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Narrative: Grades Suppose a class of 100 students took their statistics final and their grades are shown in the table below. A B C D F 25 28 34 10 3 1. {Grades narrative} Choose one student at random. What is the probability that he/she received a B or a C? 2. {Grades narrative} What is the probability that a student selected at random passed the final (where a D is considered to be a passing grade)? Narrative: Shaking hands Suppose the chances of picking up a cold from someone by shaking hands with them is .01 (assuming you don’t know whether they have a cold or not), and that each encounter you have is independent of another. 3. {Shaking hands narrative}. Suppose you shake hands with 3 people in a given day. What is the probability that you don’t pick up a cold from any of these people? Narrative: Politics Suppose a population contains 60% Republicans and 40% Democrats. 4. {Politics narrative} Suppose you take a random sample of 10 people from this population. Are you certain that you would get 6 Republicans and 4 Democrats in your sample? Explain your answer. 5. {Politics narrative} Is it possible to get a random sample that does not represent the population well, in terms of Democrats and Republicans? Explain your answer. 6. Explain (in words that a non-statistics student would understand) what is meant by a ‘95% confidence interval.’ 7. In testing hypotheses, what should we do if the consequences of rejecting the null hypothesis are very serious? 8. Large samples make it easier to detect real relationships or differences in the population than small samples. Explain how this is taken into account in the formula for the test statistic for testing a population mean. 9. Suppose you are told that “a relationship has been found between two variables” or that “a difference has been found between two groups.” What does the researcher really mean, in terms of a hypothesis test? 10. An engineer designs an improved light bulb. The previous design had an average lifetime of 1,200 hours. The new bulb had a lifetime of 1,200.2 hours, using a sample of 40,000 bulbs. Although the difference is quite small, the effect was statistically significant. What is the most likely explanation? 11. Which of the following questions does a test of significance deal with? (i) Is the difference due to chance? (ii) Is the difference important? (iii) What does the difference prove? (iv) Was the experiment properly designed? Explain briefly. 12. True or False and explain briefly. (a) A difference which is highly significant can still be due to chance. (b) A statistically significant number is big and important. (c) A p-value of 0.047 means something quite different from a p-value of 0.052 13. Suppose you are conducting a one-sided hypothesis test for a difference in two means (using 2 independent large samples), and you have a test statistic of z=1.75, which is in the direction of the alternative hypothesis. How extreme is this value and what is your conclusion for this test? 14. Suppose you suspect that a majority of the people in your neighborhood would like to start a neighborhood watch program, but you want to conduct a hypothesis test to find out. What are your null and alternative hypotheses? 15. (a) Other things being equal, which of the following P-values is best for the null hypothesis? Explain briefly. 0.001 0.03 0.17 0.32 (b) Repeat, for the alternative hypothesis. 16. In which of the following situations does the rule for sample means not apply? a. A pollster takes a random sample of 1,000 Americans and asks them to give their opinion of the President on a scale from 1 (completely disapprove) to 100 (completely approve). He is interested in the average rating. b. You take a random sample of 20 students’ scores from the ACT exam and record the average score. Assume ACT scores are bell-shaped. c. A sports fan takes a random sample of 20 NBA players and records their salaries. He wants to estimate the average salary for the entire NBA. d. The rule for sample means does not apply in any of these situations. 17. Which of the following conclusions do you draw if the p-value is smaller than the level of significance? a. Reject the null hypothesis. b. Accept the alternative hypothesis. c. The true population value is significantly different from the value in the null hypothesis. d. All of the above. 18. Suppose the p-value for your hypothesis test for a difference of two means was .001. Which of the following is an appropriate conclusion? a. You reject the null hypothesis. b. If there really was no difference in the population means, we would see results as extreme as these only .1% of the time. c. You conclude that there is a statistically significant difference between the population means. d. All of the above. 19. Which of the following statements is false? a. If the sample size is large enough, almost any null hypothesis can be rejected. b. There is almost always a slight relationship between two variables, or a difference between two groups, and if you collect enough data, you will find it. c. Larger samples sizes always produce more meaningful results than small sample sizes. d. None of the above statements are false. 20. Suppose you are examining the difference between two means, and you conduct a two-sided hypothesis test and your friend constructs a confidence interval. Assuming that both reveal that the null hypothesis should be rejected, what can the confidence interval tell you that the hypothesis test cannot? a. The magnitude of the difference in the means. b. The direction of the difference in the means. c. How much variability there was in the observed results. d. All of the above.