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Week 8 Statistical Testing in Manufacturing PPT
WEEK 8 ASSIGNMENT - CASE STUDY: STATISTICAL INFERENCEWeek 8 Assignment - Case Study: Statistical InferenceOverviewThe rese ...
Week 8 Statistical Testing in Manufacturing PPT
WEEK 8 ASSIGNMENT - CASE STUDY: STATISTICAL INFERENCEWeek 8 Assignment - Case Study: Statistical InferenceOverviewThe research department of an appliance manufacturing firm has developed a solid-state switch for its blender that the research department claims will reduce appliance returns under the one-year full warranty by 3%6%. To determine if the claim can be supported, the testing department selects a group of the blenders manufactured with the new switch and a group with the old switch and subjects them to a normal years worth of wear. Out of 250 blenders tested with the new switch, nine would have been returned. Sixteen would have been returned out of the 250 blenders with the old switch. As the manager of the appliance manufacturing process, use a statistical procedure to verify or refute the research departments claim.InstructionsCreate 810 slides, including a cover and a sources list, for a presentation to the director of the manufacturing plant in which you: Summarize the problem with the appliance manufacturing firm's blenders. Propose the statistical inference to use to solve the problem. Support your decision using a scholarly reference. Using Excel:Develop a flowchart for the proposed statistical inference, including specific steps. Compute all statistical calculations using Excel. Place your flowchart in a slide.Determine if you can verify or refute the research department's claim.Choose sources that are credible, relevant, and appropriate. Cite each source listed on your source page at least one time within your assignment. For help with research, writing, and citation, access the library or review library guides. This course requires the use of Strayer Writing Standards. For assistance and information, please refer to the Strayer Writing Standards link in the left-hand menu of your course. Check with your professor for any additional instructions. The specific course learning outcome associated with this assignment is:Develop recommendations to improve business processes using statistical tools and analysis. https://blackboard.strayer.edu/webapps/assignment/uploadAssignment?content_id=_38131813_1&course_id=_462527_1&group_id=&mode=view
Post your 4 paragraph (max) article critique, and then address the following:
read the attached article and write a four-paragraph (maximum) critique addressing the following questions:Do you believe ...
Post your 4 paragraph (max) article critique, and then address the following:
read the attached article and write a four-paragraph (maximum) critique addressing the following questions:Do you believe the results? Why or why not?State the reported confidence interval(s) estimate in two ways: the upper and lower limitsthe sample proportion ± the error term(don’t forget to include the confidence level)State the meaning of the confidence interval(s).Can a reported mean of your article data be treated as a value from a population having a normal distribution? Why or why not?https://files.eric.ed.gov/fulltext/EJ1174957.pdf Link to articlehttps://eric.ed.gov/?q=Confidence+Interval&pr=on&f...
stats homework using R
#Name:
#Student ID:
rm(list=ls())
source('Rallfun-v33.txt')
#PART 1
#A company claims that, when exposed to their to ...
stats homework using R
#Name:
#Student ID:
rm(list=ls())
source('Rallfun-v33.txt')
#PART 1
#A company claims that, when exposed to their toothpaste, 45% of all bacteria related to gingivitis are killed, on average. You run 10 tests and ???nd that the percentages of bacteria killed in each test was:
# 38%, 44%, 62%, 72%, 43%, 40%, 43%, 42%, 39%, 41%
# Assuming normality, you will test the hypothesis that the average percentage of bacteria killed was 45% at alpha=0.05.
#1.1) Write out the Null and Alternative hypotheses
#1.2) Calculate the T-statistic and use Method 1 (we saw in class) to determine if the average bacteria killed was 45%. Do it by "hand".
#Hint: Method 1 is to compare T to a critical value "c".
#1.3) Do you reject or fail to reject the null?
################################################
#PART 2
#Now, let's not assume normality
#2.1) Using the same data as in Part 1, test the hypothesis that the 20% trimmed mean is 45%?
#2.2) Do you reject or fail to reject the null?
#2.3) Assuming your test in 2.1 is the truth, what type of error did you make in #1.3?
################################################
#PART 3
#In a study of court administration, the following times to disposition (in minutes) were determined for twenty cases and found to be:
# 42, 90, 84, 87, 116, 95, 86, 99, 93, 92, 121, 71, 66, 98, 79, 102, 60, 112, 105, 98
#Assuming normality, you will test the hypothesis that the average time to disposition was 99 minutes at alpha=0.05.
#3.1) Write out the Null and Alternative hypotheses
#3.2) Calculate the T-statistic and use Method 2 (we saw in class) to determine if the average time to disposition was 99? Do it by "hand".
