Q 8.83 - Elementary Statistics

Question

Prices of New Mobile Homes. Recall that a simple random sample of 36 new mobile homes yielded the prices, in thousands of dollars, shown in Table $8.1$ on page 315 . We found the mean of those prices to be $\backslash (63.28$ thousand.

a. Use this information and Procedure $8.1$ on page 322 to find a $95\%$ confidence interval for the mean price of all new mobile homes. Recall that $\sigma =\backslash )7.2$ thousand.

b. Compare your $95\%$ confidence interval in part (a) to the one found in Example 8.2(c) on page 317 and explain any discrepancy that you observe.

Step-by-Step Solution

Answer

(a) $95\%$ confidence interval for $\mu$ is $(60.928, 65.632)$

(b) Above example provides $95\%$ confidence interval.

Step 1: Given Information

It is given that $\overline{x}=563.28, n=36, \sigma =7.2$

Step 2: Determination of 95 % Confidence Interval of μ

From table, value of $z_{\frac{\alpha }{2}}=1.96$ with $95\%$ confidence interval.

Hence, $\overline{x}\pm z_{\frac{a}{2}}\left(\frac{\sigma }{\sqrt{n}}\right)=63.28\pm 1.96\left(\frac{7.2}{\sqrt{36}}\right)$

$=63.28\pm 2.352$

$=(60.928,65.632)$

Hence, $95\%$ confidence interval of $\mu$ is $=(60.928,65.632)$

Step 3: Comparison of confidence interval here with that of example 8 . 2 (c)

The confidence level from example is $(60.88, 65.68)$

In above part, confidence interval is $\left(60.88, 65.88\right)$

Hence, it can be concluded that example provides approximately $95\%$ confidence interval.