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Questions 1 (15pt)

A study was performed in Basel, Switzerland, on the relationship between the concentration of plasma antioxidant vitamins and cancer risk. The data set has π = 20 subjects and shows the plasma vitamin-A concentration for stomach-cancer cases has a mean π₯Μ = 2.65 and a variance π 2 = 0.28. Assume that the mean plasma vitamin-A concentration among controls (i.e. individuals without stomach-cancer) is known to be 2.88.

(a) (4pt) We wish to test whether the mean concentration is the same for stomach-cancer cases and controls. Specify the null and alternative hypotheses in this case and comment on what test can be used. Be sure to define any variables that you use.

(b) (6pt) Perform the test in (a) at a significance level of πΌ = 0.05, and report a p-value (two-sided). Clearly state your conclusions to the problem of interest.

(c) (5pt) How many stomach-cancer cases are needed to achieve 80% power if the mean plasma vitamin- A concentration among controls is known without error, the true difference in mean concentration is

0.20 πmol/L, and a two-sided test is used with πΌ?

Questions 2 (15pt)

Osteoporosis is an important cause of morbidity in middle-aged and elderly women. Several drugs are currently used to prevent fractures in postmenopausal women. Suppose the incidence rate of fractures over a 4-year period is known to be 4% among untreated postmenopausal women with no previous fractures. A pilot study conducted among 100 women without previous fractures aims to determine whether a new drug affects the rate of fractures. The results of this 4-year trial can be found in the file osteoporosis.csv on the course website. Use that information to answer the following questions.

(a) (2pt) What are the null and two-sided alternative hypotheses in this case? Be sure to define any variables that you use.

(b) (6pt) Given the significant level of πΌ = 0.05, is there a significant difference between the fracture rate in individual treated women and the fracture rate in the untreated women? Answer this question using both exact test method and normal approximation test method.

(c) (3pt) Based on previous natural history studies, 1.5% of the women would be expected to develop abdominal pain during the trial. Using the normal approximation method, provide a 95% CI for the rate of abdominal pain among the active treated women. Interpret the results compared with previous natural history studies.

(d) (4pt) Suppose the new drug yields a fracture rate of 2% over a 4-year period. How many subjects need to be studied to have an 90% chance of detecting a significant difference between the incidence rate of fractures in treated women and the incidence rate of fractures in untreated women?

Question 3 (15pt)

The distribution of diastolic blood pressure (DBP) for the population of female diabetics between the ages of 30 and 34 has an unknown mean π and standard deviation π = 9.2 mm Hg. It may be useful to physicians to know whether the mean of this population is equal to the mean DBP = 74.4 mm Hg of the general population of females in this age group.

(a) (3pt) Set up the hypotheses to compare the mean DBP of this group of diabetic females to its corresponding general population.

(b) (6pt) A sample of ten diabetic women is selected; their mean diastolic blood pressure is π₯Μ = 78.6 mm Hg. Conduct a statistical test at the πΌ = 0.05 level of significance. What is the p-value of the test? Explain the meaning of πΌ and the meaning of the p-value that you obtained in the context of this study.

(c) (6pt) If we do not want to assume that the standard deviation is known and we obtain a sample standard deviation of 9.6 mm Hg from the same sample of 10 diabetic women in (b), repeat the statistical test and answer the question in (b) again, and also construct a 95% confidence interval for the true mean DBP. Carefully interpret this confidence interval.

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