Statistics I Problem Set 6
1. Assume that statistics test scores at ABC university are normally distributed with a known
SD σ = 8. A random sample of the statistics test scores of 100 students at ABC University
has a sample mean of X̄ = 55.8.
(a) Construct a 95% confidence interval for the population mean µ.
(b) Reconstruct the confidence interval for a confidence level of α = 99%. What happens
to the size of the interval?
(c) Reconstruct the confidence interval for a confidence level of α = 90%. What happens
to the size of the interval?
(d) Assume now that the random sample now contains 400 students (and the other quantities are as in question 1). Reconstruct the confidence interval for a confidence level of
α = 95%. What happens to the size of the interval?
(e) Assume now that σ = 16 (and the other quantities are as in question 1). Reconstruct
the confidence interval for a confidence level of α = 95%. What happens to the size of
the interval?
2. Annual starting salaries for college graduates with degrees in economics are assumed to be
normally distributed with a known SD of $4000.
(a) Construct a 95% CI of the population mean of annual starting salaries for college graduates with degrees in economics if a random sample of 100 graduates has a sample
mean of $58000.
(b) How large a sample should be taken if the desired margin of error (ME) for a 95% CI
is $500?
(c) How large a sample should be taken if the desired margin of error for a 95% CI is $200?
(d) How large a sample should be taken if the desired margin of error for a 95% CI is $100?
(e) Would you recommend trying to obtain the $100 margin of error? Explain.
3. According to a study based on two datasets (one for kids, the other one for adults), the
sample mean daily sleep time (in minutes) for kids is X̄K = 510 minutes, whereas for adults
it is X̄A = 420.
(a) Based on X̄K and X̄A , what is the difference between kids and adults in terms of weekly
sleep time?
(b) According to the study, the datasets had the following sample sizes: for kids nK = 400;
for adults nA = 625. Moreover, the sample standard deviations were as follows: for
kids, sK = 120 minutes; for adults, sA = 100 minutes. Construct the 95% confidence
intervals for the population mean µK and µA , assuming that both sample sizes are large
enough to apply the central limit theorem.
(c) Based on the 95 confidence intervals, can we really be sure that µK , µA , i.e that kids
and adults have different mean daily sleep times?
1
4. According to a cabinet approval rating survey, the sample approval rating for men is p̂ M =
0.58, whereas for women it is p̂W = 0.62.
(a) Based on p̂ M and p̂W , what is the difference between men and women in terms of the
cabinet approval rating?
(b) According to the study, the sample size for men and women is n M = 400 and nW =
625, respectively. Construct the 95% confidence intervals for the population cabinet
approval rating p M and pW , assuming that both sample sizes are large enough to apply
the central limit theorem.
(c) Based on the 95 confidence intervals, can we really be sure that p M , pW , i.e that men
and women have different cabinet approval rating?
5. A simple random sample provided a sample mean of 22.5 and a sample standard deviation
of 4.4. Use the values summarized in lecture notes for the 97.5 percentiles of t distributions.
(a) Develop a 95% CI for the population mean µ if n = 11
(b) Develop a 95% CI for the population mean µ if n = 21
(c) Develop a 95% CI for the population mean µ if n = 31
(d) Develop a 95% CI for the population mean µ if n = 41
(e) What happens to the margin of error and the confidence interval as the number of
sample is increased?
6. The following sample data are from a normal population: 10, 8, 12, 15, 13, 11, 6, 5, 11, 9, 8.
(a) What is the point estimate of the population mean µ?
(b) What is the point estimate of the population standard deviation σ?
(c) With 95% confidence, what is the margin of error for the estimation of the population
mean?
(d) What is the 95% confidence interval for the population mean?
Use the data in ps6 stat1.xlsx and solve the following questions.
7. Worksheet named Rent contains data for the monthly rents of 1 bedroom apartments. Answer the following questions, assuming that rents follows normal distribution.
(a) What is the point estimate for the population mean for the monthly rents of 1BR apartments?
(b) What is the standard error for the above estimate?
(c) What is the sampling distribution of the sample mean?
(d) What is the 97.5 percentile of the sampling distribution?
(e) With 95% confidence, what is the margin of error?
(f) What is the 95% CI of the population mean?
(g) With 90% confidence, what is the margin of error?
(h) What is the 90% CI of the population mean?
2
8. The mean cost of a meal for two in a midrange restaurant in Tokyo is assumed to be $35.
Worksheet named HK meals contains the costs for a sample of 42 recent meals for two
in Hong Kong midrange restaurants. Answer the following questions, assuming that costs
follows normal distribution.
(a) What is the point estimate for the population mean for meals for two in midrange
restaurants in Hong Kong?
(b) What is the standard error for the above estimate?
(c) What is the sampling distribution of the sample mean?
(d) What is the 97.5 percentile of the sampling distribution?
(e) With 95% confidence, what is the margin of error?
(f) What is the 95% CI of the population mean?
(g) How do prices for meals for two in midrange restaurants in Hong Kong compare to
prices for comparable meals in Tokyo restaurants based on the 95% CI?
(h) What is the 99.5 percentile of the sampling distribution?
(i) With 99% confidence, what is the margin of error?
(j) What is the 99% CI of the population mean?
(k) How do prices for meals for two in midrange restaurants in Hong Kong compare to
prices for comparable meals in Tokyo restaurants based on the 99% CI?
3