8.1.11 population mean, confidence interval

For the provided sample​ mean, sample​ size, and population standard​ deviation, complete parts​ (a) through​ (c) below

\(\bar{x}=37, n=64, \sigma=3\)

(a). Find a​ 95% confidence interval for the population mean.

First, we need to get the data from the question

x = 37
n = 64
sigma = 3

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Empirical rule:

• 99.7% - 1 standard deviation

• 95% - 2 standard deviation

• 68% - 3 standard deviation

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To find 95% confidence interval, we run

x - 2 * sigma/sqrt(n)

## [1] 36.25

x + 2 * sigma/sqrt(n)

## [1] 37.75

Round to two decimal places as needed

round(x - 2 * sigma/sqrt(n), 2)

## [1] 36.25

round(x + 2 * sigma/sqrt(n), 2)

## [1] 37.75

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(b). Identify and interpret the margin of error.

Margin of error = 1/2 distance of two endpoints of confidence interval

(round(x + 2 * sigma/sqrt(n), 2) - round(x - 2 * sigma/sqrt(n), 2)) /2

## [1] 0.75

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(c). Express the endpoints of the confidence interval in terms of the point estimate and the margin of error.

We can find margin of error

round( 2 * sigma/sqrt(n), 2)

## [1] 0.75

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Hope that helps!