11/17/05 Dr. Lunsford
MATH 171 Test 2
Name:__________________ 100 Points Possible
I. Multiple Choice, True/False, Short Answer. (3 points each, 30 points total) 1. If we want to compute a 93% confidence interval for a population mean, then the value of z ∗ (rounded to three decimal places) will be: (a) 1.812
(b) 1.476
(c) 1.645
(d) None of the above
2. The test statistic for a two-sided significance test for a population mean is z = –2.12. What is the corresponding p-value? (a) 0.017
(b) 0.483
(c) 0.034
(d) 0.966
3. The test statistic for a significance test for a population mean is z = –2.12. The hypotheses are H0: μ = 10 versus Ha: μ < 10. What is the corresponding p-value? (a) 0.017
(b) 0.483
(c) 0.034
(d) 0.983
4. An engineer has designed an improved light bulb. The previous design had an average lifetime of 1200 hours. Based on a sample of 2000 of these new bulbs, the average lifetime was found to be 1201 hours. Although the difference is quite small, the effect was statistically significant. What is the best explanation (circle one)? (a) New designs typically have more variability than standard designs. (b) The sample size is very large, so that even a small difference can be detected. (c) The relative improvement in average lifetime is 0.000083, which is much smaller than 0.05. (d) The mean of 1200 is large. 5. Write “true” or “false” next to each statement according to which is correct. _____ The margin of error for a 95% confidence interval for the mean μ increases as the sample size increases. _____The margin of error for a confidence interval for the mean μ, based on a specified sample size n, increases as the confidence level decreases. _____The margin of error for a 95% confidence interval for the mean μ decreases as the population standard deviation decreases. 6. In tests of significance about an unknown parameter, what does the test statistic represent? (circle the best answer) (a) (b) (c) (d)
The value of the unknown parameter under the null hypothesis. The value of the unknown parameter under the alternative hypothesis. A measure of compatibility between the null and alternative hypotheses. A measure of compatibility between the null hypothesis and the data.
I. Multiple Choice, True/False, Short Answer, continued. 7. Suppose a multiple choice test has 20 questions, each with 4 possible answers and a rather unprepared student decides to randomly guess the answers to all 20 questions. Let X be the number of questions the student answers correctly on the test. Then X has a binomial distribution with the parameters n = ______ and p = _____. Problem II. A one-question survey is to be distributed to a random sample of 1000 adults in Ohio. The question asks if they support an increase in the state sales tax from 5% to 6%, with the additional revenue going to education. Let pˆ denote the proportion of adults in the sample who say they support the increase and let X denote the number in the sample who support the increase. Suppose that 40% of all adults in Ohio actually support the increase. (23 points total) (a) If 450 voters in the sample support the increase, then provide the values of the following parameters and statistics: (2 points each, 8 total) p = _____
X = ______
pˆ =_____
n =_____
(b) The random variable X has a binomial distribution with what parameters? (4 points)
(c) On average, how many voters in the sample will support the tax increase? (3 points)
(d) How likely is it that at least 45% of the voters in the random sample will favor the tax increase? Clearly indicate how you compute this value and show all calculator input. Clearly indicate your answer. (5 points)
(e) A certain political group believes that the percent of all adults in Ohio that actually support the tax increase is more than 40%. Do you think the data obtained supports their belief or not (Hint: Consider the probability that you computed in part (d))? Why? (3 points)
Problem III. The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures the motivation, attitude, and study habits of college students. Scores range from 0 to 200 and follow (approximately) a normal distribution with mean 115 and standard deviation 25. You suspect that incoming freshman at your school have a mean μ, which is different from 115 because they are often excited yet anxious about entering college. To test your suspicion, you decide to test the hypotheses H0: μ = 115 versus Ha: μ ≠ 115. Please answer the following: (27 points total) (a) You give the SSHA to 25 incoming freshman and find their mean score to be 116.2. What test will you use to test these hypotheses and why? (3 points)
(b) Find the value of your test statistic. Please show all calculator input. (4 points)
(c) What is the p-value of the test? Draw a graph that shows the test statistic computed in part (b) and the p-value of the test. (3 points)
(d) What is your conclusion in the context of this problem? You should correctly (in a statistical sense) use the word “significant” in your explanation. (4 points)
(e) Use the data in part (a) to compute a 95% confidence interval for μ . Write a complete English sentence giving the meaning of this interval. Be careful to explicitly write the meaning of μ . Be sure to show all calculator input. (4 points)
(f) What is the margin of error for the confidence interval you computed in part (e)? (3 points) (g) What is the smallest sample size of freshman you could use to get a margin of error less than 5 (with 95% confidence)? (4 points)
(h) Explain how the confidence interval you computed in part (e) makes sense with the conclusion of the test in part (d). You should write something like, “At the alpha = XX level we would reject/fail to reject the null hypothesis for the two sided alternative, consequently, the XX% confidence interval for μ computed from this data….” (2 points)
Problem IV. Suppose the heights of American women (in inches) are normally distributed with a mean of 63 inches and a standard deviation of 3 inches. Please answer the following questions (clearly show all calculator input): (20 points total) (a) Dr. L. is 5 feet, 8 inches tall. According to this model she is taller than what percent of the population of American women? (4 points)
(b) How likely is it that the average height of 10 randomly chosen American women is greater than 67 inches? (4 points)
(c) Dr. L. recently visited the remote (and unheard of) tropical island of Amazonia. While there she noticed that she did not seem to be much taller than most of the Amazonian women, and was shorter than many! So, she decided to run a test to determine if the mean height, μ , of Amazonian women was more than that of American women. Please formulate the appropriate hypotheses for this test. (4 points)
(d) To conduct her test, Dr. L., randomly selected 10 Amazonian women and found their average height to be 67 inches. Conduct the appropriate hypothesis test using this data and your hypotheses above. Clearly show the value of your test statistic and the p-value of the test. You will need to make some assumptions in order to conduct this test, clearly indicate what they are! (5 points)
(e) What, if anything, can Dr. L. conclude about the heights of the women on Amazonia? (3 points)
