Question

In each 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 mean. (Note: You may want to review Example $8.2$, which begins on the 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.

$\overline{x}=30,n=25,\sigma =4$

Short answer

Part (a) The $95\%$ confidence interval for the population mean is $(28.4,31.6)$

Part (b) The population means $\left(\mu \right)$to within $1.6$with $95\%$confidence.

Part (c) The endpoint of the confidence interval is $30\pm 1.6$

Step-by-step solution

Part (a) Step 1: Given information

$\overline{x}=30,n=25,\sigma =4$

Part (a) Step 2: Concept

The formula used:$\overline{x}\pm 2\frac{\sigma }{\sqrt{n}}$

Part (a) Step 3: Calculation

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

Consider $\overline{x}=30,n=25$, and $\sigma =4$

Empirical rule:

Property 1: Around $68\%$of the data set is located between $(\overline{x}-s,\overline{x}+s)$

Property 2: Approximately $95\%$of the data set is located between $(\overline{x}-2s,\overline{x}+2s)$

Property 3: Approximately $99.7\%$of the data set is located between $(\overline{x}-3s,\overline{x}+3s)$

By using Property $2$, the $95\%$of all observations lie within two standard deviations to either side of the mean.

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

$$\begin{gathered} \overline{x}\pm 2\frac{\sigma }{\sqrt{n}}=30\pm 2\left(\frac{4}{\sqrt{25}}\right) \\ =30\pm 1.6 \\ =(30-1.6,30+1.6) \\ =(28.4,31.6) \end{gathered}$$

Thus, the confidence interval for the population mean is $(28.4,31.6)$

Part (b) Step 1: Explanation

From part (a), the margin of error is $1.6$

With $95\%$ confidence, the population means $\left(\mu \right)$ may be predicted to within $1.6$

Part (c) Step 1: Calculation

The confidence interval's endpoints should be expressed.

$$\begin{gathered} \text{endpoints}=\text{Point estimate}\pm \text{Margin of error} \\ =30\pm 1.6 \end{gathered}$$

Thus, the endpoints of the confidence interval is $30\pm 1.6$