#Hint: Method 2 is to evaluate the confidence interval.
#3.3) Do you reject or fail to reject the null?
################################################
#PART 4
#Now, let's not assume normality
#4.1) Using the same data as in Part 3, test the hypothesis that the 20% trimmed mean is 99?
#4.2) Do you reject or fail to reject the null?
#4.3) Assuming your test in 4.1 is the truth, what type of error did you make in #3.3?
################################################
#PART 5
#Suppose you run an experiment, and observe the following values:
# 12, 20, 34, 45, 34, 36, 37, 50, 11, 32, 29
#You will test the hypothesis that the average was 25 at alpha=0.05.
#5.1) Write out the Null and Alternative hypotheses. Conduct the hypothesis test assuming normality. Use the "t.test" function. Do you reject or fail to reject the null?
#5.2) Conduct the hypothesis test without assuming normality. Do you reject or fail to reject the null?
#5.3) Assuming the answer in #5.2 is the truth, what type of error (if any) did you make in #5.1 by assuming normality?
------------------------------------------------------------------------------------------
Lab 7- Lecture Notes (FOR YOUR REFERENCE)
#Lab 7-Contents
#1. Formulating Hypotheses
#2. T-statistics by Hand
#3. Alpha Level
#4. Evaluating Our Results
#5. Using the t.test function
#6. T-tests with Trimmed Means (trimci function)
#7. Type 1 and Type 2 Errors
# Last week we talked about computations for when the Population
#Variance is known and unknown.
# Given that we rarely know the population variance,
#we will use the T-distribution for all of today's lab.
#We will primarily work with the dataset brfss09_lab7.txt:
#########################################################################################################################
#Behavioral Risk Factors Surveilance Survey 2009 (BRFSS09) Data Dictionary:
#------------------------------------------------------------------------------------------------------------------------
#id: "Subject ID"Values[1,998]
#physhlth: "# Days past month phsycial health poor" Values[1,30]
#menthlth: "# Days past month mental health poor"Values[1,30]
#hlthplan: "Have healthcare coverage?"Values 1=Yes, 2=No
#age:"Age in Years"Values[18,99]
#sex:"Biologic Sex"Values 0=Female, 1=Male
#fruit_day: "# of servings of fruit per day"Values[0,20]
#alcgrp: "Alcohol Consumption Groups"Values 1=None, 2= 1-2 drinks/day 3= 3 or more drinks/day
#smoke:"Smoking Status"Values 0=Never, 1=Current EveryDay, 2=Current SomeDays, 3=Former
#bmi:"Body Mass Index"Values[14,70]
#mi:"Myocardial Infarction (heart attack)"Values 0=No, 1=Yes
#------------------------------------------------------------------------------------------------------------------------
# For today's lab, let's start by reading in our datafile
# 'brfss09_lab7.txt' into an object called mydata
mydata=read.table('brfss09_lab7.txt', header=T)
#This file contains:
dim(mydata)#100 Subjects, 11 variables
#With the following variables:
names(mydata)
# We have collected this data and would like to know
#if the values we have found in our sample are different
#from the reported values in the literature.
# For example, it has been reported that the average BMI
# in the population is 27.5. We would like to know if the
#values in our sample are somehow different than this value.
#---------------------------------------------------------------------------------
# 1. Formulating Hypotheses
#---------------------------------------------------------------------------------
#Step 1 of determining if our BMI values differ from the
#national average of 27.5 is to formulate our hypotheses
#We have TWO hypotheses
#1) The Null Hypothesis: H0: mu = 27.5
#2) The Alternative Hypothesis: HA: mu != 27.5
#NOTE: mu=Population Mean
#The above hypotheses are Two-Sided.
#By this I mean that we are looking to see if our sample values of
#BMI are greater than (>) OR less than (<) 27.5.
# A one-sided hypothesis test would look like:
#H0: mu < 27.5
#HA: mu > 27.5
#OR
#H0: mu > 27.5
#HA: mu < 27.5
#We will always use two-sided tests in this class,
#and similarly in the real world two-sided tests dominate.
#Once we have our hypotheses we will evaluate them
#and determine one of two outcomes:
# A) Reject the Null Hypothesis
# B) Fail to Reject the Null Hypothesis
#---------------------------------------------------------------------------------
# 2. T-statistics by Hand (well..with help from the computer)
#---------------------------------------------------------------------------------
#Recall from the last lab, that the formula for a T-statistic is:
# T = (SampleMean - PopMean) / (SampleSD/sqrt(N))
#Another way to write this would be:
# T = (xbar - mu) / (s/sqrt(N))
#In this instance PopMean (mu) is the NULL hypothesis
#value we are testing against.
