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# Briefly describe the relationship between confidence intervals and p-values during one-sample and 2-sample hypothesis testing.

INSTRUCTIONS TO CANDIDATES
###### ANSWER ALL QUESTIONS

Instructions: You must perform all calculations by hand. You should write each step of your calculations on your answer sheets and include the measurement units with your quantitative answers. MATLAB is not necessary. Submit your answer sheets in the Google Classroom by the deadline. Total points: 20

Question 1 (1 pt.): Explain the analogy between Type I α and Type II β errors in hypothesis testing and the false positive and false negative results that occur in diagnostic testing.

Question 2 (1.5 pts.): Briefly describe the relationship between confidence intervals and p-values during one-sample and 2-sample hypothesis testing.

Question 3 (1 pt.): When would you use the one-way analysis of variance? What two measures of variation are being compared, and how are these measures used for hypothesis testing?

Question 4 (1.5 pts.): What factors affect the p-value of a test and how? What factors affect the power of a test and how?

Question 5 (1 pt.): Suppose that you are designing a randomized clinical trial to assess the efficacy of a novel cancer treatment regimen that you believe will reduce the size of a tumor much more than the conventional treatment. You would like at least an 80% power to detect a difference of tumor size reduction by 1 cm between the new treatment group and the conventional treatment group. What parameter does 1 cm represent in your sample size formula?

Question 6 (2 pts.): E. canis infection is a tick-borne disease of dogs that is sometimes contracted by humans. Among infected humans, the distribution of white blood cell counts has unknown mean µ and standard deviation σ. In the general population, the mean white blood cell count is 7250/mm3. It is believed that a person infected with E. canis has a lower white blood cell count than an uninfected person.

(a) What type of test must be performed to determine whether the white blood cell counts of infected people are significantly lower than those of the general population (i.e., two-sided, one-sided right-tail, or one-sided left-tail test)? State the null and alternative hypotheses for the test.

(b) In a sample of 15 infected people, the mean white blood cell count is 4,767/mm3 and the standard deviation is 3,204/mm3. Conduct the test at the α = 0.05 level. State the p-value and provide your conclusion.

Question 7 (3 pts): Body mass index (BMI) is calculated by dividing a person's weight by the square of his or her height. It is a measure of the extent to which the individual is overweight. For the population of middle-aged men who later develop diabetes mellitus, the distribution of baseline BMIs is approximately normal with an unknown mean µ and standard deviation σ. A sample of 58 men

selected from this group has mean, 88̅ = 25.0 kg/m2 and standard deviation, s = 2.7 kg/m2. (a) Based on the statistics of the sample, construct a 95% confidence interval for the population mean of baseline BMIs.

(b) At the 5% significance level, test whether the mean baseline BMI for the population of middle aged men who do develop diabetes is equal to 24.0 kg/m2, which is the mean for the population of men who do not. What is the p-value of the test? What can be interpreted from the test results?

(c) Based on the 95% confidence interval, would you have expected to reject or not to reject the

null hypothesis? Why?

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Question 8 (2 pts): In an investigation of pregnancy-induced hypertension, one group of women with this disorder was treated with low-dose aspirin, and a second group was given a placebo. A sample consisting of 23 women who received aspirin has a mean arterial blood pressure of 111 mmHg and   a standard deviation of 8 mmHg; a sample of 24 women who were given the placebo has a mean blood pressure of 109 mmHg and a standard deviation of 8 mmHg. The data were assumed to come from populations that were normally-distributed and have equal variances.

(a) At the 5% level of significance, test the null hypothesis that the two populations of women have the same mean arterial blood pressure. State the p-value and your conclusion.

(b) Construct a 95% confidence interval for the true difference in population means. Does this interval contain the value 0? Based on the 95% confidence interval, would you have expected to reject or not to reject the null hypothesis? Justify your answer.

Question 9 (4 pts): Data from the Framingham Study allow us to compare the distributions of initial serum cholesterol levels for two populations of males: those who go on to develop coronary heart disease and those who do not. The mean serum cholesterol level of the population of men who do NOT develop heart disease is µ=219 mg/100 ml and the standard deviation is σ=41 mg/100 ml. Suppose, however, that you do not know the true population mean; instead, you hypothesize that µ is equal to 244 mg/100 ml, which is the mean initial serum cholesterol level of men who go on to develop the disease. Because it is believed that the mean serum cholesterol level for the men who do not develop heart disease cannot be higher than the mean level for men who do, a one-sided test conducted at the 5% level of significance is appropriate.

(a) What is the probability of making a Type I error?

(b) If a sample of size 31 is selected from the population of men who do NOT develop coronary heart disease (sample mean serum cholesterol level of 219 mg/100 ml), what is the probability of making a Type II error?

(c) What is the power of the test? How could you increase the power?

(d) You would like to test the null hypothesis H0: µ ≥ 244 mg/100 ml against the alternative hypothesis HA: µ < 244 mg/100 ml at the α = 0.05 level of significance. If the true population mean is as low as 219 mg/100 ml and you are willing to risk a 10% chance of failing to reject a false null hypothesis. How large a sample would be required?

Question 10 (3 pts): Investigators at Apollo Hospitals Ahmedabad are interested in examining the effects of the transition from fetal to postnatal circulation among premature infants. For each of 10 healthy newborns, the respiratory rate is measured at two different times – once when the infant is less than 15 days old, and again, when the infant is more than 25 days old. The table below shows the respiratory rates (breaths/minute) of these infants. Suppose that the data at the two time points came from populations that are not normally-distributed.

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