Q 8.19RP. Millionaires. Dr. Thomas Stanley... [FREE SOLUTION]

Chapter 8: Q 8.19RP. (page 347)

Millionaires. Dr. Thomas Stanley of Georgia State University has surveyed millionaires since $1973$ Among other information, Stanley obtains estimates for the mean age, $渭$ , of all U.S. millionaires. Suppose that $36$ randomly selected U.S. millionaires are the following ages, in years.
Determine a $95 %$ confidence interval for the mean age $渭$ of all U.S. millionaires. Assume that the standard deviation of ages of all U.S. millionaires is $13.0$ years. (Note: The mean of the data is $58.53$ years)

Short Answer

Expert verified

We are $95 %$ confident that mean age, $渭$ of all us. Millionaires, lies some where between $54.28$ years and $62.78$ years

Step by step solution

Given information

The figure is given:

31	45	79	64	48	38	39	68	52
59	68	79	42	79	53	74	66	66
71	61	52	47	39	54	67	55	71
77	64	60	75	42	69	48	57	48

Concept

The formula used: $(\overset{炉}{x} - z_{伪 / 2} \frac{蟽}{\sqrt{n}}, \overset{炉}{x} + z_{伪 / 2} \frac{蟽}{\sqrt{n}})$

Calculation

Let the population mean age be $渭$

Population S.D $蟽 = 13$ years

Sample size $n = 36$

Sample mean $\overset{炉}{x} = 58.53$

$\overset{炉}{x}$ is roughly normally distributed because the sample is huge.

The $100 (1 - 伪) % Cl$ of $渭$ is calculated using the $z -$ interval technique.

$(\overset{炉}{x} - z_{伪 / 2} \frac{蟽}{\sqrt{n}}, \overset{炉}{x} + z_{伪 / 2} \frac{蟽}{\sqrt{n}})$

Here confidence level $= 95 % = 100 脳 0.95 %$

鈭� 1 - 伪 = 0.95 鈬� 伪 = 1 - 0.95 鈬� 伪 = 0.05 鈬� 伪 / 2 = 0.025 鈭� z_{伪 / 2} = z^{125} = 1.96

Calculation

$鈭� 95 % Cl of 渭 = (\overset{炉}{x} - Z_{0.025} \frac{蟽}{\sqrt{n}}, \overset{炉}{x} + z_{0.025} \frac{蟽}{\sqrt{n}}) = (58.53 - 1.96 脳 \frac{13}{\sqrt{36}}, 58.53 + 1.96 脳 \frac{13}{\sqrt{36}}) = (58.53 - 4.25, 58.53 + 4.25) = (54.28, 62.78)$

i.e., we are $95 %$ certain that the average age of all of us is $渭$ Millionaires are born between the ages of $54.28$ and $62.78$

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Step by step solution

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31	45	79	64	48	38	39	68	52
59	68	79	42	79	53	74	66	66
71	61	52	47	39	54	67	55	71
77	64	60	75	42	69	48	57	48

31	45	79	64	48	38	39	68	52
59	68	79	42	79	53	74	66	66
71	61	52	47	39	54	67	55	71
77	64	60	75	42	69	48	57	48

31	45	79	64	48	38	39	68	52
59	68	79	42	79	53	74	66	66
71	61	52	47	39	54	67	55	71
77	64	60	75	42	69	48	57	48