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.