5 5 3 5 7 9 6 4 8 8 7 8 8 8 9 9 9 4 5 5 5 6 7 8 9 9 9 9 9 Leaves © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 45 Statistics for Business and Economics (13e) Stretched Stem-and-Leaf Display • Whenever a stem value is stated twice, the first value corresponds to leaf values of 0 - 4, and the second value corresponds to leaf values of 5 - 9. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 46 Statistics for Business and Economics (13e) Stretched Stem-and-Leaf Display • Example: Hudson Auto Repair 5 5 6 6 7 7 8 8 9 9 10 10 2 7 2 5 1 5 0 5 1 7 1 5 2 6 1 5 0 8 3 7 4 5 2 7 2 5 2 9 2 8 8 8 9 9 9 2 3 4 4 6 7 8 9 9 9 3 7 8 9 9 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 47 Statistics for Business and Economics (13e) Stem-and-Leaf Display • Leaf Units • A single digit is used to define each leaf. • In the preceding example, the leaf unit was 1. • Leaf units may be 100, 10, 1, 0.1, and so on. • Where the leaf unit is not shown, it is assumed to equal 1. • The leaf unit indicates how to multiply the stem-and-leaf numbers in order to approximate the original data. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 48 Statistics for Business and Economics (13e) Stem-and-Leaf Display • Example: Leaf Unit = 0.1 8.6 If we have data with values such as 11.7 9.4 9.1 10.2 11.0 8.8 Leaf Unit = 0.1 8 9 10 11 6 8 1 4 2 0 7 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 49 Statistics for Business and Economics (13e) End of Chapter 2, Part A © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 50
. . . . . . . . . . . . SLIDES BY John Loucks St. Edward’s Univ. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 1 Chapter 3, Part A Discrete Probability Distributions Introduction to probability Random Variables Discrete Probability Distributions Binomial Probability Distribution Poisson Probability Distribution .40 .30 .20 .10 0 1 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2 3 4 Slide 2 Uncertainties Managers often base their decisions on an analysis of uncertainties such as the following: What are the chances that sales will decrease if we increase prices? What is the likelihood a new assembly method will increase productivity? What are the odds that a new investment will be profitable? © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 3 Probability Probability is a numerical measure of the likelihood that an event will occur. Probability values are always assigned on a scale from 0 to 1. A probability near zero indicates an event is quite unlikely to occur. A probability near one indicates an event is almost certain to occur. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 4 Probability as a Numerical Measure of the Likelihood of Occurrence Increasing Likelihood of Occurrence Probability: 0 The event is very unlikely to occur. .5 The occurrence of the event is just as likely as it is unlikely. 1 The event is almost certain to occur. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 5 Statistical Experiments In statistics, the notion of an experiment differs somewhat from that of an experiment in the physical sciences. In statistical experiments, probability determines outcomes. Even though the experiment is repeated in exactly the same way, an entirely different outcome may occur. For this reason, statistical experiments are sometimes called random experiments. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 6 Assigning Probabilities ◼ Basic Requirements for Assigning Probabilities 1. The probability assigned to each experimental outcome must be between 0 and 1, inclusively. 0 < P(Ei) < 1 for all i Where: Ei is the ith experimental outcome and P(Ei) is its probability 2. The sum of the probabilities for all experimental outcomes must equal 1. P(E1) + P(E2) + . . . + P(En) = 1 Where: n is the number of experimental outcomes © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 7 1. Random Variables A random variable is a numerical description of the outcome of an experiment. A discrete random variable may assume either a finite number of values or an infinite sequence of values. A continuous random variable may assume any numerical value in an interval or collection of intervals. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 8 Random Variables Examples of Random Variables The first, second, and fourth variables above are discrete, while the third one is continuous. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 9 Random Variables Examples of Random Variables Question Family size Type Random Variable x x = Number of dependents in family reported on tax return Discrete Distance from x = Distance in miles from home to store home to the store site Continuous Own dog or cat Discrete x = 1 if own no pet; = 2 if own dog(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s) © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 10 Example: JSL Appliances Discrete random variable with a finite number of values Let x = number of TVs sold at the store in one day, where x can take on 5 values (0, 1, 2, 3, 4) © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 11 Example: JSL Appliances Discrete random variable with an infinite sequence of values Let x = number of customers arriving in one day, where x can take on the values 0, 1, 2, . . . We can count the customers arriving, but there is no finite upper limit on the number that might arrive. