HW07

Homework 7 STAT651 Monday, March 25, 2024 Please submit the relevant graphs and JMP outputs ( remove / close those unnecessary ones ). Every test should include H 0 , H A , justification of your conclusion and conclusion. 1. The hippocampus shrinks in late adulthood, leading to impaired memory and increased risk for demen- tia. In a study of evaluating whether exercise training expands the hippocampus and improves spatial memory, 60 healthy but sedentary participants, aged 58 to 70, joined a one-year aerobic exercise pro- gram. Magnetic resonance images were collected before the intervention and again after the completion of the program and participants’ hippocampal volumes (in cm 3 ) were measured. The data are in the file, memory.jmp. JMP commands: Select Analyze from menu bar −→ Specialized Modeling −→ Matched Pairs −→ Select Before then After for Y, Paired Response in the Matched Pairs dialog box −→ OK. Please remove the graph by clicking ▼ then un-checking Plot Dif By Mean . (a) What does the 95% confidence interval suggest about the effect of exercise training? (b) Does exercise training increase the size of hippocampus? Give a test at 1% level of significance. (c) Is there sufficient evidence in the data to support that exercise training increases the average size of hippocampus by more than 0.1 cm 3 ? Give a test at 5% level of significance. Questions 2 and 3 require the data file NormTemp.jmp. It contains data on body temperatures and heart rates for 130 randomly selected subjects. The first column (Temp) contains the temperatures themselves. For each subject, this temperature, in degrees Fahrenheit, represents an average of several readings taken over the course of two consecutive days. The second column (Gender) is M for male, F for female, and the third column (HeartRate) is measured in beats per minute. 2. We focus on body temperature first, the summary statistics for men and women are shown here. F M Level 65 65 Number 98.39 98.10 Mean 0.743 0.699 Std Dev 0.092 0.087 Std Err Mean 98.210 97.931 Lower 95% 98.578 98.278 Upper 95% Means and Std Deviations (a) From the above JMP output, what is the sample mean for the Female group? Male group? What is the 95% CI for the true average body temperature of females? of males? To analyze gender differences between in temperature, please use the instructions at the end of this question to generate relevant JMP outputs. (b) From the side-by-side boxplots for the temperatures of males and females, do there seem to be any differences? (c) What assumption you would make about the two population variances based on the F -test for equal variance? Please state the null H 0 and alternative H A hypotheses, the test result and conclusion. 1

Homework 7 STAT651 Monday, March 25, 2024 (d) Write down the 95% CI for the mean difference in body temperature. Does this confidence interval suggest any gender differences in body temperature? Explain. (e) From the output, what is the estimate of the mean difference in body temperature? What is the estimated standard error of the sample mean difference? (f) Give a formal test for the difference in mean body temperature between males and females. Can the null hypothesis be rejected at level .05? At level .01? JMP commands: Select Analyze from menu bar −→ Fit Y by X −→ Select Temp for Y, Response and Gender for X, Factor in the Fit Y by X dialog box −→ OK . A dialog window labelled “Oneway Analysis of TEMP by GENDER” would show. First get boxplots and descriptive statistics for each group. Click ▼ at top left corner −→ Display Options −→ All Graphs . Click ▼ −→ Box Plots . Next, get a test for equality of variances. Click ▼ −→ Unequal Variances . (You don’t need to hand in this variance test output, clicking ▼ to close when print). Based on the result from F Test 2-sided with α = 0.05, continue with one of the following t tests. Equal variance assumption: Click ▼ −→ Means/Anova/Pooled t −→ Display Options −→ se- lect Box Plots and un-select Means Diamonds . You need only the t Test output and the Means for Oneway Anova output. Remove the unwanted output by clicking ▼ next to its title. Unequal variance assumption: Click ▼ −→ t Test . 3. We will analyze heart rates for men and women in this question. The summary statistics of heart rate for male and female groups are shown here. F M Level 65 65 Number 74.15 73.37 Mean 8.105 5.875 Std Dev 1.005 0.729 Std Err Mean 72.145 71.913 Lower 95% 76.162 74.825 Upper 95% Means and Std Deviations Generate the relevant output and answer the same questions, (a)-(f), as in Questions 2 . 4. Ex. 10.20 (pages 536-537) of Ott and Longnecker. 2