Stat 301 - HW 8

Due noon, Friday, March 13

 

Please remember to submit a separate file for each problem (with the problem number in the file name and your name inside the file) and to integrate all relevant computer output.  Also remember the Extensions assignments have their own submission page in Canvas.

 

JMP Note: To run a “matched pairs analysis” in JMP (using the two columns of data), it’s now under Analyze > Specialized Modelling > Matched Pairs

 

0) Confirm with me if you requested a switch to final exam days

(2pm section: Friday 1:10-4pm, 3pm section: Monday 1:10-4pm in 10-215)

Interest in review sessions? 3/15, 3/19

Two online course evaluations

 

1) Suppose you want to compare student’s performances on the first two exams in a course to see whether there is convincing evidence that students (in general) tend to perform better on the first exam.

(a)  Would you recommend a paired design or an independent samples design? Justify your recommendation, clarify how each design would differ in this context.

(b)  Consider the following two classes. In one class, pairing appears to be useful, but not in the other class.  Explain how you can tell, and what this implies about students’ exam performances in the two classes.

Class A

 

 

 

 

Class B

 

 

 
Exam 1

n1 = 12

1 = 86.4

s1 = 9.5

 
Exam 3

n3 = 12

3 = 86.4

s3 = 9.5

Exam 2

n2 = 12

2 = 83.3

s2 = 12.3

 

Exam 4

n4 = 12

4 = 83.3

s4 = 12.3

Differences

nd = 12

d = 3.2

sd = 4.5

 

Differences

nd = 12

d = 3.2

sd = 18.0

(c)  Below is output of a power analysis in JMP for an independent samples design. 

Explain the output to a non-statistician.  Your explanation should include

·       Definitions of Alpha, Std Dev, Difference to detect, Sample Size

·       Interpretation of the calculated power value in context

·       Sketches (probably by hand) of the null and alternative sampling distributions, showing the area under the curve representing power

Hint: If you are having trouble visualizing this process, perhaps check out this applet: http://www.rossmanchance.com/applets/TwoPopulations.html?population=model

Also keep in mind that calculation of power is a two-step process.

(d)  Below is output of a power analysis in JMP for a paired samples design

Explain to a non-statistician why the set up of the calculation has changed, whether this is reasonable, and why and how the power has changed so much.

 

2) hw8problem2.Rmd new 3/11

Does the path that you take to “round” first base make much of a difference? Hollander and Wolfe (1999) report on a Master’s Thesis by W. F. Woodward (1970) that investigated different base running strategies. For example, you could take a “narrow angle” or a “wide angle” around first base.

In Woodward’s study, he used a stopwatch to time 22 different runners going from a spot 35 feet past home to a spot 15 feet before second. He had each runner use each method, with a rest period in between, randomizing which method they used first. The data in BaseRunning.txt shows the time (in seconds) for each running using the narrow angle and the wide angle.

(a) Compute the differences in times (wide – narrow). Produce, include, and comment on relevant graphical displays and numerical summaries for investigating the question of whether there is an advantage for taking wide angles or narrow angles.

(b) Define the parameter of interest in this study and write the null and alternative hypotheses for testing whether there is an advantage for taking wide angles or narrow angles.

(c) Conduct a paired t-test or use the Matched Pairs applet to determine whether the data suggest a genuine difference in times for wide angles and narrow angles. If you use the t-test, make sure comment on whether you believe the test procedure is valid and how you are deciding. (Remember to include your output.)

(d) Construct, include, and interpret a 95% confidence interval for estimating the population mean difference in base running times.  (Be sure it’s clear in your interpretation which method is faster.)

(e) Summarize the conclusions you would draw from this study. Make sure you comment on significance, estimation, generalizability, and causation.

(f) Using a sign test (e.g., Investigation 2.7, Example 3.4).  State the corresponding hypotheses (in symbols and in words) and p-value (exact binomial or normal approximation).  Compare your conclusions to those in question (c).

 

3) Researchers investigated a possible link between having a tonsillectomy and developing Hodgkin’s disease.  They studied a sample of 85 Hodgkin’s patients who had a sibling of the same sex within 5 years of age who was free of the disease (170 individuals total, 85 pairs).  Taking into account the paired nature of these data produces the following table:

 

Hodgkin’s patient,

had tonsillectomy

Hodgkin’s patient,

did not have tonsillectomy

Total

Control, had tonsillectomy

26

7

33

Control, did not have tonsillectomy

15

37

52

Total

41

44

85

 

(a) Identify the observational units in this study. Clearly identify the population of interest.

(b) For how many pairs was the tonsillectomy outcome differ?  How many “successes” and how many “failures” (being clear how you are defining each of these.)

(c) Identify the parameter of interest in this study.

(d) State appropriate null and alternative hypotheses for determining whether the Hodgkin’s sibling was more likely to have a tonsillectomy, in symbols and in words.

(e) Apply McNemar’s Test to calculate an appropriate p-value including the name of the appropriate probability distribution and also specify its input values.  Report the p-value (include output) and summarize your conclusion in context.

(f) Calculate and interpret a 95% confidence interval for the parameter in (c).  Do you think your interval procedure is valid?

 

 

Possible Extension Assignments

·       Use the Two Populations applet to verify/approximate the power calculation in 1c. First, determine the rejection region, then estimate the power.  Be very clear your process and include some screen captures. (Hint: Are you using a one-sided or two-sided test? How do you use the rejection region in step 1, to estimate the power in step 2?)

·       Find the “Ask Marilyn” column in Parade Magazine from Feb. 21, 2016.  Review the discussion and how it relates to this class.

·       Read, summarize, and critique Kwok et al. (2015) “Face touching: A frequent habit that has implications for hand hygiene,” American Journal of Infection Control, 43(112-4).

·       If you performed poorly on a HW or Exam problem earlier in the quarter, include your reworked solution.

·       Review and reflect on the ASA Press Release on p-values!

https://www.causeweb.org/cause/caption-contest/march/2020/submissions