Question

In each of the Exercises $8.11-8.16$, we provide a sample mean, sample size, and population standard deviation. In each case, perform the following tasks.

a. Find a $95\%$confidence interval for the population means. (Note: You may want to review Example 8.2 , which begins on page $316$

b. Identify and interpret the margin of error.

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

$\hat{x}=25,n=36,\sigma =3$

Short answer

Part (a) The$95\%$confidence interval for the population mean is from $24$to$26$

Part (b) The population mean $\left(\mu \right)$to within $1$with $95\%$confidence.

Part (c) The endpoints of the confidence interval is$25\pm 1$

Step-by-step solution

Part (a) Step 1: Given information

$\hat{x}=25,n=36,\sigma =3$

Part (a) Step 2: Concept

The formula used:$\overline{x}-\frac{2\sigma }{\sqrt{n}}\text{to}\overline{x}+\frac{2\sigma }{\sqrt{n}}$

Part (a) Step 3: Calculation

Find a $95\%$confidence interval for the population mean.

Consider $\overline{x}=25,n=36$, and $\sigma =3$

Empirical rule:

Property 1: Approximately $68\%$the data set is contained within the range

Property 2: Approximately $95\%$the data set is located between

Property 3: Approximately $99.7\%$the data set is located between

Using Property $2$, $95\%$all observations are within two standard deviations of the mean on either side.

$$\begin{gathered} \overline{x}-\frac{2\sigma }{\sqrt{n}}\text{to}\overline{x}+\frac{2\sigma }{\sqrt{n}}=25-\frac{2\left(3\right)}{\sqrt{36}}\text{to}25+\frac{2\left(3\right)}{\sqrt{36}} \\ =25-\frac{6}{6}\text{to}25+\frac{6}{6} \\ =25-1\text{to}25+1 \\ =24\text{to}26 \end{gathered}$$

The $95\%$confidence interval for the population mean is,

Thus, the $95\%$confidence interval for the population mean is from $24$to$26$

Part (b) Step 1: Explanation

Identify the margin of error.

From part a., the margin of error is $1$

Interpretation:

It can be estimated that the population mean $\left(\mu \right)$to within $1$with $95\%$confidence.

Part (c) Step 1: Calculation

In terms of the point estimate and the margin of error, express the confidence interval's endpoints.

Endpoints $=$Point estimate $\pm$Margin of error

$=25\pm 1$

The endpoint of the confidence interval is$25\pm 1$