**Stat 301 - HW 8**

**Due midnight, Friday, March 15**

*Remember to
put your names in this file and to include and integrate all relevant computer
output. *

**1)** In
baseball, when running from say home plate to second base, 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 planned to use a stopwatch to time
runners going from a spot 35 feet past home to a spot 15 feet before second
based. He had access to 22 different
runners. Woodard wanted to test H_{0}: m_{narrow} - m_{wide} = 0
vs. H_{a}: m_{narrow} - m_{wide} ≠
0.

(a) Suppose he tells you that the standard deviation of
running speeds among such runners is about 0.30 seconds. Give a one-sentence interpretation of this
value.

(b) According to somatechnology.com,
the average human eye blink is 0.10 seconds.
If Woodward randomly assigns these 22 runners to
two groups of 11, use the Normal Probability Calculator applet (or R’s pnorm and
qnorm functions or iscaminvnorm and iscamnormprob)
to approximate the power he will detect a difference in mean running time in a
two-sided test:

Include
a screen capture of your results.

Include a screen capture of your results.

What is the probability that Woodard
will correctly reject the null hypothesis in this case?

(c) Instead, Woodward conducted a paired-design
with his 22 runners, asking each runner to use
each method, with a rest period in between, randomizing which method they used
first. Should the variability in the *time differences *be larger or
smaller than the variability in the *times*? Explain your reasoning.

The data in BaseRunning.txt shows the time (in seconds) for each running using
the narrow angle and the wide angle. His
original hypotheses are equivalent to testing H_{0}: m_{diff} = 0
vs. H_{a}: m_{diff} ≠
0.

(e) Carry out a ** simulation analysis** of the
paired data using the Matched
Pairs applet (the data are preloaded into the applet). Include a short
description of the simulation process and what it represents. Include a screen
capture of the simulated null distribution with the two-sided p-value.

(f) Use R or the Applet to carry out the ** one-sample
t-test** on the differences (aka a matched-pairs

R br <- read.delim("http://www.rossmanchance.com/iscam3/data/BaseRunning.txt", sep="\t") t.test(br$wide, br$narrow, paired=TRUE) |
Applet Change the statistic to Enter
the observed |

Include a screen capture of the results and report the test
statistic and two-sided p-value.

(g) Does the *t*-test appear to be valid for these
data? You should comment on the validity conditions of the paired *t*-test
as well as how the results in (e) and (f) compare.

(h) Carry out a ** sign test** on the paired data:

1. Calculate the time differences
(narrow – wide) (You can use the dotplot in the
applet.)

2. How many of the differences are
positive? How many are negative? How many are zero?

3.
Consider the non-zero differences, what proportion are positive?

4. Use the
binomial distribution to determine whether there is a statistically significant
majority of the differences are positive (define the parameter of interest,
state the hypotheses, and determine the exact binomial p-value – be sure to
include a screen capture)

(i) Does the sign test provide stronger or weaker evidence
that one base-running method tends to be faster than the other? (Note: You
should compare the two-sided p-values to each other.)

(j) Would a one-sample *z*-test be appropriate in
(h)? Explain how you are deciding.

(k) Determine, include, and interpret in context a 95%
confidence interval for . This time, consider your answer to (j) in
deciding which interval procedure to use.

**2) **Based on your responses to Question 4 on
HW 7, I have created two datasets:

containing your original water usage value and your
“realistic” adjusted water usage value, depending on whether you were using the
short form or the long form.

Use R or the Matched Pairs applet.

(a) Examine the data with the short form. One observation stands out to me as
unusual. What graph did you look at to
spot it? I believe there is an error in
the data reported – what do you think happened? (*Hint*:
See (c) as well?)

(b) Remove the unusual observation in (a) (document how you do so, you can replace a value with *
or remove both observations and reload the data?) and provide a dotplot and numerical summaries for the differences between
the original and adjusted values. Also
produce and interpret a 95% *t*-confidence interval using the differences.
Include your output.

(c) Now consider the data from the long form. Again we have one strange observation, perhaps with the same
explanation? Remove this observation and
produce numerical and graphical summaries of the differences. Produce a *t*-confidence interval and
discuss the similarities and differences compared to the interval in (b). Include your output!

**Reminder:**
Course Evaluations due Friday night as well!