#We can solve for the other values that we don't yet know:
mu=27.5
xbar=mean(mydata$bmi) #28.22
s=sd(mydata$bmi) #6.32
N=100
T = (xbar - mu) / (s/sqrt(N))
T #1.14
#We end up with a T value of ~ 1.14
#But how does this tell us if our mean is different from 27.5 ???!!!
#Before we move on, I want us to think about why we need
#to evaluate if our mean of 28.22 is different from 27.5.
#Certainly we can see that these are different numbers,
#so what are we really asking here?
#One way to think about it is that we are asking if our
#sample mean of 28.22 is different from 27.5 simply due to chance.
#Think of a coin tossing example:
#Your friend tosses a coin in the air and it lands on heads
#3 times in a row!
#While, kinda cool, seems like that is probably random chance.
#What about if it landed on heads 100 times in a row?!
#You would probably think she was cheating somehow!
#Though it is possible to have 100 heads in a row
#by chance alone, it is very unlikely
#The point at which we say that something is random vs not
#is determined by our alpha level.
#---------------------------------------------------------------------------------
# 3. Alpha Level
#---------------------------------------------------------------------------------
# The alpha level is determined a priori (a head of time)
#and used to set the threshold by which we consider something
#to be random chance
# A common alpha level is 0.05.
# We typically reject the null (think something is not chance)
#when the result we have (eg. 28.22) would only be
#that extreme < 5% of the time by chance.
#Recall from Lab 6, that we use the alpha level
#to help figure out critical values (c)
# c=qt(1-(alpha/2), df)
#---------------------------------------------------------------------------------
# 4. Evaluating our Results
#---------------------------------------------------------------------------------
# There are 3 ways to evaluate if our mean of 28.22
# is different from the null of 27.5
# All three ways will yield the same conclusion.
#1) Compare T to a critical value (c)
#2) Evaluate the Confidence interval
#3) Compare the p value to our alpha level
###########################################################
#1) Compare T to a critical value (c)
#In order to compute the critical value (c),
#we must know the alpha level.
#We will choose a value of 0.05 (which is standard)
alpha=0.05
df=100-1
c=qt(1-(alpha/2), df)
#We can then compare the abosulte value of T (|T|)
#to the critical value c
#A) If |T| > c, then Reject the Null Hypothesis
#B) If |T| < c, then Fail to Reject the Null Hypothesis
#Let's look at T can c
abs(T)
c
#What decision do we make about the Null Hypothesis????
###########################################################
#2) Evaluate the Confidence interval
#Rather than compare T to c,
#we could instead compute the confidence interval.
#Recall the formula for the Confidence interval is:
#LB= xbar - c*(s/sqrt(N))
#UB= xbar + c*(s/sqrt(N))
LB = xbar - c*(s/sqrt(N))
UB= xbar + c*(s/sqrt(N))
#A) If mu is not within the Confidence Interval,
#then Reject the Null Hypothesis
#B) If mu is within the Confidence Interval,
#then Fail to Reject the Null Hypothesis
#Let's look at LB and UB
LB
UB
mu
#What decision do we make about the Null Hypothesis????
###########################################################
#3) Compare the p value to our alpha level
#Lastly, we could find the probability value (or p-value)
#for the T statistic we created.
#We can do this by using the pt() function we learned
#about last week in lab 6.
#There is a forumla for computing P values from T-statitics:
# pval = 2*(1-pt(abs(T), df))
pval = 2*(1-pt(abs(T), df))
#We then compare the p-value to our alpha level
#A) If pval < alpha, then Reject the Null Hypothesis
#B) If pval > alpha, then Fail to Reject the Null Hypothesis
#Let's look at our p-value.
pval
alpha
#What decision do we make about the Null Hypothesis????
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
#Exercise 4-1:
#Evaluate if the mean age from our sample (mydata) is different
#than the populatiuin mean age of 56
# A) Write down the Null and Alternative Hypotheses
# B) Calculate the T-statistic by hand
# C) Evaluate the Null hypothesis by using ALL 3 methods that
# we just discussed
# D) Based on the results in C, do you Reject or Fail to Reject
# the Null Hypothesis?
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
#A)
#B)
#C)
#Method 1: Compare T to a critical value (c)
#Method 2: Evaluate the Confidence interval
#Method 3: Compare the p value to our alpha level
#D)
#---------------------------------------------------------------------------------
# 5. Using the t.test function
#---------------------------------------------------------------------------------
#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^#
# One Sample T-Test : t.test(data$variable, mu)
#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^#
#It was really awesome that we figured out T by hand!
