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Confidence Intervals and Introduction to Inference Free Response Practice Problems for AP Statistics

By — McGraw-Hill Professional
Updated on Feb 4, 2011

Review the following concepts if necessary:

Problems

  1. You attend a large university with approximately 15,000 students. You want to construct a 90% confidence interval estimate, within 5%, for the proportion of students who favor outlawing country music. How large a sample do you need?
  2. The local farmers association in Cass County wants to estimate the mean number of bushels of corn produced per acre in the county. A random sample of 13 1-acre plots produced the following results (in number of bushels per acre): 98, 103, 95, 99, 92, 106, 101, 91, 99, 101, 97, 95, 98. Construct a 95% confidence interval for the mean number of bushels per acre in the entire county. The local association has been advertising that the mean yield per acre is 100 bushels. Do you think it is justified in this claim?
  3. Two groups of 40 randomly selected students were selected to be part of a study on drop-out rates. Members of one group were enrolled in a counseling program designed to give them skill needed to succeed in school and the other group received no special counseling. Fifteen of the students who received counseling dropped out of school, and 23 of the students who did not receive counseling dropped out. Construct a 90% confidence interval for the true difference between the drop-out rates of the two groups. Interpret your answer in the context of the problem.
  4. A hotel chain claims that the average stay for its business clients is 5 days. One hotel believes that the true stay may actually be fewer than 5 days. A study conducted by the hotel of 100 randomly selected clients yields a mean of 4.55 days with a standard deviation of 3.1 days. What is the probability of getting a finding as extreme, or more extreme than 4.55, if the true mean is really 5 days? That is, what is the P-value of this finding?
  5. One researcher wants to construct a 99% confidence interval as part of a study. A colleague says such a high level isn't necessary and that a 95% confidence level will suffice. In what ways will these intervals differ?
  6. A 95% confidence interval for the true difference between the mean ages of male and female statistics teachers is constructed based on a sample of 95 males and 62 females. Consider each of the following intervals that might have been constructed:
    1. (–4.5, 3.2)
    2. (2.1, 3.9)
    3. (–5.2, –1.7)
  7. For each of these intervals,

    1. Interpret the interval, and
    2. Describe the conclusion about the difference between the mean ages that might be drawn from the interval.
  8. A 99% confidence interval for a population mean is to be constructed. A sample of size 20 will be used for the study. Assuming that the population from which the sample is drawn is approximately normal, what is the upper critical value needed to construct the interval?
  9. A university is worried that it might not have sufficient housing for its students for the next academic year. It's very expensive to build additional housing, so it is operating under the assumption (hypothesis) that the housing it has is sufficient, and it will spend the money to build additional housing only if it is convinced it is necessary (that is, it rejects its hypothesis).
    1. For the university's assumption, what is the risk involved in making a Type-I error?
    2. For the university's assumption, what is the risk involved in making a Type-II error?
  10. A flu vaccine is being tested for effectiveness. Three hundred fifty randomly selected people are given that vaccine and observed to see if they develop the flu during the flu season. At the end of the season, 55 of the 350 did get the flu. Construct and interpret a 95% confidence interval for the true proportion of people who will get the flu despite getting the vaccine.
  11. A research study gives a 95% confidence interval for the proportion of subjects helped by a new anti-inflammatory drug as (0.56, 0.65).
    1. Interpret this interval in the context of the problem.
    2. What is the meaning of "95%" confidence interval as stated in the problem?
  12. A study was conducted to see if attitudes toward travel have changed over the past year. In the prior year, 25% of American families took at least one vacation away from home. In a random sample of 100 families this year, 29 families took a vacation away from home. What is the P-value of getting a finding this different from expected?
  13. (Note: is computed somewhat differently for a hypothesis test about a population proportion than for constructing a confidence interval to estimate a population proportion. Specifically, for a confidence interval,

    and, for a hypothesis test,

    where p0 is the hypothesized value of p in H0: p = p0 (p0 = 0.25 in this exercise). We do more with this in the next chapter, but you should use

    for this problem.)

  14. A study was conducted to determine if male and female 10th graders differ in performance in mathematics. Twenty-three randomly selected males and 26 randomly selected females were each given a 50-question multiple-choice test as part of the study. The scores were approximately normally distributed. The results of the study were as follows:
  15. Construct a 99% confidence interval for the true difference between the mean score for males and the mean score for females. Does the interval suggest that there is a difference between the true means for males and females?

  16. Under H0 : μ = 35, HA: μ > 35, a decision rule is decided upon that rejects H0 for > 36.5. For the sample, = 0.99. If, in reality, μ = 38, what is the power of the test?
  17. You want to estimate the proportion of Californians who want outlaw cigarette smoking in all public places. Generally speaking, by how much must you increase the sample size to cut the margin of error in half?
  18. The Mathematics Department wants to estimate within five students, and with 95% confidence, how many students will enroll in Statistics next year. They plan to ask a sample of eligible students whether or not they plan to enroll in Statistics. Over the past 5 years, the course has had between 19 and 79 students enrolled. How many students should they sample? (Note: assuming a reasonably symmetric distribution, we can estimate the standard deviation by Range/4.)
  19. A hypothesis test is conducted with α = 0.05 to determine the true difference between the proportion of male and female students enrolling in Statistics (H0: p1p2 = 0). The P-value of is determined to be 0.03. Is this finding statistically significant? Explain what is meant by a statistically significant finding in the context of the problem.
  20. Based on the 2000 census, the population of the United States was about 281.4 million people, and the population of Nevada was about 2 million. We are interested in generating a 95% confidence interval, with a margin of error of 3%, to estimate the proportion of people who will vote in the next presidential election. How much larger a sample will we need to generate this interval for the United States than for the state of Nevada?
  21. Professor Olsen has taught statistics for 41 years and has kept the scores of every test he has ever given. Every test has been worth 100 points. He is interested in the average test score over the years. He doesn't want to put all of the scores (there are thousands of them) into a computer to figure out the exact average so he asks his daughter, Anna, to randomly select 50 of the tests and use those to come up with an estimate of the population average. Anna has been studying statistics at college and decides to create a 98% confidence interval for the true average test score. The mean test score for the 50 random selected tests she selects is 73.5 with a standard deviation of 7.1. What does she tell her father?
  22. A certain type of pen is claimed to operate for a mean of 190 hours. A random sample of 49 pens is tested, and the mean operating time is found to be 188 hours with a standard deviation of 6 hours.
    1. Construct a 95% confidence interval for the true mean operating time of this type of pen. Does the company's claim seem justified?
    2. Describe the steps involved in conducting a hypothesis test, at the 0.05 level of significance, that the true mean differs from 190 hours. Do not actually carry out the complete test, but do state the null and alternative hypotheses.
  23. A young researcher thinks there is a difference between the mean ages at which males and females win Oscars for best actor or actress. The student found the mean age for all best actor winners and all best actress winners and constructed a 95% confidence interval for the mean difference between their ages. Is this an appropriate use of a confidence interval? Why or why not?
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