Q. 37 Refer exercise 35.Suppose we sel... [FREE SOLUTION]

91影视

The Practice of Statistics for AP

David Moore,Daren Starnes,Dan Yates

$Math Studyset 91影视 Explanations$ Math

4th Edition 路 809 pages

ISBN: 9781319113339

Chapter 10: Q. 37 (page 652)

Refer exercise $35 .$ Suppose we select independent SRSs of $25$ men aged $20$ to $34$ and $36$ boys aged $14$ and calculate the sample mean heights $x_{M}$ and $x_{B}$ .
(a) Describe the shape, center, and spread of the sampling distribution of $x_{M} - x_{B}$ .
(b) Find the probability of getting a difference in sample means $x_{M} - x_{B}$ that鈥檚 less than $0$ mg/dl. Show your work.
(c) Should we be surprised if the sample mean cholesterol level for the $14$ -year-old boys exceeds the sample mean cholesterol level for the men? Explain.

Short Answer

Expert verified

(a) The shape, center, and spread of the sampling distribution is normal with $渭_{{\overset{炉}{x}}_{M} - {\overset{炉}{x}}_{B}} = 18$ and $蟽_{{\overset{炉}{x}}_{M} - {\overset{炉}{x}}_{B}} 鈮� 9.6042$

(b) The probability of getting a difference in sample means that鈥檚 less than $0$ mg/dl.

(c) Yes we should be surprised if the sample mean cholesterol level for the $14$ -year-old boys exceeds the sample mean cholesterol level for the men.

Step by step solution

Part (a) Step 1: Given information

Select independent SRSs of $24$ men aged $20$ to $34$ and $36$ boys aged $14$ and calculate the sample mean heights.

Part (a) Step 2: Explanation

From exercise $35,$

Distribution M: Normal with $渭_{M} = 188 a n d 蟽_{M} = 41$

Distribution B: Normal with $渭_{B} = 170 a n d 蟽_{B} = 30$

$n_{M} = 25$

$n_{B} = 36$

The sample mean $x$ is normally distributed with mean $渭$ and standard deviation $蟽 / \sqrt{n}$ (if the population distribution is normal with mean $渭$ and standard deviation $蟽$ )

Properties mean and standard deviation

$渭_{a X + b Y} = a 渭_{X} + b 渭_{Y}$

$蟽_{a X + b Y} = \sqrt{a^{2} 蟽_{X}^{2} + b^{2} 蟽_{Y}^{2}}$

Therefore we get,

localid="1650363687592" $渭_{{\overset{炉}{x}}_{M} - {\overset{炉}{x}}_{B}} = 渭_{{\overset{炉}{x}}_{M}} - 渭_{{\overset{炉}{x}}_{B}} = 188 - 170 = 18$

localid="1650363704750" $蟽_{{\overset{炉}{x}}_{M} - {\overset{炉}{x}}_{B}} = \sqrt{蟽_{{\overset{炉}{x}}_{M}}^{2} + 蟽_{{\overset{炉}{x}}_{B}}^{2}} = \sqrt{\frac{41^{2}}{25} + \frac{30^{2}}{36}} 鈮� 9.6042$

The distribution is normal with $渭_{{\overset{炉}{x}}_{M} - {\overset{炉}{x}}_{B}} = 18$ and $蟽_{{\overset{炉}{x}}_{M} - {\overset{炉}{x}}_{B}} 鈮� 9.6042 .$

Part (b) Step 1: Given information

Select independent SRSs of $24$ men aged $20$ to $34$ and $36$ boys aged $14$ and calculate the sample mean heights.

Part (b) Step 2: Explanation

From the result of part (a),

$渭_{{\overset{炉}{x}}_{M} - {\overset{炉}{x}}_{B}} = 18$ and $蟽_{{\overset{炉}{x}}_{M} - {\overset{炉}{x}}_{B}} 鈮� 9.6042$

The z-value is the difference between the population mean and the standard deviation, divided by the population mean:

localid="1650514978192" $z = \frac{x - 渭}{蟽} = \frac{0 - 18}{9.6042} = - 1.87$

Find the probability using table A

localid="1650363838861" $P ({\overset{炉}{x}}_{M} - {\overset{炉}{x}}_{B} < 0) = P (Z < - 1.87) = 0.0307$

$P ({\overset{炉}{x}}_{M} - {\overset{炉}{x}}_{B} < 0) = 0.0307 .$

Part (c) Step 1: Given information

Select independent SRSs of $24$ men aged $20$ to $34$ and $36$ boys aged $14$ and calculate the sample mean heights.

Part (c) Step 2: Explanation

From part (b)

$P ({\overset{炉}{x}}_{M} - {\overset{炉}{x}}_{B} < 0) = 0.0307 = 3.07 %$

Because the chance is less than , it's improbable that the sample mean for boys will be higher than the sample mean for men, therefore we shouldn't be astonished.