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STATISTICS 101 - Homework 8 Due Friday, November 9, 2001 • Homework is due by 5:00 PM on the due date in your course instructor’s oﬃce. You can always hand in your homework at the end of lecture. • You may talk with others about the homework problems but please write your solutions up independently. • Please answer homework questions in complete sentences. Make sure to staple the pages of your assignment together. Be sure to indicate your lab section on your paper. • You will have an opportunity to get help on homework during lab. Reading: Oct. 29 Nov. 2 - Nov. 2 - Nov. 9 Section 6.1 Sections 6.2, 6.3 Assignment: 1. Read pages 318-349 and do exercises 6.28, 6.30, and 6.40 in the text. 2. A high school principal knows that over the past 10 years the average ACT score (the ACT is a test used in college admission decisions) of seniors in her school has been 21. This year 36 members of the senior class take an ACT preparation course during the summer prior to their senior year. The average ACT score for this group after the preparation course is X = 22. Assume that the national standard deviation of ACT scores, σ = 4.5 is thought to be appropriate for the population of students taking the ACT preparation course. Let µ represent the mean ACT score that would be observed if all seniors at this school took the preparation course. (a) The principal wonders whether this sample provides evidence that a summer preparation course can raise the mean score of all seniors (the population of interest). Explain why a one-sided alternative hypothesis is appropriate. (b) Formally state the null and alternative hypotheses for addressing the principal’s question. Find the P-value for testing the null hypothesis and state your conclusion. (c) Explain the meaning of the P-value to the principal - the principal doesn’t know much about statistics so you’ll have to make it easy to understand. (d) Suppose that 200 members of the senior class had taken the preparation course and had obtained the same sample average X = 22. Test the null hypothesis again using this larger sample. How does the larger sample size aﬀect the P-value? (Note: It is almost always possible to obtain a very small P-value by obtaining a large enough sample. This does not mean that the result is important in a practical sense.) (e) The data suggest that students who took the ACT preparation course did better than students from this school have done in the past (whether the result is statistically signiﬁcant or not depends on the sample size we look at). The study is not very convincing because it is not well designed. Explain what is wrong with the study and how you might improve on the design. 1 3. In the past, the average number of on-the-job accidents per day at a factory was 3. A new safety program is instituted at the factory and the daily number of accidents is monitored. For a random sample of 36 days under the new program, the average number of accidents is 2.1. Does this sample provide suﬃcient statistical evidence that with the new safety program the true mean number of accidents per day is less than before? Assume that the distribution of the number of accidents is slightly skewed to the right with a standard deviation σ=1.3. (a) Set up a null and alternative hypothesis. Be sure to clearly identify what µ is. (b) Calculate the value of the test statistic and compute the P-value for the test. (c) Do you need the Central Limit Theorem to justify your answer in (b)? Explain brieﬂy. (d) Use the P-value from part (b) to reach a decision and state a conclusion taking into account the context of the problem. 4. Below are the heights (in inches) of a random sample of 16 women from Stat 101. 63 67 69 65 63 67 62 66 67 66 69 67 62 62 70 65 It is assumed that the distribution of heights for women enrolled in Stat 101 is normally distributed with a spread of σ = 2.7. (a) Construct a 95% conﬁdence interval for the mean height of all women enrolled in Stat 101. (b) Based on the conﬁdence interval, does a test of Ho : µ = 66 Ha : µ = 66 reject Ho at the 5% level? Explain your reasoning brieﬂy. (c) Actually perform the test of hypotheses given in (b) by calculating a value of the z-test statistic, a P-value, and comparing that P-value to the 5% level. Do you arrive at the same conclusion as in (b)? 2