#And then figured out the confidence intervals and P values!
#From now on, let's just use a program to do all this for us.
#The function t.test will presume an alpha level of 0.05 by default.
t.test(mydata$age, mu=56)
# t.test(mydata$bmi, mu=27.5)
#Much simpler!
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
#Exercise 5-1: Use the t.test function to evaluate if
#A) the mean days of physical health (physhlth) is different
# than the population mean of 10? Reject the Null?
#B) the mean fruits per day (fruit_day) is different than
# the populatiuin mean of 4? Reject the Null?
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
#A)
#B)
#---------------------------------------------------------------------------------
# 6. T-test with Trimmed Means
#---------------------------------------------------------------------------------
#To use the T-test with trimmed means,
#we will need to load in the source code 'Rallfun-v33.txt'
#The trimmed mean T-test is beneficial in that it does not
#presume a perfect Normal Distribution
#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^#
# Trimmed Mean T-Test:
# trimci(data$variable, tr=0.2, alpha=0.05, null.value=0)
#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^#
#For example, if I wanted to test if the age was equal to 56
#using Trimmed Means I could do:
trimci(mydata$age, null.value=56)
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
#Exercise 6-1: Use the trimci function to evaluate if
#A) the 20% trimmed mean of days of physical health (physhlth) is
# different than the populatiuin mean of 10? Reject the Null?
#B) the 20% trimmed mean fruits per day (fruit_day) is different
#than the populatiuin mean of 4? Reject the Null?
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
#A)
#B)
#---------------------------------------------------------------------------------
# 7. Type 1 and Type 2 Errors
#---------------------------------------------------------------------------------
#Notice that we had very different answers to the same
#questions in Ex. 5-1 and 6-1
#Depending upon the method that we used.
#This brings us to discussing Type 1 and Type 2 Error
#A Type 1 error is when our test tells us to reject the null,
#but in truth we should not have
#A Type 2 error is when our test tells us to fail to reject the
#null, but in truth we should have rejected the null
#The following 2x2 square might make this easier to see.
# Truth
#------------------------------------
#| H0 | HA |
#-------------- |-------|-----------|
#My Test: H0 | H0 Type 2|
#-------------- |-------|-----------|
#My Test: HA Type 1 | HA |
#------------------------------------
#For the next exercise, let's presume that our test of the trimmed mean is the Truth
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
#Exercise 7-1:
#A) What type of error did we make when evaluating the mean
#of physhlth in exercise 5-1?
#B) What type of error did we make when evaluating the mean
#of fruit_day in exercise 5-1?
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
#A)
#B)
Pre-Calculus help with parabola
The parabola in the figure below has an equation of the form y=ax^2+bx-4. Find the equation of this parabola in two differ ...
Pre-Calculus help with parabola
The parabola in the figure below has an equation of the form y=ax^2+bx-4. Find the equation of this parabola in two different ways, by hand and with technology (graphing utility or computer software). Write a paragraph describing the methods you used and comparing the results of the two methods.
Colorado Technical University Patient Safety in AKT Hospital Presentation
Individual Project5 Unit: Communicating Statistical Data Outcomes and Strategic Planning ...
Colorado Technical University Patient Safety in AKT Hospital Presentation
Individual Project5 Unit: Communicating Statistical Data Outcomes and Strategic Planning Due Date: Wed,11/11/20 Grading Type: Numeric Points Possible: 250 Points Earned: Points Earned not available Deliverable Length: 15-20 slide presentation View objectives for this assignment Go To:Assignment Details Learning MaterialsReading Assignment My Work: Online Deliverables: Submissions Looking for tutoring? Go to Smarthinking Details Learn Read My Work Online Deliverables: Submissions Assignment Details Assignment DescriptionIn Week 4, you identified some immediate areas of concern that you were able to effectively address. You must present the final phase of your improvement plan to your staff and upper-level management. You will create a presentation of 15-20 slides. Each slide will have 4-6 bullets and 100-150 words of speakers notes. The presentation will address the following areas:In preparation for the accreditation visit for AKT, choose 1 health care accrediting and credentialing organization. Select a quality improvement focus (QIF) area to improve patient outcomes in beyond the 3 issues that you identified and addressed in Week 4. Discuss the selected accreditation agency related to the QIF and why the organization is seeking this particular agency for credentialing. As part of the quality improvement initiative, select 3-4 related accrediting standards that the organization will use as the basis for the quality improvement plan. Provide a clear mission statement and set of 3-4 specific, measurable, attainable, realistic, and timely (SMART) goals for the QIF initiative. Using the online database provided the by the organization you selected conduct an analysis. Provide general statistical data related to the QIF. Discuss specific health care examples of local, state, and national policies that have been developed to improve this QIF based on evidence-based practice research. What internal policies do you plan to implement based on evidence-based practice approaches to ensure your organization meets these standards? Develop a plan that includes strategies for your facility to improve patient outcomes regarding the QIF. Describe how the QIF initiative can be incorporated to the organization’s overall strategic plan. Describe how you plan to evaluate the effectiveness of the initiative. Each slide will have 4-6 bullets and 100-150 words of speaker’s notes and pictures.
