explain a hypothesis

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timer Asked: Oct 20th, 2018
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Question Description

a. State

the null hypothesis.

1 pt

b. State the alternative hypothesis.

1 pt.

c. State the significance level

.

1 pt.

d. Perform the calculations.

3 pts.

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Slides by John Loucks St. Edward’s University © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 1 Chapter 7 Sampling and Sampling Distributions ◼ Selecting a Sample ◼ Point Estimation ◼ Introduction to Sampling Distributions ◼ Sampling Distribution of 𝑥ҧ ◼ Sampling Distribution of 𝑝ҧ ◼ Other Sampling Methods © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 2 Introduction An element is the entity on which data are collected. A population is a collection of all the elements of interest. A sample is a subset of the population. The sampled population is the population from which the sample is drawn. A frame is a list of the elements that the sample will be selected from. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 3 Introduction The reason we select a sample is to collect data to answer a research question about a population. The sample results provide only estimates of the values of the population characteristics. The reason is simply that the sample contains only a portion of the population. With proper sampling methods, the sample results can provide “good” estimates of the population characteristics. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 4 Selecting a Sample Sampling from a Finite Population Sampling from an Infinite Population © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 5 Sampling from a Finite Population Finite populations are often defined by lists such as: • Organization membership roster • Credit card account numbers • Inventory product numbers A simple random sample of size n from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 6 Sampling from a Finite Population ◼ Replacing each sampled element before selecting subsequent elements is called sampling with replacement. ◼ Sampling without replacement is the procedure used most often. ◼ In large sampling projects, computer-generated random numbers are often used to automate the sample selection process. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 7 Sampling from a Finite Population Example: St. Andrew’s College St. Andrew’s College received 900 applications for admission in the upcoming year from prospective students. The applicants were numbered, from 1 to 900, as their applications arrived. The Director of Admissions would like to select a simple random sample of 30 applicants. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 8 Sampling from a Finite Population Example: St. Andrew’s College Step 1: Assign a random number to each of the 900 applicants. The random numbers generated by Excel’s RAND function follow a uniform probability distribution between 0 and 1. Step 2: Select the 30 applicants corresponding to the 30 smallest random numbers. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 9 Sampling from a Finite Population Using Excel Excel Formula Worksheet A Applicant 1 Number 2 1 3 2 4 3 5 4 6 5 7 6 8 7 9 8 B Random Number =RAND() =RAND() =RAND() =RAND() =RAND() =RAND() =RAND() =RAND() Note: Rows 10-901 are not shown. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 10 Sampling from a Finite Population Using Excel Excel Value Worksheet A B Applicant Random 1 Number Number 2 1 0.61021 3 2 0.83762 4 3 0.58935 5 4 0.19934 6 5 0.86658 7 6 0.60579 8 7 0.80960 9 8 0.33224 Note: Rows 10-901 are not shown. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 11 Sampling from a Finite Population Using Excel Put Random Numbers in Ascending Order Step 1 Step 2 Step 3 Step 4 Select any cell in the range B2:B901 Click the Home tab on the Ribbon In the Editing group, click Sort & Filter Choose Sort Smallest to Largest © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 12 Sampling from a Finite Population Using Excel Excel Value Worksheet (Sorted) A B Applicant Random 1 Number Number 2 12 0.00027 3 773 0.00192 4 408 0.00303 5 58 0.00481 6 116 0.00538 7 185 0.00583 8 510 0.00649 9 394 0.00667 Note: Rows 10-901 are not shown. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 13 Sampling from an Infinite Population ◼ Sometimes we want to select a sample, but find it is not possible to obtain a list of all elements in the population. ◼ As a result, we cannot construct a frame for the population. ◼ Hence, we cannot use the random number selection procedure. ◼ Most often this situation occurs in infinite population cases. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 14 Sampling from an Infinite Population Populations are often generated by an ongoing process where there is no upper limit on the number of units that can be generated. ◼ Some examples of on-going processes, with infinite populations, are: • parts being manufactured on a production line • transactions occurring at a bank • telephone calls arriving at a technical help desk • customers entering a store © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 15 Sampling from an Infinite Population ◼ In the case of an infinite population, we must select a random sample in order to make valid statistical inferences about the population from which the sample is taken. A random sample from an infinite population is a sample selected such that the following conditions are satisfied. • Each element selected comes from the population of interest. • Each element is selected independently. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 16 Point Estimation Point estimation is a form of statistical inference. In point estimation we use the data from the sample to compute a value of a sample statistic that serves as an estimate of a population parameter. We refer to 𝑥ҧ as the point estimator of the population mean . s is the point estimator of the population standard deviation . 𝑝ҧ is the point estimator of the population proportion p. