Suppose that the mean value of interpupillary distance (the distance between thepupils of the left and right eyes) for adult males is 65 mm and that the populationstandard deviation is 5 mm.a. If the distribution of interpupillary distance is normal and a random sample ofn = 25 adult males is to be selected, what is the probability that the sample meandistance x̄ for these 25 will be between 64 and 67 mm? at least 68 mm? (Hint: SeeExamples 8.5 and 8.6.)b. Suppose that a random sample of 100 adult males is to be obtained. Withoutassuming that interpupillary distance is normally distributed, what is theapproximate probability that the sample mean distance will be between 64 and67 mm? at least 68 mm?

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...