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 12 2. Discrete Probability Distributions The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable. We can describe a discrete probability distribution with a table, graph, or equation. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 13 Discrete Probability Distributions The probability distribution is defined by a probability function, denoted by f(x), which provides the probability for each value of the random variable. The required conditions for a discrete probability function are: f(x) > 0 f(x) = 1 For a discrete probability distribution we calculate the probability of being less than some value x, i.e. P(X < x), by simply summing up the probabilities of the values less than x. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 14 Example: DiCarlo Motors, Inc. Using past data on daily car sales, … a tabular representation of the probability distribution for car sales was developed. x 0 1 2 3 4 5 f(x) .18 .39 .24 .14 .04 .01 1.00 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 15 Example: DiCarlo Motors, Inc. Graphical Representation of the Probability Distribution Probability .50 .40 .30 .20 .10 0 1 2 3 4 5 Values of Random Variable x (car sales) © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 16 Example: DiCarlo Motors, Inc. The probability distribution provides the following information. • There is a 0.18 probability that no cars will be sold during a day. • The most probable sales volume is 1, with f(1) = 0.39. • There is a 0.05 probability of an outstanding sales day with four or five cars being sold. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 17 Discrete Uniform Probability Distribution The discrete uniform probability distribution is the simplest example of a discrete probability distribution given by a formula. The discrete uniform probability function is f(x) = 1/n the values of the random variable are equally likely where: n = the number of values the random variable may assume © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 18 Expected Value and Variance The expected value, or mean, of a random variable is a measure of its central location. E(x) =  = xf(x) The variance summarizes the variability in the values of a random variable. Var(x) =  2 = (x - )2f(x) The standard deviation, , is defined as the positive square root of the variance. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 19 Example: DiCarlo Motors, Inc. Expected Value of a Discrete Random Variable x 0 1 2 3 4 5 f(x) xf(x) .18 .00 .39 .39 .24 .48 .14 .42 .04 .16 .01 .05 E(x) = 1.50 expected number of cars sold in a day © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 20 Example: DiCarlo Motors, Inc. Variance and Standard Deviation of a Discrete Random Variable x x- (x - )2 0 1 2 3 4 5 .18 -1.5 2.25 .4050 .39 -0.5 0.25 .0975 .24 0.5 0.25 .0600 .14 1.5 2.25 .3150 .04 2.5 6.25 .2500 .01 3.5 12.25 .1225 Variance of daily sales =  2 = 1.2500 f(x) (x - )2f(x) cars squared Standard deviation of daily sales = 1.118 cars © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 21 3. Binomial Probability Distribution Four Properties of a Binomial Experiment 1. The experiment consists of a sequence of n identical trials. 2. Two outcomes, success and failure, are possible on each trial. 3. The probability of a success, denoted by p, does not change from trial to trial. stationarity assumption 4. The trials are independent. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 22 Binomial Probability Distribution Our interest is in the number of successes occurring in the n trials. We let x denote the number of successes occurring in the n trials. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 23 Binomial Probability Distribution Binomial Probability Function n! f (x) = p x (1 − p)( n − x ) x !(n − x )! where: f(x) = the probability of x successes in n trials p = the probability of success on any one trial n = the number of trials x = number of successes in n trials n! =n( n - 1)( n - 2) . . . (2)(1) © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 24 Binomial Probability Distribution Binomial Probability Function n! f (x) = p x (1 − p)( n − x ) x !(n − x )! Number of experimental outcomes providing exactly x successes in n trials Probability of a particular sequence of trial outcomes with x successes in n trials © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 25 Example: Nastke Clothing Store Binomial Probability Distribution The store manager estimates that the probability of a customer making a purchase is 0.30. What is the probability that 2 of the next 3 customers entering the store make a purchase? Let: p = .30 (success), n = 3, x = 2 n! f ( x) = p x (1 − p ) (n − x ) x !( n − x )! 3! f (2) = (0.3)2 (0.7)1 = 3(.09)(.7) = .189 2!