A variable of two populations has a mean of 30 and a standard deviation of 24 for one of the populations and a mean of 30 and a standard deviation of 40 for the other population. Complete parts​ (a) through​ (c).   a. For independent samples of size 9 and 4​, ​respectively, find the mean and standard deviation of x1−x2. (Assume that the sampling is done with replacement or that the population is large​ enough.)   The mean of x1−x2 is _________ ​(Type an integer or a decimal. Do not​ round.)   The standard deviation of x1−x2 is _________ ​(Round to four decimal places as​ needed.)   b. Must the variable under consideration be normally distributed on each of the two populations for you to answer part​ (a)? Choose the correct answer below.   A. No, the variable must be approximately normally distributed on at least one of the two populations for the formulas for the mean and standard deviation of x1−x2to​ hold, as long as the sampling is done with replacement.   B. No, the variable does not need to be normally distributed for the formulas for the mean and standard deviation of x1−x2 to hold as long as the sample sizes are large​ enough, as long as the sampling is done with replacement.   C. Yes, the variable must be approximately normally distributed on each of the two populations for the formulas for the mean and standard deviation of x1−x2 to hold.   D. ​No, the formulas for the mean and standard deviation of x1−x2 hold regardless of the distributions of the variable on the two​ populations, as long as the sampling is done with replacement or that the population is large enough.   c. Can you conclude that the variable x1−x2 is normally​ distributed? Explain your answer.   Choose the correct answer below.   A. No, since x1−x2 must be greater than or equal to​ 0, the distribution is right​ skewed, so cannot be normally distributed.   B. ​Yes, x1−x2 is always normally distributed because of the central limit theorem.   C. No, x1−x2 is normally distributed only if x is normally distributed on each of the two populations or if the sample sizes are large enough.   D. Yes, x1−x2 is always normally distributed because it is calculated using parameters

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 + ...