Q 8.12. - Elementary Statistics

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.

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

Step-by-Step Solution

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 $25\pm 1$

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 $25\pm 1$