Chapter 18, Problem 39SE

a.

To determine

Check whether the median sales price in Metropolitan Area S is significantly lower than the national median of $180,000 or not.

Answer to Problem 39SE

The median sale price for single-family homes in Metropolitan Area S is less than the national median price of $180,000.

Explanation of Solution

Calculation:

The national median sales price for single-family homes is $180,000. The level of significance is 0.05.

The hypotheses are given below:

Null hypothesis:

$H_0:Median\ge 180,000$

That is, the median sales price in Metropolitan Area S is significantly greater than or equal to the national median of $180,000.

Alternative hypothesis:

$H_a\text{: Median <180,000}$

That is, the median sales price in Metropolitan Area S is significantly lower than the national median of $180,000.

For a sign test, the mean $\mu =0.5n$, standard deviation $\sigma =\sqrt{0.25n}$, and $n=32+18=50$.

Substitute $n=50$.

The mean is obtained as follows:

$\begin{array}{c}\mu =0.5\left(50\right)\\ =25\end{array}$

Thus, the mean is 25.

The standard deviation is obtained as follows:

$\begin{array}{c}\sigma =\sqrt{0.25\left(50\right)}\\ =\sqrt{12.5}\\ =3.5355\end{array}$

Thus, the standard deviation is 3.5355.

There are18 plus signs in the lower tail.

The probability of 18 plus signs in the lower tail can be obtained using the continuity correction factor and normal approximation. Hence, the p-value is obtained using normal distribution with $\mu =25\text{ and }\sigma =3.5355$

$\begin{array}{c}P\left(\text{18 or fewer plus signs}\right)=P\left(x\le 18.5\right)\\ =P\left(z\le \frac{18.5-25}{3.5355}\right)\\ =P\left(z\le -1.84\right)\end{array}$

Procedure:

Step-by-step procedure to obtain the above probability using Table 1 of Appendix B:

• Locate the value –1.8 in the left column.
• Go through the row corresponding to the value –1.8 and column corresponding to the value 0.04.

Thus, $P\left(z<-1.84\right)=0.0329$.

$P\left(\text{18or fewer plus signs}\right)=0.0329$

Rejection rule:

If $p\text{-value}\le \alpha$, then reject the null hypothesis $\left(H_a\right)$.

Conclusion:

Here, the level of significance $\alpha =0.05$.

Here, $p\text{-value}\left(=0.0329\right)<\alpha \left(=0.05\right)$.

That is, the p-value is less than the significance level.

Therefore, the null hypothesis is rejected.

Hence, it can be concluded that the median sale price for single-family homes in Metropolitan Area S is less than the national median price of $180,000.

b.

To determine

Check whether the median sales price in Metropolitan Area D is significantly higher than the national median of $180,000 or not.

Answer to Problem 39SE

The median sale price for single-family homes in Metropolitan Area D is greater than the national median price of $180,000.

Explanation of Solution

Calculation:

The hypotheses are given below:

Null hypothesis:

$H_0:Median\le 180,000$

That is, the median is less than or equal to 180,000.

Alternative hypothesis:

$H_a\text{: Median >180,000}$

That is, the median is greater than 180,000.

There are $13+27=40$ observations in the data set.

For a sign test, the mean $\mu =0.5n$, standard deviation $\sigma =\sqrt{0.25n}$, and $n=13+27=40$.

Substitute $n=40$.

The mean is obtained as follows:

$\begin{array}{c}\mu =0.5\left(40\right)\\ =20\end{array}$

Thus, the mean is 20.

The standard deviation is obtained as follows:

$\begin{array}{c}\sigma =\sqrt{0.25\left(40\right)}\\ =\sqrt{10}\\ =3.1623\end{array}$

Thus, the standard deviation is 3.1623.

There 27 plus signs in the upper tail.

The probability of 27 plus signs in the upper tail can be obtained using the continuity correction factor and normal approximation. Hence, the p-value is obtained using normal distribution with $\mu =20\text{ and }\sigma =3.1623$.

$\begin{array}{c}P\left(\text{27 or more plus signs}\right)=P\left(x\ge 26.5\right)\\ =P\left(z\ge \frac{26.5-20}{3.1623}\right)\\ =P\left(z\ge 2.06\right)\\ =1-P\left(z<2.06\right)\end{array}$

Procedure:

Step-by-step procedure to obtain the above probability using Table 1 of Appendix B:

• Locate the value 2.0 in the left column.
• Go through the row corresponding to the value 2.0 and column corresponding to the value 0.06.

Thus, $P\left(z<2.06\right)=0.9803$.

$\begin{array}{c}P\left(\text{27 or more plus signs}\right)=1-0.9803\\ =0.0197\end{array}$

The upper tail p-value is 0.0197.

Rejection rule:

If $p\text{-value}\le \alpha$, then reject the null hypothesis $\left(H_a\right)$.

Conclusion:

Here the level of significance $\alpha =0.05$.

Here, $p\text{-value}\left(=0.0197\right)<\alpha \left(=0.05\right)$.

That is, the p-value is less than the significance level.

Therefore, the null hypothesis is rejected.

Hence, it can be concluded that the median sale price for single-family homes in Metropolitan Area D is greater than the national median price of $180,000.