Homework Assignment
Michael J. Culbertson UW–Madison STAT 371 Fall 2021
HOMEWORK 8:NON-PARAMETRIC, 1-GRP
Instructions
This assignment is due in Canvas by Wednesday, November 3 at 11:59pm.
Your submission needs to include your R code, the corresponding R output, and your
narrative interpretation/responses to the questions (in complete sentences). The easiest way to do
this is to work in an R Notebook, as described in the first homework guide and demonstrated in the
corresponding video. If you use an R Notebook, you will submit the notebook’s HTML file to Canvas.
Set Up
Copy and paste the following code (as is) into your R Notebook. Execute the code to gain access
to the boot.ci() and boot.p() functions for bootstrap confidence intervals and p-values:
boot.ci <- function(x, f, conf.level = .95, R = 5000, … ) { D <- sapply(1:R, function(i) f(sample(x, length(x),
replace = T), …) )
quantile(D, c((1-conf.level)/2, (1+conf.level)/2))
}
boot.p <- function(x, f, mu = 0, R = 5000,
alternative = c(‘two.sided’, ‘less’, ‘greater’), … ) {
alternative <- match.arg(alternative)
n <- length(x) D <- sapply(1:R, function(i) f(sample(x, n, replace = T), …)) switch(alternative,
two.sided = 2*min(mean(D <= mu), mean(D >= mu)), less = mean(D >= mu),
greater = mean(D <= mu) )
}
Recall from lecture that you supply your data (using $ style) as the first argument and the function
that computes your statistic of interest (e.g., sd) as the second argument. The third argument is either
the confidence level (for boot.ci) or the hypothesized parameter value (for boot.p). For example:
# Bootstrap 95% confidence interval for the standard deviation
boot.ci(pigs$litter.size, sd, .95)
# P-value for testing whether the population S.D. is 2.42
boot.p(pigs$litter.size, sd, 2.42)
Exercise
Refer to the piglets and hurricane datasets on Canvas for this assignment.
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1. The climatologist you were consulting with last week has moved on to look at the effects
of hurricanes. The hurricanes dataset includes the estimated damage caused by each hurricane in millions of 2013 dollars (damage_norm). The climatologist would like to know
whether the average damage is over $1 billion (i.e., a value of 1000 for damage_norm).
The climatologist decides to use a � = .05 significance level with a two-sided test.
a. Examine a histogram of the damage amounts and comment on its characteristics.
b. Why might it be helpful to use the median damage, rather than the mean damage
caused by the hurricanes? From the perspective of statistical inference, what
might be the drawback of using the median, instead of the mean?
c. What is the best guess for the median damage caused by hurricanes?
d. What test statistic will you use for a test of the median? (Explain how you found
your answer.)
e. Find the p-value for the test of the median. Explain what this p-value represents
in the context of this research scenario.
f. What conclusion should the climatologist draw about hurricane damage?
g. The climatologist doesn’t understand why his best guess for the damage of hurricanes seems to conflict with the conclusion you said he should draw from the
statistical test. How would you explain this apparent conflict to him?
2. A hog farmer is trying to improve the consistency of the newborn weight of the piglets
on his farm. Last year, the standard deviation of the newborn piglet weight was 0.44 kg.
This year, the farmer has attempted to selectively breed sows that produced particularly
uniform litters of piglets. The piglets dataset on Canvas includes the weights in kilograms
of 85 randomly selected piglets born on the farm this year.
a. What are the population, sample, parameter, and statistic for this study?
b. Use the boot.ci() function to calculate a 99 percent confidence interval for the
population standard deviation of newborn piglet weights this year.
c. Does the confidence interval suggest that the farmer has improved the consistency of newborn piglet weights compared with last year? Explain how you
know.
d. What is the significance level of the implicit statistical test that you conducted in
part (c)?
e. Find the p-value for this statistical test using the boot.p() function.
f. If the farmer had used a significance level of � = .05, would he have come to a
different conclusion? Explain.
g. Why did we use the bootstrap procedure for this research scenario?
3. The mean cholesterol level in the U.S. population is approximately 180, with a standard
deviation of 41. A pharmaceutical company thinks that the new drug they have developed
will reduce cholesterol levels by 10 percent. They are planning to run a study of the drug,
but they are looking to redesign their study, after determining that their initial plans
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would yield very low statistical power. They will conduct a two-sided test with a significance level of � = .01.
a. What sample size would they need in order to obtain 90 percent power, if the
drug reduces cholesterol by 10 percent?
b. At a planning meeting, an executive wonders what would be the consequences
to the study if the drug was only half as effective as they think (i.e., it reduces
cholesterol by 5 percent, instead of 10 percent). Explain what would happen to
statistical power if this were the case.
c. What sample size would they need in order to obtain 90 percent power, if the
drug reduces cholesterol by only 5 percent?
d. How many times larger does the sample size need to be in order to detect an
effect that is half as big? Explain.
e. Why is it sometimes a good idea to plan a research study based on a smaller effect
size than you expect you might find? What is the downside to planning based on
a very small expected effect size?
hw-08-quantile-bootstrap
APA
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The post The climatologist you were consulting with last week has moved on to look at the effects of hurricanes. The hurricanes dataset includes the estimated damage caused by each hurricane in millions of 2013 dollars (damage_norm). The climatologist would like to know whether the average damage is over $1 billion (i.e., a value of 1000 for damage_norm). The climatologist decides to use a � = .05 significance level with a two-sided test. a. Examine a histogram of the damage amounts and comment on its characteristics. b. Why might it be helpful to use the median damage, rather than the mean damage caused by the hurricanes? From the perspective of statistical inference, what might be the drawback of using the median, instead of the mean? c. What is the best guess for the median damage caused by hurricanes? d. What test statistic will you use for a test of the median? (Explain how you found your answer.) e. Find the p-value for the test of the median. Explain what this p-value represents in the context of this research scenario. f. What conclusion should the climatologist draw about hurricane damage? g. The climatologist doesn’t understand why his best guess for the damage of hurricanes seems to conflict with the conclusion you said he should draw from the statistical test. How would you explain this apparent conflict to him? appeared first on Apax Researchers.
