Q 8.26. - Elementary Statistics

Question

New Mobile Homes. Refer to Examples $8.1$ and $8.2$ Use the data in Table $8.1$ on page $315$ to obtain a $68\%$ confidence interval for the mean price of all new mobile homes. (Hint: Proceed as in Example $8.2$, but use Property $1$ of the empirical rule on page $271$ instead of Property $2$ .)

Step-by-Step Solution

Answer

The mean price of all new mobile homes $\left(\mu \right)$ lie within the $68\%$ confidence interval obtained.

Step 1: Given information

$n=36$ and $\overline{x}=63.28$

Step 2: Concept

The formula used: $\overline{x}\pm \frac{1\sigma }{\sqrt{n}}$

Step 3: Explanation

Using Property $1.$, calculate a $68\%$ confidence interval for the mean price of all new mobile homes.

From Examples $8.1$ and $8.1, n=36$ and $\overline{x}=63.28$

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

$(\overline{x}-3s,\overline{x}+3s)$

Using Property $1,$ $68$ percent of the observations are within one standard deviation of the mean on either side.

Step 4: Calculation

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

$$\begin{gathered} \overline{x}\pm \frac{1\sigma }{\sqrt{n}}=63.28\pm \frac{1(7.2)}{\sqrt{36}} \\ =63.28\pm 1.2 \\ =(63.28-1.2,63.28+1.2) \\ =(62.08,64.48) \end{gathered}$$

By considering Property $1,$ the $68$ percent confidence interval for the mean price of all new mobile homes is $(62.08,64.48)$

As a result, the $\left(\mu \right)$ mean price of all new mobile homes falls within the $68\%$ confidence interval.