(3 − 2)! © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 26 Binomial Probability Distribution TABLE 3.6 PROBABILITY DISTRIBUTION FOR THE NUMBER OF CUSTOMERS MAKING A PURCHASE X f (x) 0 0.343 1 0.441 2 0.189 3 0.027 Total 1.000 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 27 Binomial Probability Distribution Expected Value E(x) =  = np Variance Var(x) =  2 = np(1 − p) Standard Deviation s = np(1- p) © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 28 Example: Nastke Clothing Store Binomial Probability Distribution • Expected Value E(x) =  = 3(.3) = .9 customers out of 3 • Variance Var(x) =  2 = 3(.3)(.7) = .63 • Standard Deviation  = 3(.3)(.7) = .794 customers © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 29 Poisson Probability Distribution Poisson Probability Distribution A Poisson distributed random variable is often useful in estimating the number of occurrences over a specified interval of time or space It is a discrete random variable that may assume an infinite sequence of values (x = 0, 1, 2, . . . ). © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 30 Poisson Probability Distribution Poisson Probability Distribution Examples of Poisson distributed random variables: the number of knotholes in 14 linear feet of pine board the number of vehicles arriving at a toll booth in one hour Bell Labs used the Poisson distribution to model the arrival of phone calls. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 31 Poisson Probability Distribution Poisson Probability Distribution Two Properties of a Poisson Experiment 1. The probability of an occurrence is the same for any two intervals of equal length. 2. The occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 32 Poisson Probability Distribution Poisson Probability Function 𝜇 𝑥 𝑒 −𝜇 𝑓 𝑥 = 𝑥! where: x = the number of occurrences in an interval f(x) = the probability of x occurrences in an interval  = mean number of occurrences in an interval e = 2.71828 x! = x(x – 1)(x – 2) . . . (2)(1) © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 33 Poisson Probability Distribution Poisson Probability Function Since there is no stated upper limit for the number of occurrences, the probability function f(x) is applicable for values x = 0, 1, 2, … without limit. In practical applications, x will eventually become large enough so that f(x) is approximately zero and the probability of any larger values of x becomes negligible. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 34 Poisson Probability Distribution Example: Mercy Hospital Patients arrive at the emergency room of Mercy Hospital at the average rate of 6 per hour on weekend evenings. What is the probability of 4 arrivals in 30 minutes on a weekend evening? © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 35 Poisson Probability Distribution Poisson Probability Distribution Example: Mercy Hospital  = 6/hour = 3/half-hour, x = 4 𝑓 4 = 34 (2.71828)−3 4! Using the probability function = .1680 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 36 Chapter 3, Part B Continuous Probability Distributions Uniform Probability Distribution Normal Probability Distribution Exponential Probability Distribution f (x) f (x) Exponential Uniform f (x) Normal x x x © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 37 Continuous Random Variables Examples of continuous random variables include the following: • The flight time of an airplane traveling from Chicago to New York • The lifetime of the picture tube in a new television set • The drilling depth required to reach oil in an offshore drilling operation © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 38 Continuous Probability Distributions A continuous random variable can assume any value in an interval on the real line or in a collection of intervals. It is not possible to talk about the probability of the random variable assuming a particular value. Instead, we talk about the probability of the random variable assuming a value within a given interval. For a continuous probability distribution we calculate the probability of being less than some value x, i.e. P(X < x), by calculating the area under the curve to the left of x. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 39 Continuous Probability Distributions The probability of the random variable assuming a value within some given interval from x1 to x2 is defined to be the area under the graph of the probability density function between x1 and x2. f (x) f (x) Exponential Uniform f (x) x1 x 2 Normal x x1 x1 x2 xx12 x2 x x © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 40 Normal Probability Distribution The normal probability distribution is the most important distribution for describing a continuous random variable. It is widely used in statistical inference. