1. A 2004 survey of randomly sampled Utah residents found that 10.5% (0.105) of them smoked regularly.
(a) Suppose you survey 100 randomly sampled Utah residents. Use the Central Limit Theorem to sketch and label the sampling distribution of the sample proportion of Utah smokers Calculate the z-score for the sample proportion ˆp = 0.125.
(b) Calculate the probability that a sample proportion of smokers from this population exceeds 12.5%.
2. A statistics professor has asked his students to flip coins over the years. He has kept track of how many flips land heads and how many land tails. Combining the results of his students over many years, he has formed a 95% confidence interval for the long-run population proportion of heads to be (0.501, 0.513).
(a) Why is this interval so narrow?
(b) Suppose he were to conduct a hypothesis test of whether the long-run population proportion of heads difers from one-half. Based on this interval (do not conduct the test), would he reject the null hypothesis at the α= .05 significance level? Explain briefly (no more than one sentence).
(c) Does the interval provide strong evidence that the long-run population proportion of heads is much different from one-half? Explain briefly.
3. A student wants to assess whether her dog Muffin tends to chase her blue ball more often than she chases her red ball. The student rolls both a blue ball and a red ball at the same time and observes which ball Muffin chooses to chase. Repeating this process a total of 96 times, the student finds that Muffin chased the blue ball 52 times and the red ball 44 times.
(a) What is the variable in this study?
(b) State the appropriate null and alternative hypotheses, in words and in symbols.
(c) Calculate the test statistic and p-value.
(d) Would you reject the null hypothesis at the α = .10 significance level? Explain.
(e) Write a one-sentence conclusion to the student, summarizing what the data reveal about whether her dog Muffin tends to chase her blue ball more often than her red ball. Include an explanation of what the p-value means in the context of this study.
4. Students in an introductory statistics class were asked to report the age of their mothers when they were born. Summary statistics include: n = 28 students, ¯x = 29.643 years, s = 4.564 years. The ages were normally distributed.
(a) Calculate the standard error of this sample mean.
(b) Determine and interpret a 90% confidence interval for the mothers mean age (at student's birth) in the population of all students at this university.
(c) How would a 99% confidence interval compare to the 90% interval in terms of its midpoint and half-width?
(d) Would you expect 90% of the ages in the sample to be within the 90% confidence interval? Explain why or why not.
(e) Even if the distribution of mothers ages were somewhat skewed, would this confidence interval procedure still be valid with these data? Explain why or why not.
5. Reconsider the previous problem about the age of a college students mother when the student was born. Now suppose you want to conduct a significance test of whether the sample data provide strong evidence that the population mean of mothers age is less than 30 years.
(a) State the appropriate null and alternative hypotheses in symbols.
(b) Calculate the test statistic.
(c) Determine (as accurately as possible) the p-value of the test.
(d) State your test decision at the α = .10 significance level.
(e) Summarize your conclusion in context, and explain the reasoning process by which you reached this conclusion.
6. Suppose you want to estimate the proportion of people in your community who would choose Thin Mints as their favorite Girl Scout cookie. Also suppose previous studies have shown this proportion to be around .3.
(a) Determine how many people you would have to sample in order to estimate this proportion to within .025 with 90% confidence.
If your boss says that this number (your answer to part a) is too large to be practical, you might respond that a smaller sample size would suffice if either the margin-of-error (.025) or the confidence level (90%) were changed.
(b) In which direction would the margin-of-error need to change in order for a smaller sample size to suffice?
! Larger margin-of-error
! Smaller margin-of-error
(c) In which direction would the confidence level need to change in order for a smaller sample size to suffice?
! Larger confidence level
! Smaller confidence level
7. The following questions all focus on definitions and concepts. You can easily answer each of these questions without a calculator, since only basic arithmetic is involved. Read each question carefully and focus on what is being asked.
(a) Suppose Alejandro studies a random sample of data on a categorical variable and calculates a 95% confidence interval for the population proportion to be (0.546, 0.674). Determine what the sample proportion must have been, and explain why.
(b) Suppose Brad and Carly plan to collect separate random samples, with Brad using a sample size of 500 and Carly using a sample size of 1500. If Brad plans to construct a 99% confidence interval for the population proportion and Carly plans to construct a 90% confidence interval, who is more likely to obtain an interval that succeeds in capturing the population proportion? Explain.
8. Suppose a school has 20 classes: 16 with 25 students in each, three with 100 students in each, and one with 300 students for a total of 1000 students.
(a) What is the average class size?
(b) Select one of the 1000 students at random and let the random variable X equal the size of that student's class. Define the pmf of X.
(c) What is the mean of X.
9. Astronomers estimate that as many as 100 billion stars in the Milky Way galaxy may be encircled by planets. Let p denote the probability that any such solar system contains intelligent life.
How small can p be and still give a 50-50 chance that there is intelligent life in at least one other solar system in our galaxy? (Hint: Use the probability distribution designed to handle problems where the probability is small and n is large.)
A statistically-minded fraternity man keeps records on how many women he must ask before one of them says she will be his date for a Saturday football game. His school plays five home games and his five acceptances come on the third, sixth, fourth, second, and ninth women he asks. Assume that the probability, p, that any woman he asks will accept his invitation is constant from woman to woman. Estimate p.
10. Consider the population {1, 2, 5, 6, 10, 12}.
(a) Find the sampling distribution of the medians for samples of size 3 without replacement. Display this distribution in the following table
Median Frequency
2
5
6
10
(b) Compare the median of the population to the mean of the medians found in part (a).