Please type your answers in complete sentences and in the context of the question. To obtain full credit for your answer, show the major steps in your work or provide an explanation of how you obtained your answer.
1.In the Physicians Health Study over 22,000 male physicians were randomly assigned to two groups. One group took an ordinary aspirin tablet every other day while the other group took a placebo. The study was double-blinded. After a number of years, the number of heart attacks was recorded for each of the subjects. For the aspirin group the number of attacks per 1,000 doctors was 9.42. The corresponding rate for the placebo group was 17.13.
(a)Is this an experiment or an observational study? Explain yourchoice.[5 pts]
(b)What are the explanatory and the response variables in this example? Indicate whether each is qualitative or quantitative.[7 pts]
(c)A critic of the study acknowledged that the aspirin group had a substantially lower heart attack rate than the placebo group, but suggested that the difference may be explained by the fact that the aspirin group may have been significantly healthier to begin with than the placebo group.Is the critic correct? Please explain.[8pts]
2.The probability that a particular screening test for breast cancer will be positive when, in fact, the patient does, in fact, have breast cancer is 0.9. You may assume that the outcome of any one test is independent of the outcome of any other. One morning five women, each of whom does have breast cancer, take the test. What is the probability that:
(a)All five of the tests are positive?[6 pts]
(b)The first test isnegative and the others positive?[6 pts]
(c) at least one of the fivetests is negative?[8 pts]
3. At the Brigham and Women’s Hospital approximately 25% of all births are by caesarian section. You plan to record details of the next 100 births at that hospital.
(a) What is the sampling distribution of the proportion of those 100 births that are by caesarian section?[9 pts]
(b) What is the probability that more than 30% of the births are by caesarian section? Draw a picture with appropriate labeling that shows your answer.[11 pts]
4. A random sample of 540 second semester seniors from Boston-area colleges was selected last May by a colleague of mine at UMass Boston. Those selected were asked various questions about the courses they had taken. A total of 212 of the students had taken a statistics course.
(a) Compute the proportion in the sample that had taken a statistics course.[3 pts]
(b) Obtain a 90% confidence interval for the fraction of all second-semester seniors in the Boston area who had taken a statistics course.[5 pts]
(c) Provide the ‘correct’ interpretation of your interval.[6 pts]
(d) My colleague at UMass Boston offers to use the data in this study to provide you with a 90% confidence interval for the fraction of allfemalesecond-semester seniors in the Boston area who had taken a statistics course. Explain whether this interval will have a smaller or a larger margin of error than the one you computed in part (b) above. [6 pts]
5. Recently updated census figures indicate that 38% of adults in Massachusetts have a college degree. A researcher believes that the corresponding percentage for state residents that are Catholic will be higher. She selects a random sample of adults resident in the state. A total of 320 are Catholic. Of these 126 are college graduates.
(a) State the null and the alternative hypotheses. Be sure to define p in this case.[6 pts]
(b) Compute the sample proportion and hence the p-value.[7 pts]
(c) Provide a conclusion (in context please).[7 pts]
6. As part of your doctoral thesis you plan to select a random sample of pre-1980 Simmons alumnae. You are interested in estimating the percentage of pre-1980 alumnae that went on to get a graduate degree. You advisor suggests that you select a sample large enough so that you can be 90% confident of a margin of error of 4%. You know that your sample will cost you roughly $2:50 per sample member.
Compute a conservatively large sample size.[8 pts]
Please type your answers in complete sentences and in the context of the question. To obtain full credit for your answer, show the major steps in your work or provide an explanation of how you obtained your answer.
1.In the Physicians Health Study over 22,000 male physicians were randomly assigned to two groups. One group took an ordinary aspirin tablet every other day while the other group took a placebo. The study was double-blinded. After a number of years, the number of heart attacks was recorded for each of the subjects. For the aspirin group the number of attacks per 1,000 doctors was 9.42. The corresponding rate for the placebo group was 17.13.
(a)Is this an experiment or an observational study? Explain yourchoice.[5 pts]
(b)What are the explanatory and the response variables in this example? Indicate whether each is qualitative or quantitative.[7 pts]
(c)A critic of the study acknowledged that the aspirin group had a substantially lower heart attack rate than the placebo group, but suggested that the difference may be explained by the fact that the aspirin group may have been significantly healthier to begin with than the placebo group.Is the critic correct? Please explain.[8pts]
2.The probability that a particular screening test for breast cancer will be positive when, in fact, the patient does, in fact, have breast cancer is 0.9. You may assume that the outcome of any one test is independent of the outcome of any other. One morning five women, each of whom does have breast cancer, take the test. What is the probability that:
(a)All five of the tests are positive?[6 pts]
(b)The first test isnegative and the others positive?[6 pts]
(c) at least one of the fivetests is negative?[8 pts]
3. At the Brigham and Women’s Hospital approximately 25% of all births are by caesarian section. You plan to record details of the next 100 births at that hospital.
(a) What is the sampling distribution of the proportion of those 100 births that are by caesarian section?[9 pts]
(b) What is the probability that more than 30% of the births are by caesarian section? Draw a picture with appropriate labeling that shows your answer.[11 pts]
4. A random sample of 540 second semester seniors from Boston-area colleges was selected last May by a colleague of mine at UMass Boston. Those selected were asked various questions about the courses they had taken. A total of 212 of the students had taken a statistics course.
(a) Compute the proportion in the sample that had taken a statistics course.[3 pts]
(b) Obtain a 90% confidence interval for the fraction of all second-semester seniors in the Boston area who had taken a statistics course.[5 pts]
(c) Provide the ‘correct’ interpretation of your interval.[6 pts]
(d) My colleague at UMass Boston offers to use the data in this study to provide you with a 90% confidence interval for the fraction of allfemalesecond-semester seniors in the Boston area who had taken a statistics course. Explain whether this interval will have a smaller or a larger margin of error than the one you computed in part (b) above. [6 pts]
5. Recently updated census figures indicate that 38% of adults in Massachusetts have a college degree. A researcher believes that the corresponding percentage for state residents that are Catholic will be higher. She selects a random sample of adults resident in the state. A total of 320 are Catholic. Of these 126 are college graduates.
(a) State the null and the alternative hypotheses. Be sure to define p in this case.[6 pts]
(b) Compute the sample proportion and hence the p-value.[7 pts]
(c) Provide a conclusion (in context please).[7 pts]
6. As part of your doctoral thesis you plan to select a random sample of pre-1980 Simmons alumnae. You are interested in estimating the percentage of pre-1980 alumnae that went on to get a graduate degree. You advisor suggests that you select a sample large enough so that you can be 90% confident of a margin of error of 4%. You know that your sample will cost you roughly $2:50 per sample member.
Compute a conservatively large sample size.[8 pts]