West Coast University Week 7 Correlation Does Not Mean Causation Discussion
You may have heard it said before that “correlation does not imply causation.” This can also be called spurious correl ...
West Coast University Week 7 Correlation Does Not Mean Causation Discussion
You may have heard it said before that “correlation does not imply causation.” This can also be called spurious correlation, which is defined as a correlation between two variables that does not result from a direct relationship between them. Instead, it results from the variables’ relationship to other variables. One example is the relationship between crime and ice cream sales. Ice cream sales and crime rates are highly correlated. However, ice cream sales do not cause crime; instead, it is both variables’ relationship to weather and temperature.Do some research and find some interesting, or even funny, examples of spurious correlation. Share, cite your source, and discuss. Why is this an example of spurious correlation? How do you know?Do not use the same example given from another student. Make sure you read your classmates’ posts before submitting your example.Search entries or author
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Week 8 Statistical Testing in Manufacturing PPT
WEEK 8 ASSIGNMENT - CASE STUDY: STATISTICAL INFERENCEWeek 8 Assignment - Case Study: Statistical InferenceOverviewThe rese ...
Week 8 Statistical Testing in Manufacturing PPT
WEEK 8 ASSIGNMENT - CASE STUDY: STATISTICAL INFERENCEWeek 8 Assignment - Case Study: Statistical InferenceOverviewThe research department of an appliance manufacturing firm has developed a solid-state switch for its blender that the research department claims will reduce appliance returns under the one-year full warranty by 3%6%. To determine if the claim can be supported, the testing department selects a group of the blenders manufactured with the new switch and a group with the old switch and subjects them to a normal years worth of wear. Out of 250 blenders tested with the new switch, nine would have been returned. Sixteen would have been returned out of the 250 blenders with the old switch. As the manager of the appliance manufacturing process, use a statistical procedure to verify or refute the research departments claim.InstructionsCreate 810 slides, including a cover and a sources list, for a presentation to the director of the manufacturing plant in which you: Summarize the problem with the appliance manufacturing firm's blenders. Propose the statistical inference to use to solve the problem. Support your decision using a scholarly reference. Using Excel:Develop a flowchart for the proposed statistical inference, including specific steps. Compute all statistical calculations using Excel. Place your flowchart in a slide.Determine if you can verify or refute the research department's claim.Choose sources that are credible, relevant, and appropriate. Cite each source listed on your source page at least one time within your assignment. For help with research, writing, and citation, access the library or review library guides. This course requires the use of Strayer Writing Standards. For assistance and information, please refer to the Strayer Writing Standards link in the left-hand menu of your course. Check with your professor for any additional instructions. The specific course learning outcome associated with this assignment is:Develop recommendations to improve business processes using statistical tools and analysis. https://blackboard.strayer.edu/webapps/assignment/uploadAssignment?content_id=_38131813_1&course_id=_462527_1&group_id=&mode=view
Post your 4 paragraph (max) article critique, and then address the following:
read the attached article and write a four-paragraph (maximum) critique addressing the following questions:Do you believe ...
