Question

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

a. Find a $95\%$confidence interval for the population means. (Note: You may want to review Example $8.2$, which begins on page $3.16$)

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.

$\overline{x}=50,n=16,\sigma =5$

Short answer

Part (a) The $95\%$confidence interval for the population mean is from $47.5$to $52.5$

Part (b) The population mean $95\%$to within of our sample mean with $2.5$confidence.

Part (c) Point estimate $\pm$Margin of error $=50\pm 2.5$

Step-by-step solution

Part (a) Step 1: Given information

$\overline{x}=50,n=16,\sigma =5$

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 $\overline{x}=50,n=16$, and $\sigma =5$

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 contained within the range $(\overline{x}-2s,\overline{x}+2s)$

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

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

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

$$\begin{gathered} \overline{x}-\frac{2\sigma }{\sqrt{n}}\text{to}\overline{x}+\frac{2\sigma }{\sqrt{n}}=50-\frac{2\left(5\right)}{\sqrt{16}}\text{to}50+\frac{2\left(5\right)}{\sqrt{16}} \\ =50-\frac{10}{4}\text{to}50+\frac{10}{4} \\ =50-2.5\text{to}50+2.5 \\ =47.5\text{to}52.5 \end{gathered}$$

Thus, the $95\%$ confidence interval for the population mean is from $47.5$ to $52.5$

Part (b) Step 1: Explanation

Identify the margin of error.

From part a., the margin of error is $\mathbf{2}\mathbf{.}\mathbf{5}$

Interpretation:

With $95\%$ confidence, the population mean $\left(\mu \right)$ can be projected to be within $2.5$ of our sample mean.

Part (c) Step 1: Calculation

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

The point estimate and margin of error endpoints of the confidence interval are as follows: Point estimate $\pm$ Margin of error $=50\pm 2.5$