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STAT 4204/5204G: Experimental Design Homework 2

发布时间：2022-09-15

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STAT 4204/5204G: Experimental Design

Problem 1

Problem 1 will consist of 2 questions. In Question 1, you will be generating random data for an ANOVA analysis. Therefore, everyone in the class will have different answers. For this question, please create one document that contains the answers to all parts of Question 1 and upload this single document to Canvas for grading. Please create either a word document or a pdf document for upload.

Question 1

Use Minitab, JMP, or the software of your choice to generate normal random data for a one-way ANOVA analysis. Note: There are documents on Canvas to help with generating the data for this question. Use a = 5 and n = 12. The response variable represents yield of a chemical process. Treatment A should have a mean of 92.5, Treatment B should have a mean of 97.5 and the other three treatments (C, D, E) should all have means of 95. Assume that G = 2.5. Complete the following parts to the question and upload your answers in one document to the Canvas quiz for Question 1.

1. Create side-by-side boxplots and copy the graph to your homework submission. Comment on which groups seem to differ. (Remember you know which groups are different because you generated the data. Does the box plot seem to depict these differences?)

2. Provide the one-way ANOVA output and analysis. Be sure to state the hypotheses, test statistic (this is the Fob value), p-value, and conclusion.

3. Complete the appropriate residual analysis. Be sure to provide the appropriate graphs and comment on whether or not the assumptions are met. Remember to use the studentized residuals for the residual plots. You should provide the Normal Quantile plot along with the Studentized Residuals vs. Predicted (Fits) plot. You do not need the run order plot because the data were not collected in a certain order (they were randomly generated).

4. Provide the results from Tukey’ s and Fisher’s Protected LSD multiple comparison tests. Comment on these results. In particular, how well did you do relative to the known values of the treatment means?

5. Complete the following parts for a contrast that compares the mean yield for Treatment A with the mean yield for Treatment B.

i. State the population contrast.

ii. State the hypotheses for testing whether the mean yield for Treatment A differs from the mean yield for Treatment B.

iii. Calculate the estimated contrast.

iv. Calculate the t test statistic.

v. Use software to calculate the p-value and state your conclusion.

Question 2

Suppose that instead of the contrast you completed in Question 1, part 5, you did a two-sample pooled t-test to determine if the mean yield for Treatment A differs from the mean yield for Treatment B. Select the all the TRUE statements below about these two procedures.

o The contrast uses the MSE from the overall ANOVA to estimate error, while the pooled two-sample t-test uses the pooled standard deviation for Treatment A and Treatment B (Sp(2)) to estimate error.

o The conclusions from the contrast comparing Treatment A to Treatment B will always be the same as the conclusions from the two-sample pooled t-test that compares the two treatments.

o You can do two-sided hypothesis tests using the pooled two-sample t-test but not using the contrast approach.

o The contrast uses information from all the Treatments (A, B, C, D, E), while the pooled two-sample t-test only uses information from Treatments A and B.

Problem 2

Read the article “Technical Advice: Residual Plots to Check Assumptions” by Professor Vining. This article is linked on Canvas. After finishing the article, complete Questions 3 through 6.

Question 3

True or False: Dr. Vining recommends using the internally studentized residuals.

Question 4

True or False: The raw residuals do not have constant variance and are also not independent.

Question 5

True or False: The internally studentized residuals follow a t-distribution.

Question 6

True or False: The externally studentized residuals follow a t-distribution and have constant variance.

Problem 3

This set of questions covers some of the basics ofANOVA and hypothesis testing.

Question 7

A designed experiment has five treatments and 25 observations and the SSTreatment = 126 and the SST = 215. What is the value for the MSE ? Use 2 decimal places.

Question 8

A researcher has a designed experiment that has 5 treatments (A, B, C, D, E). She decided to test using a contrast of whether the average response from treatments A and B is different from the

average response from the remaining treatments. Select the appropriate set of contrast coefficients.

a) cA = 1, cB = 1, cC = − 1, cD = − 1, cE = − 1

b) cA = 1, cB = 1, cC = − , cD = − , cE = −

c) cA = , cB = , cC = − , cD = − , cE = −

d) cA = , cB = , cC = − , cD = − , cE = −

Question 9

The researcher in from Question 8 completed the analysis of the contrast to determine if the average response from treatments A and B is different from the average response from the remaining treatments. The p-value for this hypothesis test is 0.056. Note, this p-value is based on the hypothesis test that is computed by taking the average response of Treatments A and B minus the average response of the remaining treatments. If instead, the research was in interested in testing if the average response from treatments A and B is greater than the average response from the remaining treatments, what would be the p-value for this hypothesis test? Use 3 decimal places.

Question 10

True or False: The p-value is the probability the null hypothesis is true.

Question 11

You are designing a new experiment to study the effects of 4 different diets in turkeys. The experiment is to be designed so that each of the 4 diet groups will have the same sample size. Determine how many turkeys would be needed for each of the diets to declare a weight gain of 3lbs significant with a power of 0.80, a variance of 3lb2 and a=0.05.