Question

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 popularion mean. (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.

where, $\hat{x}=20,n=36,\sigma =3$

Short answer

Part (a)The $95\%$confidence interval for the population mean is $\left(19,21\right)$

Part (b) The margin of error is 1

Part (C) The endpoints of the confidence interval is $20\pm 1$

Step-by-step solution

Step 1:Given information

$\hat{x}=20,n=36,\sigma =3$

Step 2:Part (a) Calculation

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

Consider $\overline{x}=20,n=36$, and $\sigma =3$.

Empirical rule:

Property 1: Approximately $68\%$of the data set lies between ($\overline{x}-2s,\overline{x}+2s$)

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

Property 3: Approximately$99.7\%$of the data set lies 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,

$$\begin{gathered} \overline{x}\pm 2\frac{\sigma }{\sqrt{n}}=20\pm 2\left(\frac{3}{\sqrt{36}}\right) \\ =20\pm 1 \\ =(20-1,20+1) \\ =(19,21) \end{gathered}$$

Thus, the$95\%$ confidence interval for the population mean is $(19,21)$

 Step 3:part (b) Calculation

From part (a), we have found out that the margin of error is 1

Interpretation:

we can estimate that the population mean $\left(\mu \right)$to within $1$with $95\%$confidence

Step 4:Part (c) Calculation

Expressing the endpoints of the confidence interval.

Endpoints$=\text{Point estimate}\pm \text{Margin of error}$$=20\pm 1$

Thus, the endpoints of the confidence interval is $20\pm 1$