LaunchPad for Moore's Introduction to the Practice of Statistics (12 month access)
LaunchPad for Moore's Introduction to the Practice of Statistics (12 month access)
8th Edition
ISBN: 9781464133404
Author: David S. Moore, George P. McCabe, Bruce A. Craig
Publisher: W. H. Freeman
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Chapter 8.2, Problem 47UYK

(a)

To determine

To find: The formula for the mean and standard deviation of p^1and p^2.

(a)

Expert Solution
Check Mark

Answer to Problem 47UYK

Solution: The formula for mean of p^1 is as follows:

μp^1=p1

Similarly, the formula for mean of p^2 is as follows:

μp^2=p2

The formula for standard deviation of p^1 is as follows:

σp^1=p1(1p1)n1

Similarly, the formula for standard deviation of p^2 is as follows:

σp^2=p2(1p2)n2

Explanation of Solution

Calculation: The formula for mean of population proportion (p^) is as follows:

μp^=p

The formula for standard deviation of population proportion (p^) is as follows:

σp^=p(1p)n

So, the mean for p^1 and p^2 will be p1 and p2, and the standard deviation of p^1 and p^2 will be

σp^1=p1(1p1)n1 and σp^2=p2(1p2)n2.

(b)

To determine

To find: The mean of difference D=p^1p^2.

(b)

Expert Solution
Check Mark

Answer to Problem 47UYK

Solution: The mean of D is p1p2_.

Explanation of Solution

Calculation: The mean of the sum of two random variables is the sum of the individual means of those random variables.

The formula for mean of the sum of two random variables X and Y is as follows:

(μX+Y)=μX+μY

where μX and μY represent the individual mean of the random variables X and Y, respectively.

So, the mean of D is calculated as follows:

μD=μp^1p^2=μp^1+μp^2=μp^1μp^2

From part (a), the value of μp^1=p1 and μp^2=p2.

On substituting the values,

μD=μp^1μp^2=p1p2

Hence, the required mean of the difference is p1p2.

(c)

To determine

To find: The variance of D.

(c)

Expert Solution
Check Mark

Answer to Problem 47UYK

Solution: The variance of D is p1(1p1)n1+p2(1p2)n2_.

Explanation of Solution

Calculation: The variance of the sum of two random variables will be equal to the sum of the individual variances and twice the covariance between the variables. The formula for variance of the sum of two random variables X and Y is as follows:

σXY2=σX2+σY22×Cov(X,Y)

where σX2 and σY2 represent the variance of random variables X and Y, respectively, and Cov(X,Y) is the covariance of X and Y. So,

σp^1p^22=σp^12+σp^222×Cov(p^1,p^2)

But the variables are independent, so Cov(p^1,p^2)=0.

σp^1p^22=σp^12+σp^22=p1(1p1)n1+p2(1p2)n2

Hence, the required variance of the difference is p1(1p1)n1+p2(1p2)n2.

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Chapter 8 Solutions

LaunchPad for Moore's Introduction to the Practice of Statistics (12 month access)

Ch. 8.1 - Prob. 11UYKCh. 8.1 - Prob. 12ECh. 8.1 - Prob. 13ECh. 8.1 - Prob. 14ECh. 8.1 - Prob. 15ECh. 8.1 - Prob. 16ECh. 8.1 - Prob. 17ECh. 8.1 - Prob. 18ECh. 8.1 - Prob. 19ECh. 8.1 - Prob. 20ECh. 8.1 - Prob. 21ECh. 8.1 - Prob. 22ECh. 8.1 - Prob. 23ECh. 8.1 - Prob. 24ECh. 8.1 - Prob. 25ECh. 8.1 - Prob. 26ECh. 8.1 - Prob. 27ECh. 8.1 - Prob. 28ECh. 8.1 - Prob. 29ECh. 8.1 - Prob. 30ECh. 8.1 - Prob. 31ECh. 8.1 - Prob. 32ECh. 8.1 - Prob. 33ECh. 8.1 - Prob. 34ECh. 8.1 - Prob. 35ECh. 8.1 - Prob. 37ECh. 8.1 - Prob. 39ECh. 8.1 - Prob. 40ECh. 8.1 - Prob. 41ECh. 8.1 - Prob. 42ECh. 8.1 - Prob. 43ECh. 8.1 - Prob. 44ECh. 8.1 - Prob. 36ECh. 8.1 - Prob. 38ECh. 8.2 - Prob. 45UYKCh. 8.2 - Prob. 46UYKCh. 8.2 - Prob. 47UYKCh. 8.2 - Prob. 48UYKCh. 8.2 - Prob. 49UYKCh. 8.2 - Prob. 50UYKCh. 8.2 - Prob. 51UYKCh. 8.2 - Prob. 52ECh. 8.2 - Prob. 53ECh. 8.2 - Prob. 54ECh. 8.2 - Prob. 55ECh. 8.2 - Prob. 56ECh. 8.2 - Prob. 57ECh. 8.2 - Prob. 58ECh. 8.2 - Prob. 59ECh. 8.2 - Prob. 60ECh. 8.2 - Prob. 61ECh. 8.2 - Prob. 62ECh. 8.2 - Prob. 63ECh. 8.2 - Prob. 64ECh. 8.2 - Prob. 65ECh. 8.2 - Prob. 66ECh. 8.2 - Prob. 67ECh. 8.2 - Prob. 69ECh. 8.2 - Prob. 68ECh. 8.2 - Prob. 70ECh. 8.2 - Prob. 71ECh. 8 - Prob. 72ECh. 8 - Prob. 73ECh. 8 - Prob. 74ECh. 8 - Prob. 75ECh. 8 - Prob. 76ECh. 8 - Prob. 77ECh. 8 - Prob. 94ECh. 8 - Prob. 79ECh. 8 - Prob. 80ECh. 8 - Prob. 81ECh. 8 - Prob. 82ECh. 8 - Prob. 83ECh. 8 - Prob. 84ECh. 8 - Prob. 85ECh. 8 - Prob. 86ECh. 8 - Prob. 87ECh. 8 - Prob. 88ECh. 8 - Prob. 89ECh. 8 - Prob. 90ECh. 8 - Prob. 95ECh. 8 - Prob. 96ECh. 8 - Prob. 97ECh. 8 - Prob. 98ECh. 8 - Prob. 99ECh. 8 - Prob. 92ECh. 8 - Prob. 93ECh. 8 - Prob. 78ECh. 8 - Prob. 100ECh. 8 - Prob. 101E
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