Stat 301 – HW 3

Due midnight, Friday, Jan. 24

 

Please remember to put your name(s) inside the file and if submitting jointly to join a HW 3 group first. Please use Word or PDF format only. Remember to integrate your output with your discussion.  Points will be deducted if you are missing output.  No late assignments accepted.

 

1) Hill and Barton (Nature, 2005) conducted a study to investigate whether Olympic athletes in certain uniform colors have an advantage over their competitors. They noted that competitors in the combat sports of boxing, tae kwon do, Greco-Roman wrestling, and freestyle wrestling are randomly assigned to wear red or blue uniforms. For each match in the 2004 Olympics, they recorded the uniform color of the winner. They found that in 457 Olympic combat sport matches, the competitor wearing red won 248 times (successes), while the person wearing blue won 209 times (failures).

(a) Define the parameter in words (in context).

(b) State appropriate null and alternative hypotheses in symbols and/or words.

(c) Report the sample proportion, .

(d) Is the normal approximation to the binomial distribution likely to be valid for this study? Explain how you are deciding.

(e) Find the p-value from the one-sample z-test using R or JMP or the Theory-Based Inference applet.

(f) Report and interpret, in context, the standardized statistic from the output in (e).

(g) Use R or JMP or Theory-Based Inference applet to find the one-proportion z-confidence interval. Include your output, being sure it’s clear how you found your interval.

(h) Are the p-value from (e) and the confidence interval “consistent” with each other?  Explain how you are deciding.

(i) Report the half-width (aka margin of error) for the one-proportion z-confidence interval in (g). Compare this value to . Based on the formula for the margin of error in the 95% one-proportion z-confidence interval, why does this approximation make sense? 

(j) So a short-cut approximation to the one-proportion z-confidence interval is .  If anything, this interval will be wider than it needs to be (is “conservative”).  Why?  (Hint: How does the standard error formula change with ?)

 

2) In the Week 2 survey, I asked you to match a name with a face.  I randomized whether I asked you about Bob and Tim or about Tim and Bob.  Across both versions, 55 of 69 responses matched the name Tim with the face on the left.  Let’s assume your responses are identical trials from a random process with  = probability of matching Tim with the face on the left.

(a) Is the normal approximation to the binomial distribution likely to be valid for this study? Explain how you are deciding.

(b) The adjusted-Wald 95% confidence interval (aka Plus Four) procedure is discussed in Investigation 1.10. The procedure is to add two successes and two failures to the sample before computing the z-confidence interval.  Use technology to find the 95% adjusted-Wald confidence interval.  Make sure it’s clear how you are doing so.

(c) Interpret your interval in (b) in context.

(d) Use the same data to find the exact Binomial 95% confidence interval (from R or JMP or trial and error, make sure it’s clear how you find it) and the 95% Wald (one proportion z-) interval. Which of the 3 intervals would you recommend?  Explain your reasoning.

 

Keep in mind

·       In this class, we will use the term standard error to refer to an estimate of the standard deviation of the statistic (computed entire from information available in the sample data). Interpret a standard error just like you would the standard deviation of the statistic (e.g., a typical deviation of the statistic from the parameter across different samples).

·       If I don’t give you a confidence level, assume 95%.

 

Example Extension Assignments

·       Look into some of the controversy and debate over confidence intervals for a single proportion!

·       Read this Gallup Poll Methodology. What is an aspect you mostly understand already and one new thing you learned.