Suppose that the mean value of inter-pupillary distance (the distance between the pupils of the left and right eyes) for adult males is 65 mm and that the population standard deviation is 5 mm. (a) I

(a) We are given that the mean inter-pupillary distance for adult males is 65 mm and the standard deviation is 5 mm. We want to find the probability that the sample average distance for a sample of 25 adult males is between 64 and 66 mm. We know that the distribution of sample means follows a normal distribution with a mean of ? (in this case, 65 mm) and a standard deviation of ?/?n (in this case, 5 mm/?25 = 1 mm). Using the formula for z-scores, we can standardize the sample mean range of 64-66 mm: z = (x? - ?) / (? / ?n) z = (64 - 65) / (1) = -1 z = (66 - 65) / (1) = 1 The probability that the sample average distance will be between 64 and 66 mm can be found using a standard normal distribution table or calculator. P(-1 < z < 1) = 0.6827 Therefore, the probability that the sample average distance for these 25 adult males will be between 64 and 66 mm is 0.6827. (b) We are now asked to find the \approx imate probability that the sample average distance will be between 64 and 66 mm for a sample of 100 adult males, without assuming that inter-pupillary distance is normally distributed. Since we are not assuming a normal distribution, we can use the central limit theorem to \approx imate the distribution of sample means. The central limit theorem states that as sample size increases, the distribution of sample means will approach a normal distribution, regardless of the distribution of the population. So, even though we don't know the distribution of inter-pupillary distance in the population of adult males, we can \approx imate the distribution of sample means as normal with a mean of 65 mm and a standard deviation of 5mm/?100 = 0.5 mm. Using the same formula for z-scores as in part (a), we can standardize the sample mean range of 64-66 mm: z = (64 - 65) / (0.5) = -2 z = (66 - 65) / (0.5) = 2 The probability that the sample average distance will be between 64 and 66 mm can be found using a standard normal distribution table or calculator: P(-2 < z < 2) = 0.9545 Therefore, the \approx imate probability that the sample average distance will be between 64 and 66 mm for a sample of 100 adult males is 0.9545.