Question

(1) To estimate the mean height μμ of male students on your campus, you will measure an SRS of students. Heights of people of the same sex and similar ages are close to Normal. You know from government data that the standard deviation of the heights of young men is about 2.8 inches. Suppose that (unknown to you) the mean height of all male students is 70 inches.

(a) If you choose one student at random, what is the probability that he is between 64 and 69 inches tall?
(b) You measure 49 students. What is the standard deviation of the sampling distribution of their average height x¯¯¯x¯?
(c) What is the probability that the mean height of your sample is between 64 and 69 inches?

(2)Sheila's doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if the glucose level is above 140 miligrams per deciliter (mg/dl) one hour after having a sugary drink. Sheila's measured glucose level one hour after the sugary drink varies according to the Normal distribution with μμ = 125 mg/dl and σσ = 20 mg/dl.

(a) If a single glucose measurement is made, what is the probability that Sheila is diagnosed as having gestational diabetes?
(b) If measurements are made on 7 separate days and the mean result is compared with the criterion 140 mg/dl, what is the probability that Sheila is diagnosed as having gestational diabetes?

(3)  Sheila's measured glucose level one hour after a sugary drink varies according to the Normal distribution with μμ = 130 mg/dl and σσ = 20 mg/dl. What is the level L such that there is probability only 0.2 that the mean glucose level of 4 test results falls above L?

(4)  Andrew plans to retire in 30 years. He plans to invest part of his retirement funds in stocks, so he seeks out information on past returns. He learns that over the entire 20th century, the real (that is, adjusted for inflation) annual returns on U.S. common stocks had mean 8.7% and standard deviation 20.2%. The distribution of annual returns on common stocks is roughly symmetric, so the mean return over even a moderate number of years is close to Normal.

(a) What is the probability (assuming that the past pattern of variation continues) that the mean annual return on common stocks over the next 30 years will exceed 11%?
(b) What is the probability that the mean return will be less than 5%?

Answer 1