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Chapter 8: Q 8.67. (page 331)

In each of Exercises $8.63 - 8.68$ , we provide a sample mean, sample size, population standard deviation, and confidence level In each case, perform the following tasks:
a. Use the one-mean $z -$ interval procedure to find a confidence interval for the mean of the population from which the sample was drawn.
b. Obtain the margin of error by taking half the length of the confidence interval.
c. Obtain the margin of error by using Formula $8 . l$ on page $325$
$\overset{炉}{x} = 50, n = 16, 蟽 = 5$ , confidence level $= 99 %$

Short Answer

Expert verified

Part (a) The $99 %$ confidence interval for $渭$ is $(46.781, 53.219)$

Part (b) The margin of error by using the half-length of the confidence interval is $3.219$

Part (c) The margin of error by using the formula is $3.219$

Step by step solution

01

Part (a) Step 1: Given information

$\overset{炉}{x} = 50, n = 16, 蟽 = 5$ , confidence level $= 99 %$

02

Part (a) Step 2: Concept

The formula used: the confidence interval $\overset{炉}{x} 卤 z_{\frac{伪}{2}} (\frac{蟽}{\sqrt{n}})$ and $Margin of error (E) = z_{\frac{a}{2}} (\frac{蟽}{\sqrt{n}})$

03

Part (a) Step 3: Calculation

Compute the $99 %$ confidence interval for $渭$

Consider $\overset{炉}{x} = 50, n = 16, 蟽 = 5$ , and confidence level is $99 %$

The needed value of $z_{\frac{a}{2}}$ with a $99 %$ confidence level is $2.575$ , according to "Table II Areas under the standard normal curve."

Thus, the confidence interval is,

\overset{炉}{x} 卤 z_{\frac{伪}{2}} (\frac{蟽}{\sqrt{n}}) = 50 卤 2.575 (\frac{5}{\sqrt{16}}) = 50 卤 2.575 (1.25) = 50 卤 3.219 = (46.781, 53.219)

Therefore, the $99 %$ confidence interval for $渭$ is $(46.781, 53.219)$

04

Part (b) Step 1: Calculation

Using the half-length of the confidence interval, calculate the margin of error.

Margin of error = \frac{Upper limit - Lower limit}{2} = \frac{53.219 - 46.781}{2} = \frac{6.438}{2} = 3.219

Thus, the margin of error by using the half-length of the confidence interval is $3.219$

05

Part (c) Step 1: Calculation

Using a formula, calculate the margin of error.

Margin of error (E) = z_{\frac{a}{2}} (\frac{蟽}{\sqrt{n}}) = 2.575 (\frac{5}{\sqrt{16}}) = 2.575 (1.25) = 3.219

Thus, the margin of error by using formula is $3.219$

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Most popular questions from this chapter

Batting Averages. An issue of Scientific American revealed that batting averages, $x$ of major-league baseball players are normally

distributed and have a mean of $0.270$ and a standard deviation of $0.031$ For samples of $20$ batting averages, identify the distribution of each variable.

a. $\frac{\overset{炉}{x} - 0.270}{0.031 / \sqrt{20}}$

b. $\frac{\overset{炉}{x} - 0.270}{s / \sqrt{20}}$

Suppose that you take $1000$ simple random samples from a population and that, for each sample, you obtain a 95% confidence interval for an unknown parameter. Approximately how many of those confidence intervals will contain the value of the unknown parameter?

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Express the form of most of the confidence intervals that you will encounter in your study of statistics in terms of "point estimate" and "margin of error."

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a. $\frac{\overset{炉}{x} - 52.0}{17.2 / \sqrt{12}}$

b. $\frac{\hat{x} - 52.0}{s / \sqrt{12}}$

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