Question 1 - Multiple Choices Questions

1a. An unbiased estimate

A. is equal to the true parameter

B. has the smallest variance of all possible estimates

C. is never an efficient estimate

D. has mean equal to the true parameter

E. is always a maximum likelihood estimate

1b. An experimenter reports that, on the basis of a sample of size 10, he calculates the 95% confidence limits for mean height to be 66 and 74 inches. Assuming his calculations are correct, this result, is to be interpreted as meaning

A. there is a 95% probability that the population mean height lies between 66 and 74 inches.

B. we have 95% confidence that a person's height lies between 66 and 74 inches.

C. we have 95% confidence that the population mean height lies between 66 and 74 inches.

D. 95% of the population has a height between 66 and 74 inches

E. none of the above

1c. A 99% confidence interval for a mean

A. is wider than a 95% confidence interval

B. is narrower than a 95% confidence interval

C. includes the mean with 99% probability

D. excludes the mean with 99% probability

E. is obtained as the sample average plus two standard deviations

1d. A 95% confidence interval implies that

A. the t-test gives correct intervals 95% of the time

B. if we repeatedly select representative samples and construct such interval estimate, 95 out of 100 of the intervals would be expected to bracket the true mean.

C. the hypothesis will be false in 95 out of 100 such intervals

D. the probability that the interval is false is 95%

E. there is a 95% probability that the underlying distribution is normal

1e. In a sample of 100 normal women between the ages of 25 and 29 years, systolic blood pressure was found to follow a normal distribution. If the sample mean pressure was 120 mmHg and the sample standard deviation was 10 mmHg, what interval of blood pressures would represent an approximate 95% confidence interval for the true mean?

A. 118 to 122 mmHg

B. 100 to 140 mmHg

C. 119 to 121 mmHg

D. 110 to 130 mmHg

E. 90 to 150 mmHg

Question 2 -

A randomized comparative experiment studied the effect of diet on blood pressure. Researchers divided 54 healthy males at random into two groups. One group received a calcium supplement, and the other group received a placebo. At the beginning of the study, the researchers measured many variables on the subjects. The average seated systolic blood pressure of the 27 members of the placebo group was reported to be x^- = 114.9 mmHg with a sample standard deviation s = 9.3 mmHg.

2a. Calculate a 95% confidence interval for the mean blood pressure of the population from which the subjects were recruited. You may assume normality.

2b. In 1 sentence, interpret the confidence interval you obtained in 2a.

Question 3 -

A vaccine manufacturer analyzes a batch of product to check its antigen titer. Immunologic analyses are imperfect, and repeated measurements on the same batch are expected to yield slightly different titers. Assume titer measurements vary according to a Normal distribution with mean u and known σ = 0.070. Three measurements of the manufacturer's batch demonstrate titers of 7.40, 7.36, and 7.45.

3a. Calculate a 95% confidence interval estimate of the true antigen titer of the manufacturer's batch.

3b. In 1 sentence, interpret the confidence interval you obtained in 3a.

Question 4 -  

A 95% confidence interval for the mean of a Normal distribution, in the setting of known variance, (was calculated using the usual formula x^-±[z_1-α/2](σ/√n). The resulting confidence interval was 5.7 to 6.5.

4a. What is the value of the sample mean used to calculate the confidence interval?

4b. What is the value of the standard error of the estimate?

4c. Calculate a 99% confidence interval for μ.

Question 5 -

A health management firm wants to know which of two hospitals (A or B) has the higher rating among former maternity patients with respect to such areas as: service, food, cleanliness, staff friendliness, and the quality of the doctors. A questionnaire is devised and the responses for each former patient are summarized to form a single rating. The following are the data.

5a. Under the assumption that the data represent two random samples from independent normal distributions with unknown variances, and under the assumption that the two unknown variances are equal, calculate a 95% confidence interval estimate of the difference in mean rating (Hospital A -Hospital B)

5b. In 1 sentence, interpret the confidence interval you obtained in 5a.