Post your 4 paragraph (max) article critique, and then address the following:
read the attached article and write a four-paragraph (maximum) critique addressing the following questions:Do you believe the results? Why or why not?State the reported confidence interval(s) estimate in two ways: the upper and lower limitsthe sample proportion ± the error term(don’t forget to include the confidence level)State the meaning of the confidence interval(s).Can a reported mean of your article data be treated as a value from a population having a normal distribution? Why or why not?https://files.eric.ed.gov/fulltext/EJ1174957.pdf Link to articlehttps://eric.ed.gov/?q=Confidence+Interval&pr=on&f...
stats homework using R
#Name:
#Student ID:
rm(list=ls())
source('Rallfun-v33.txt')
#PART 1
#A company claims that, when exposed to their to ...
stats homework using R
#Name:
#Student ID:
rm(list=ls())
source('Rallfun-v33.txt')
#PART 1
#A company claims that, when exposed to their toothpaste, 45% of all bacteria related to gingivitis are killed, on average. You run 10 tests and ???nd that the percentages of bacteria killed in each test was:
# 38%, 44%, 62%, 72%, 43%, 40%, 43%, 42%, 39%, 41%
# Assuming normality, you will test the hypothesis that the average percentage of bacteria killed was 45% at alpha=0.05.
#1.1) Write out the Null and Alternative hypotheses
#1.2) Calculate the T-statistic and use Method 1 (we saw in class) to determine if the average bacteria killed was 45%. Do it by "hand".
#Hint: Method 1 is to compare T to a critical value "c".
#1.3) Do you reject or fail to reject the null?
################################################
#PART 2
#Now, let's not assume normality
#2.1) Using the same data as in Part 1, test the hypothesis that the 20% trimmed mean is 45%?
#2.2) Do you reject or fail to reject the null?
#2.3) Assuming your test in 2.1 is the truth, what type of error did you make in #1.3?
################################################
#PART 3
#In a study of court administration, the following times to disposition (in minutes) were determined for twenty cases and found to be:
# 42, 90, 84, 87, 116, 95, 86, 99, 93, 92, 121, 71, 66, 98, 79, 102, 60, 112, 105, 98
#Assuming normality, you will test the hypothesis that the average time to disposition was 99 minutes at alpha=0.05.
#3.1) Write out the Null and Alternative hypotheses
#3.2) Calculate the T-statistic and use Method 2 (we saw in class) to determine if the average time to disposition was 99? Do it by "hand".
#Hint: Method 2 is to evaluate the confidence interval.
#3.3) Do you reject or fail to reject the null?
################################################
#PART 4
#Now, let's not assume normality
#4.1) Using the same data as in Part 3, test the hypothesis that the 20% trimmed mean is 99?
#4.2) Do you reject or fail to reject the null?
#4.3) Assuming your test in 4.1 is the truth, what type of error did you make in #3.3?
################################################
#PART 5
#Suppose you run an experiment, and observe the following values:
# 12, 20, 34, 45, 34, 36, 37, 50, 11, 32, 29
#You will test the hypothesis that the average was 25 at alpha=0.05.
#5.1) Write out the Null and Alternative hypotheses. Conduct the hypothesis test assuming normality. Use the "t.test" function. Do you reject or fail to reject the null?
#5.2) Conduct the hypothesis test without assuming normality. Do you reject or fail to reject the null?
#5.3) Assuming the answer in #5.2 is the truth, what type of error (if any) did you make in #5.1 by assuming normality?
------------------------------------------------------------------------------------------
Lab 7- Lecture Notes (FOR YOUR REFERENCE)
#Lab 7-Contents
#1. Formulating Hypotheses
#2. T-statistics by Hand
#3. Alpha Level
#4. Evaluating Our Results
#5. Using the t.test function
#6. T-tests with Trimmed Means (trimci function)
#7. Type 1 and Type 2 Errors
# Last week we talked about computations for when the Population
#Variance is known and unknown.
# Given that we rarely know the population variance,
#we will use the T-distribution for all of today's lab.
#We will primarily work with the dataset brfss09_lab7.txt:
#########################################################################################################################
#Behavioral Risk Factors Surveilance Survey 2009 (BRFSS09) Data Dictionary:
#------------------------------------------------------------------------------------------------------------------------
#id: "Subject ID"Values[1,998]
#physhlth: "# Days past month phsycial health poor" Values[1,30]
#menthlth: "# Days past month mental health poor"Values[1,30]
#hlthplan: "Have healthcare coverage?"Values 1=Yes, 2=No
#age:"Age in Years"Values[18,99]
#sex:"Biologic Sex"Values 0=Female, 1=Male
#fruit_day: "# of servings of fruit per day"Values[0,20]
#alcgrp: "Alcohol Consumption Groups"Values 1=None, 2= 1-2 drinks/day 3= 3 or more drinks/day
#smoke:"Smoking Status"Values 0=Never, 1=Current EveryDay, 2=Current SomeDays, 3=Former
#bmi:"Body Mass Index"Values[14,70]
#mi:"Myocardial Infarction (heart attack)"Values 0=No, 1=Yes
#------------------------------------------------------------------------------------------------------------------------
# For today's lab, let's start by reading in our datafile
# 'brfss09_lab7.txt' into an object called mydata
mydata=read.table('brfss09_lab7.txt', header=T)
#This file contains:
dim(mydata)#100 Subjects, 11 variables
#With the following variables:
names(mydata)
# We have collected this data and would like to know
#if the values we have found in our sample are different
#from the reported values in the literature.
