A variable of two populations has a mean of 45 and a standard deviation of 24 for one of the populations and a mean of 45 and a standard deviation of 40 for the other population. Moreover, the variable is normally distributed on each of the two populations. Complete parts (a) through (c). a. For independent samples of size 16 and 25, respectively, determine the mean and standard deviation of x1−x2. 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. Can you conclude that the variable x1−x2 is normally distributed? Explain your answer. Choose the correct answer below. A. Yes, since the variable is normally distributed on each of the two populations, x1−x2 is normally distributed. B. No, since x1−x2 must be greater than or equal to 0, the distribution is right skewed, so cannot be normally distributed. C. Yes, x1−x2 is always normally distributed because of the central limit theorem. D. No, x1−x2 is normally distributed only if the sample sizes are large enough. c. Determine the percentage of all pairs of independent samples of sizes 16 and 25, respectively, from the two populations with the property that the difference x1−x2 between the sample means is between −10 and 10.

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter10: Statistics

Section10.5: Comparing Sets Of Data

Problem 13PPS

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A variable of two populations has a mean of 45 and a standard deviation of 24 for one of the populations and a mean of 45 and a standard deviation of 40 for the other population. Moreover, the variable is normally distributed on each of the two populations. Complete parts (a) through (c).

a. For independent samples of size 16 and 25, respectively, determine the mean and standard deviation of x1−x2.

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. Can you conclude that the variable x1−x2 is normally distributed? Explain your answer. Choose the correct answer below.

A. Yes, since the variable is normally distributed on each of the two populations, x1−x2 is normally distributed.

B. No, since x1−x2 must be greater than or equal to 0, the distribution is right skewed, so cannot be normally distributed.

C. Yes, x1−x2 is always normally distributed because of the central limit theorem.

D. No, x1−x2 is normally distributed only if the sample sizes are large enough.

c. Determine the percentage of all pairs of independent samples of sizes

16 and 25, respectively, from the two populations with the property that the difference x1−x2 between the sample means is between −10 and

10.

_________%

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 + ...

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