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 41 Normal Probability Distribution It has been used in a wide variety of applications: Heights of people Test scores Amounts of rainfall Scientific measurements © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 42 Normal Probability Distribution Normal Probability Density Function f (x) = 1 s 2p e -(x-m )2 /2s 2 where:  = mean  = standard deviation  = 3.14159 e = 2.71828 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 43 Normal Probability Distribution Characteristics The distribution is symmetric, and is bell-shaped. x © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 44 Normal Probability Distribution Characteristics The entire family of normal probability distributions is defined by its mean  and its standard deviation  . Standard Deviation  Mean  x © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 45 Normal Probability Distribution Characteristics The highest point on the normal curve is at the mean, which is also the median and mode. x © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 46 Normal Probability Distribution Characteristics The mean can be any numerical value: negative, zero, or positive. x -10 0 20 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 47 Normal Probability Distribution Characteristics The standard deviation determines the width of the curve: larger values result in wider, flatter curves.  = 15  = 25 x © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 48 Normal Probability Distribution Characteristics Probabilities for the normal random variable are given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and .5 to the right). .5 .5 x © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 49 Normal Probability Distribution Characteristics 68.26% of values of a normal random variable are within +/- 1 standard deviation of its mean. 95.44% of values of a normal random variable are within +/- 2 standard deviations of its mean. 99.72% of values of a normal random variable are within +/- 3 standard deviations of its mean. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 50 Normal Probability Distribution Characteristics 99.72% 95.44% 68.26%  – 3  – 1  – 2   + 3  + 1  + 2 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. x Slide 51 Standard Normal Probability Distribution -Use probability tables that have been calculated on a computer. - Only one special Normal distribution, N(0, 1), has been tabulated. - If we want to calculate probabilities from different Normal distributions we convert the probability to one involving the standard Normal distribution. → standardization A random variable having a normal distribution with a mean of 0 and a standard deviation of 1 is said to have a standard normal probability distribution. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 52 Standard Normal Probability Distribution The letter z is used to designate the standard normal random variable. =1 z 0 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 53 Standard Normal Probability Distribution Converting to the Standard Normal Distribution z= x−  We can think of z as a measure of the number of standard deviations x is from . © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 54 Example: Pep Zone Standard Normal Probability Distribution Pep Zone sells auto parts and supplies including a popular multi-grade motor oil. When the stock of this oil drops to 20 gallons, a replenishment order is placed. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 55 Example: Pep Zone Standard Normal Probability Distribution The store manager is concerned that sales are being lost due to stockouts while waiting for an order. It has been determined that demand during replenishment lead-time is normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons. The manager would like to know the probability of a stockout, P(x > 20). © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 56 Example: Pep Zone Solving for the Stockout Probability Step 1: Convert x to the standard normal distribution. z = (x - )/ = (20 - 15)/6 = .83 Step 2: Find the area under the standard normal curve between the mean and z = .83. see next slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 57 Example: Pep Zone Probability Table for the Standard Normal Distribution z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 . . . . . . . . . . . .5 .1915 .1695 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224 .6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549 .7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852 .8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133 .9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389 . . . . . . . . . . . P(0 < z < .83) © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 58 Example: Pep Zone Solving for the Stockout Probability Step 3: Compute the area under the standard normal curve to the right of z = .83. P(z > .83) = .5 – P(0 < z < .83) = .5- .2967 = .2033 Probability of a stockout P(x > 20) © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 59 Example: Pep Zone Solving for the Stockout Probability Area = .5 - .2967 Area = .2967 = .2033 0 .83 z © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 60 End of Chapter 3 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 61

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