Q1. A study has shown that the interpupillary distance for all adult males is normally distributed with a mean of 65 mm and a standard deviation of 5 mm.
(a) What is the probability that the interpupillary distance for a randomly selected male will be between 64 and 67 mm?
(b) What is the probability that the mean interpupillary distance for a simple random sample of 25 males will be between 64 and 67 mm? What is the standard deviation of the sample mean here?
(c) What is the probability that the mean interpupillary distance for a simple random sample of 100 males will be between 64 and 67 mm? What is the standard deviation of the sample mean here?
(d) For each of the previous parts (a) through (c) of this question, indicate whether it is necessary to assume that interpupillary distance is normally distributed in order to arrive at an answer, and explain why.
Q2. For a population of 17 year old boys, the mean subscapular skinfold thickness is 9.7 mm and the standard deviation is 6.0 mm. What is the probability that the mean skinfold thickness for a simple random sample of 40 boys will be (a) greater than 11 mm?, (b) no more than 7.5 mm?, (c) between 7 and 10.5 mm?, (d) within 1 mm of the mean?
Q3. Recall that when the empirical rule was discussed for X, it was argued that if X is exactly or approximately normal, then the probability that X will be within kσX of μ is given by P(μ - kσX ≤ X ≤ μ + kσX) = P(-k ≤ Z ≤ k). Find the value of k such that the probability that X will be within kσX of μ is (a) 0.80, (b) 0.90, (c) 0.95, (d) 0.98, (e) 0.99.