Economics 102
Problem Set 2
Due in Canvas on SATURDAY, July 7 before 10 p.m.
Department of Economics Professor Siegler
UC Davis Summer 2018
Instructions: Please submit a self-contained and well-formatted Word (or similar) document in
Canvas containing all of your written answers, calculations, and graphs for the questions below.
Also, submit any R scripts you used. Start early.
1. Sampling Distributions and Standard Errors
The following data represent the number of days absent per year in a population
of all four employees of a small company (N=4):
Employee Days Absent from Work
(Xi)
A 1
B 3
C 5
D 7
Assuming sampling with replacement (that is, once an employee is chosen, he or she
is returned to the population and can be selected again in the same sample of
two), select all possible samples of n = 2 and construct the sampling distribution
of the mean. Hint: there 16 possible samples of g1866 = 2, each with the same
probability of occurring.
A. Sketch an accurate histogram of both the population distribution and the
sampling distribution of the sample means. Clearly label each histogram.
You can also use R if you’d like.
B. Compute the mean of all the possible sample means and also compute the
population mean. Are they equal? Explain why or why not.
C. What is the value of the standard error of the sampling distribution?
Show your work.
D. Intuitively explain in words what a standard error represents.
2
2. A Monte Carlo Experiment in R for an Alternative Estimator for the
Population Mean
Suppose that individual scores in a population are uniformly distributed from 20
to 100.
A. What is the population mean g4666g2020g4667and population standard deviation g4666g2026g4667?
B. What is the standard deviation of the sampling distribution of sample
means (standard error) with n = 30?
C. Instead of using sample means to estimate the population mean, suppose
that someone proposes the following IQR estimator for the population
mean, which will be called g1850g3367:
g1850g3367 = g1835g1843g1844 + 30
where IQR is the interquartile range as described in lecture. Hint: the
command in R for this estimator is:
IQR(runif(30, 20, 100), type=6)+30)
Following the example posted under Modules, write and run an R script.
that simulates the properties of this estimator with 100,000 samples of n =
30. Report the histogram from this Monte Carlo experiment.
D. Is this estimator a biased or unbiased estimator of the population mean?
Explain.
E. Is the sampling distribution for this estimator normally distributed or not?
Explain.
F. Is the standard error of this estimator larger, smaller, or the same as the
standard error of the sampling distribution of sample means of n = 30
drawn from the same uniform. distribution? Referring to the theoretical
results from Part B and the simulation results from Part C, which
estimator has a smaller standard error? All in all, which estimator is
preferred: the estimator in Part C above or the sample mean? Explain.
3
3. Applying the Central Limit Theorem
A nationwide study found that the average yearly claim per automobile
insurance policy (after deductible) is $1,000 with a standard deviation of $400.
Consider these parameter values to be the true population parameters.
Suppose that you are working for a small, start-up insurance company with only
625 automobile insurance policies, and the company has $635,000 to pay for
claims next year. Suppose that your policies are a random sample of policies
nationwide. What is the probability that your company will not have enough
money set aside to pay off the claims next year? If the company does not have
enough money set aside, it will declare bankruptcy and you will lose your job.
Should you start working on your résumé? Show your work and explain.
4. One-Sample Univariate Hypothesis Testing of a Mean
Consider a random sample of six college students at a particular university (n=6),
where H is the number of hours of study on the day prior to the final exam.
g1834 = g46681,3,3,5,8,10g4669
Suppose that a professor claims that the typical college student studies more
than 8 hours on the day prior to the final exam.
A. What is the null hypothesis? What is the alternative hypothesis?
B. Can you reject the null hypothesis at the 5-percent level of significance?
Can you reject the null hypothesis at the 1-percent level of significance?
Use the critical value approach. You can use R for critical values, but you
must show all of your calculations and explain. Use R, however, to check
your work.
C. What is the 95-percent confidence interval for hours of study? Provide a
written interpretation explaining your answer.
4
5. One-Sample Univariate Hypothesis Testing with Proportions
For this question, show the results “by hand”, but you can use R to check your
work. Suppose that the 4-year graduation rate at a large, public university is 60
percent (this is the population proportion of successes). In an effort to increase
graduation rates, the university randomly selected 300 incoming freshman to
participate in a peer-advising program. After 4 years, 192 of these students
graduated. What are the null and alternative hypotheses? Can you conclude
that this program was a success at the 5-percent level of significance? Show your
work and explain.
6. Testing Human Body Temperatures using R
You must use R for this question since the data set is too large to expect to do the
calculations by hand. The data for this problem set are derived from an article in
the Journal of the American Medical Association entitled “A Critical Appraisal of
98.6 Degrees F, the Upper Limit of the Normal Body Temperature, and Other
Legacies of Carl Reinhold August Wunderlich” (Mackowiak, Wasserman, and
Levine 1992). The data are contained in the Excel spreadsheet
temperatures.xlsx. To get the data into R, install and load the rio package.
Change your directory so R is looking in the correct folder to find
temperatures.xlsx, and then the command in R is
import(“temperatures.xlsx”)
A. Is the true population mean really 98.6 degrees Fahrenheit? What are the
null and alternative hypotheses? Test this hypothesis using the critical
value, p-value, and confidence interval approaches. It is up to you to
justify your chosen level of confidence and your significance level. Be
sure to intuitively explain what your results mean. Based on these data, at
what temperature should we consider someone’s temperature to be
“abnormal”?
B. Suppose that someone claims that the true population standard deviation
of body temperatures is greater than 0.6 degrees. What are the null and
alternative hypotheses? What is the value of this test statistic? Can you
reject the null hypothesis at the 1-percent level of significance? Show your
work and explain. Hint: The test statistic is for the variance, and not for
the standard deviation, so square the standard deviation to yield the
variance.