Question

According to the National Association of Realtors, it took an average of three weeks to sell a home in 2017. Suppose data for the sale of 39 randomly selected homes sold in Greene County, Ohio, in 2017 showed a sample mean of 3.6 weeks with a sample standard deviation of 2 weeks. Conduct a hypothesis test to determine whether the number of weeks until a house sold in Greene County differed from the national average in 2017. Use α = 0.05 for the level of significance, and state your conclusion.

a. State the null and alternative hypothesis.
b. Find the value of the test statistic.
c. Find the p-value.
d. State your conclusion.

Answer 1

Solution :

Here, given :

Sample size, n = 39

Sample mean, $\bar X$ = 3.6

Standard deviation of the sample, s =2

The population mean, $\mu_0 = 3$

The significance level, $\alpha = 0.05$

a). Therefore the hypothesis is :

  $H_0 : \mu = 3 \text{ Vs} \ H_a: \mu \neq 3$

b). The test statics is given as :

  $t = \frac{\bar X - \mu_0}{\frac{s}{\sqrt n}} \rightarrow t_{n-1}$

  $t = \frac{3.6-3}{\frac{2}{\sqrt {39}}}$

     = 1.873

c). The p- value is given by :

 $P(t_{d.f}>|t_{stat}|)$

$=P(t_{39-1}> 1.873)$

$=0.0688$

d). The conclusion :

  In this case, the p-value is $0.688 > \alpha=0.05$

So, we do not reject $H_0$.

Therefore, we conclude that it is not a statistically significant difference between national average time for selling a home and the mean time for selling in Greene County.