Q 8.16. - Elementary Statistics

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 $3l6$)

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:

$\hat{x}=55,n=16,\sigma =5$

Step-by-Step Solution

Answer

Part (a) The $95\%$ confidence interval for the population mean is $(52.5,57.5)$

Part (b) The population means $\left(\mu \right)$ to within $2.5$ with $95\%$ confidence.

Part (c) The endpoint of the confidence interval is $55\pm 2.5$

Part (a) Step 1: Given information

$\hat{x}=55, n=16, \sigma =5$

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 $\overline{x}=55, n=16$, 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 located between $(\overline{x}-2s,\overline{ x}+2s)$

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

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

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

$$\begin{gathered} \overline{x}\pm 2\frac{\sigma }{\sqrt{n}}=55\pm 2\left(\frac{5}{\sqrt{16}}\right) \\ =55\pm 2.5 \\ =(55-2.5,55+2.5) \\ =(52.5,57.5) \end{gathered}$$

Thus, the $95\%$ confidence interval for the population mean is $(52.5,57.5)$

Part (b) Step 1: Explanation

From part (a), the margin of error is $2.5$

It can be estimated that the population mean $\left(\mu \right)$ to within $2.5$ with $95\%$ confidence.

Part (c) Step 1: Calculation

The confidence interval's endpoints should be expressed.

$$\begin{gathered} \text{ endpoints }=\text{ Point estimate }\pm \text{ Margin of error } \\ =55\pm 2.5 \end{gathered}$$

Thus, the endpoints of the confidence interval are $55\pm 2.5$