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 17 Point Estimation Example: St. Andrew’s College Recall that St. Andrew’s College received 900 applications from prospective students. The application form contains a variety of information including the individual’s Scholastic Aptitude Test (SAT) score and whether or not the individual desires on-campus housing. At a meeting in a few hours, the Director of Admissions would like to announce the average SAT score and the proportion of applicants that want to live on campus, for the population of 900 applicants. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 18 Point Estimation Example: St. Andrew’s College However, the necessary data on the applicants have not yet been entered in the college’s computerized database. So, the Director decides to estimate the values of the population parameters of interest based on sample statistics. The sample of 30 applicants is selected using computer-generated random numbers. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 19 Point Estimation Using Excel Excel Value Worksheet (Sorted) A B C Applicant Random SAT Score 1 Number Number 2 12 0.00027 1207 3 773 0.00192 1143 4 408 0.00303 1091 5 58 0.00481 1108 6 116 0.00538 1227 7 185 0.00583 982 8 510 0.00649 1363 9 1108 394 0.00667 D On-Campus Housing No Yes Yes No Yes Yes Yes No Note: Rows 10-31 are not shown. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 20 Point Estimation 𝑥ҧ as Point Estimator of  50,520 𝑥ҧ = = = 1684 30 30 σ 𝑥𝑖 s as Point Estimator of  𝑠= σ(𝑥𝑖 − 𝑥)ҧ 2 = 29 210,512 = 85.2 29 𝑝ҧ as Point Estimator of p 𝑝ҧ = 20Τ30 = .67 Note: Different random numbers would have identified a different sample which would have resulted in different point estimates. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 21 Point Estimation Once all the data for the 900 applicants were entered in the college’s database, the values of the population parameters of interest were calculated. Population Mean SAT Score σ 𝑥𝑖 𝜇= = 1697 900 Population Standard Deviation for SAT Score 𝜎= σ(𝑥𝑖 −𝜇)2 = 87.4 900 Population Proportion Wanting On-Campus Housing 𝑝 = 648/900 = .72 © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 22 Summary of Point Estimates Obtained from a Simple Random Sample Population Parameter Parameter Value Point Estimator Point Estimate  = Population mean 1697 𝑥ҧ = Sample mean SAT score 1684 87.4 s = Sample standard deviation for SAT score 85.2 .72 𝑝ҧ = Sample proportion wanting campus housing SAT score  = Population std. deviation for SAT score p = Population proportion wanting campus housing .67 © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 23 Practical Advice The target population is the population we want to make inferences about. The sampled population is the population from which the sample is actually taken. Whenever a sample is used to make inferences about a population, we should make sure that the targeted population and the sampled population are in close agreement. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 24 Sampling Distribution of 𝑥ҧ Process of Statistical Inference Population with mean =? The value of 𝑥ҧ is used to make inferences about the value of . A simple random sample of n elements is selected from the population. The sample data provide a value for the sample mean 𝑥ҧ . © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 25 Sampling Distribution of 𝑥ҧ The sampling distribution of 𝑥ҧ is the probability distribution of all possible values of the sample mean 𝑥.ҧ • Expected Value of 𝑥ҧ E(𝑥)ҧ =  where:  = the population mean When the expected value of the point estimator equals the population parameter, we say the point estimator is unbiased. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 26 Sampling Distribution of 𝑥ҧ • Standard Deviation of 𝑥ҧ We will use the following notation to define the standard deviation of the sampling distribution of 𝑥.ҧ 𝜎𝑥ҧ = the standard deviation of 𝑥ҧ  = the standard deviation of the population n = the sample size N = the population size © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 27 Sampling Distribution of 𝑥ҧ • Standard Deviation of 𝑥ҧ Finite Population 𝜎𝑥ҧ = 𝑁−𝑛 𝑁−1 𝜎 𝑛 Infinite Population 𝜎 𝜎𝑥ҧ = 𝑛 • A finite population is treated as being infinite if n/N < .05. • (𝑁 − 𝑛)/(𝑁 − 1) is the finite population correction factor. • 𝜎𝑥ҧ is referred to as the standard error of the mean. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 28 Sampling Distribution of 𝑥ҧ When the population has a normal distribution, the sampling distribution of 𝑥ҧ is normally distributed for any sample size. In most applications, the sampling distribution of 𝑥ҧ can be approximated by a normal distribution whenever the sample is size 30 or more. In cases where the population is highly skewed or outliers are present, samples of size 50 may be needed. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 29 Sampling Distribution of 𝑥ҧ The sampling distribution of 𝑥ҧ can be used to provide probability information about how close the sample mean 𝑥ҧ is to the population mean  . © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 30 Central Limit Theorem When the population from which we are selecting a random sample does not have a normal distribution, the central limit theorem is helpful in identifying the shape of the sampling distribution of 𝑥.ҧ CENTRAL LIMIT THEOREM In selecting random samples of size n from a population, the sampling distribution of the sample mean 𝑥ҧ can be approximated by a normal distribution as the sample size becomes large. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 31 Sampling Distribution of 𝑥ҧ Example: St. Andrew’s College Sampling Distribution of 𝑥ҧ for SAT Scores 𝜎𝑥ҧ = 𝜎 𝑛 = 87.4 30 = 15.96 𝑥ҧ 𝐸 𝑥ҧ = 1697 © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 32 Sampling Distribution of 𝑥ҧ Example: St. Andrew’s College What is the probability that a simple random sample of 30 applicants will provide an estimate of the population mean SAT score that is within +/-10 of the actual population mean  ? In other words, what is the probability that 𝑥ҧ will be between 1687 and 1707? © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, 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 for classroom use. 33 Sampling Distribution of 𝑥ҧ Example: St. Andrew’s College Step 1: Calculate the z-value at the upper endpoint of the interval. z = (1707 - 1697)/15.96 = .63 Step 2: Find the area under the curve to the left of the upper endpoint. P(z < .63) = .7357 © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwi ...
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