# For example, it has been reported that the average BMI
# in the population is 27.5. We would like to know if the
#values in our sample are somehow different than this value.
#---------------------------------------------------------------------------------
# 1. Formulating Hypotheses
#---------------------------------------------------------------------------------
#Step 1 of determining if our BMI values differ from the
#national average of 27.5 is to formulate our hypotheses
#We have TWO hypotheses
#1) The Null Hypothesis: H0: mu = 27.5
#2) The Alternative Hypothesis: HA: mu != 27.5
#NOTE: mu=Population Mean
#The above hypotheses are Two-Sided.
#By this I mean that we are looking to see if our sample values of
#BMI are greater than (>) OR less than (<) 27.5.
# A one-sided hypothesis test would look like:
#H0: mu < 27.5
#HA: mu > 27.5
#OR
#H0: mu > 27.5
#HA: mu < 27.5
#We will always use two-sided tests in this class,
#and similarly in the real world two-sided tests dominate.
#Once we have our hypotheses we will evaluate them
#and determine one of two outcomes:
# A) Reject the Null Hypothesis
# B) Fail to Reject the Null Hypothesis
#---------------------------------------------------------------------------------
# 2. T-statistics by Hand (well..with help from the computer)
#---------------------------------------------------------------------------------
#Recall from the last lab, that the formula for a T-statistic is:
# T = (SampleMean - PopMean) / (SampleSD/sqrt(N))
#Another way to write this would be:
# T = (xbar - mu) / (s/sqrt(N))
#In this instance PopMean (mu) is the NULL hypothesis
#value we are testing against.
#We can solve for the other values that we don't yet know:
mu=27.5
xbar=mean(mydata$bmi) #28.22
s=sd(mydata$bmi) #6.32
N=100
T = (xbar - mu) / (s/sqrt(N))
T #1.14
#We end up with a T value of ~ 1.14
#But how does this tell us if our mean is different from 27.5 ???!!!
#Before we move on, I want us to think about why we need
#to evaluate if our mean of 28.22 is different from 27.5.
#Certainly we can see that these are different numbers,
#so what are we really asking here?
#One way to think about it is that we are asking if our
#sample mean of 28.22 is different from 27.5 simply due to chance.
#Think of a coin tossing example:
#Your friend tosses a coin in the air and it lands on heads
#3 times in a row!
#While, kinda cool, seems like that is probably random chance.
#What about if it landed on heads 100 times in a row?!
#You would probably think she was cheating somehow!
#Though it is possible to have 100 heads in a row
#by chance alone, it is very unlikely
#The point at which we say that something is random vs not
#is determined by our alpha level.
#---------------------------------------------------------------------------------
# 3. Alpha Level
#---------------------------------------------------------------------------------
# The alpha level is determined a priori (a head of time)
#and used to set the threshold by which we consider something
#to be random chance
# A common alpha level is 0.05.
# We typically reject the null (think something is not chance)
#when the result we have (eg. 28.22) would only be
#that extreme < 5% of the time by chance.
#Recall from Lab 6, that we use the alpha level
#to help figure out critical values (c)
# c=qt(1-(alpha/2), df)
#---------------------------------------------------------------------------------
# 4. Evaluating our Results
#---------------------------------------------------------------------------------
# There are 3 ways to evaluate if our mean of 28.22
# is different from the null of 27.5
# All three ways will yield the same conclusion.
#1) Compare T to a critical value (c)
#2) Evaluate the Confidence interval
#3) Compare the p value to our alpha level
###########################################################
#1) Compare T to a critical value (c)
#In order to compute the critical value (c),
#we must know the alpha level.
#We will choose a value of 0.05 (which is standard)
alpha=0.05
df=100-1
c=qt(1-(alpha/2), df)
#We can then compare the abosulte value of T (|T|)
#to the critical value c
#A) If |T| > c, then Reject the Null Hypothesis
#B) If |T| < c, then Fail to Reject the Null Hypothesis
#Let's look at T can c
abs(T)
c
#What decision do we make about the Null Hypothesis????
###########################################################
#2) Evaluate the Confidence interval
#Rather than compare T to c,
#we could instead compute the confidence interval.
