Question

A confidence interval for a population mean has a margin of error of $10.7$

a. Obtain the length of the confidence interval.

b. If the mean of the sample is $75.2$, determine the confidence interval.

Short answer

Part (a) $21.4$

Part (b) $(64.5,85.9)$

Step-by-step solution

Part (a) Step 1: Given information

The margin of error for a population mean confidence interval is$10.7$

Part (a) Step 2: Concept

The formula used: Length of the $\mathrm{Cl}$ for population mean $=2\times$ margin of error

Part (a) Step 3: Calculation

Margin of error $E=10.7$

We know that,

$\mathrm{Cl}$length for the population mean$=2\times$ margin of error

$$\begin{gathered} =2\times 10.7 \\ =21.4 \end{gathered}$$

Part (b) Step 1: Calculation

Given that, the sample mean $\overline{x}=75.2$

The confidence interval of the population mean

$$\begin{gathered} =(\overline{x}-E,\overline{x}+E) \\ =(75.2-10.7,75.2+10.7) \\ =(64.5,85.9) \end{gathered}$$