#Recall the formula for the Confidence interval is:
#LB= xbar - c*(s/sqrt(N))
#UB= xbar + c*(s/sqrt(N))
LB = xbar - c*(s/sqrt(N))
UB= xbar + c*(s/sqrt(N))
#A) If mu is not within the Confidence Interval,
#then Reject the Null Hypothesis
#B) If mu is within the Confidence Interval,
#then Fail to Reject the Null Hypothesis
#Let's look at LB and UB
LB
UB
mu
#What decision do we make about the Null Hypothesis????
###########################################################
#3) Compare the p value to our alpha level
#Lastly, we could find the probability value (or p-value)
#for the T statistic we created.
#We can do this by using the pt() function we learned
#about last week in lab 6.
#There is a forumla for computing P values from T-statitics:
# pval = 2*(1-pt(abs(T), df))
pval = 2*(1-pt(abs(T), df))
#We then compare the p-value to our alpha level
#A) If pval < alpha, then Reject the Null Hypothesis
#B) If pval > alpha, then Fail to Reject the Null Hypothesis
#Let's look at our p-value.
pval
alpha
#What decision do we make about the Null Hypothesis????
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
#Exercise 4-1:
#Evaluate if the mean age from our sample (mydata) is different
#than the populatiuin mean age of 56
# A) Write down the Null and Alternative Hypotheses
# B) Calculate the T-statistic by hand
# C) Evaluate the Null hypothesis by using ALL 3 methods that
# we just discussed
# D) Based on the results in C, do you Reject or Fail to Reject
# the Null Hypothesis?
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
#A)
#B)
#C)
#Method 1: Compare T to a critical value (c)
#Method 2: Evaluate the Confidence interval
#Method 3: Compare the p value to our alpha level
#D)
#---------------------------------------------------------------------------------
# 5. Using the t.test function
#---------------------------------------------------------------------------------
#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^#
# One Sample T-Test : t.test(data$variable, mu)
#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^#
#It was really awesome that we figured out T by hand!
#And then figured out the confidence intervals and P values!
#From now on, let's just use a program to do all this for us.
#The function t.test will presume an alpha level of 0.05 by default.
t.test(mydata$age, mu=56)
# t.test(mydata$bmi, mu=27.5)
#Much simpler!
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
#Exercise 5-1: Use the t.test function to evaluate if
#A) the mean days of physical health (physhlth) is different
# than the population mean of 10? Reject the Null?
#B) the mean fruits per day (fruit_day) is different than
# the populatiuin mean of 4? Reject the Null?
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
#A)
#B)
#---------------------------------------------------------------------------------
# 6. T-test with Trimmed Means
#---------------------------------------------------------------------------------
#To use the T-test with trimmed means,
#we will need to load in the source code 'Rallfun-v33.txt'
#The trimmed mean T-test is beneficial in that it does not
#presume a perfect Normal Distribution
#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^#
# Trimmed Mean T-Test:
# trimci(data$variable, tr=0.2, alpha=0.05, null.value=0)
#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^#
#For example, if I wanted to test if the age was equal to 56
#using Trimmed Means I could do:
trimci(mydata$age, null.value=56)
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
#Exercise 6-1: Use the trimci function to evaluate if
#A) the 20% trimmed mean of days of physical health (physhlth) is
# different than the populatiuin mean of 10? Reject the Null?
#B) the 20% trimmed mean fruits per day (fruit_day) is different
#than the populatiuin mean of 4? Reject the Null?
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
#A)
#B)
#---------------------------------------------------------------------------------
# 7. Type 1 and Type 2 Errors
#---------------------------------------------------------------------------------
#Notice that we had very different answers to the same
#questions in Ex. 5-1 and 6-1
#Depending upon the method that we used.
#This brings us to discussing Type 1 and Type 2 Error
#A Type 1 error is when our test tells us to reject the null,
#but in truth we should not have
#A Type 2 error is when our test tells us to fail to reject the
#null, but in truth we should have rejected the null
#The following 2x2 square might make this easier to see.
# Truth
#------------------------------------
#| H0 | HA |
#-------------- |-------|-----------|
#My Test: H0 | H0 Type 2|
#-------------- |-------|-----------|
#My Test: HA Type 1 | HA |
#------------------------------------
#For the next exercise, let's presume that our test of the trimmed mean is the Truth
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
#Exercise 7-1:
#A) What type of error did we make when evaluating the mean
#of physhlth in exercise 5-1?
#B) What type of error did we make when evaluating the mean
#of fruit_day in exercise 5-1?
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
#A)
#B)
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