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. Useα = 0.05for the level of significance, and state your conclusion.(a)State the null and alternative hypothesis. (Enter != for ≠ as needed.)H0:_____________Ha:__________(b)Find the value of the test statistic. (Round your answer to three decimal places.)Find the p-value. (Round your answer to four decimal places.)p-value =___________.

Solution :Here, given :Sample size, n = 39Sample mean, [tex]$\bar X$[/tex] = 3.6Standard deviation of the sample, s =2The population mean, [tex]$\mu_0 = 3$[/tex]The significance level, [tex]$\alpha = 0.05$[/tex]a). Therefore the hypothesis is :   [tex]$H_0 : \mu = 3 \text{ Vs} \ H_a: \mu \neq 3$[/tex]b). The test statics is given as :   [tex]$t = \frac{\bar X - \mu_0}{\frac{s}{\sqrt n}} \rightarrow t_{n-1}$[/tex]   [tex]$t = \frac{3.6-3}{\frac{2}{\sqrt {39}}} $[/tex]      = 1.873c). The p- value is given by :  [tex]$P(t_{d.f}>|t_{stat}|)$[/tex][tex]$=P(t_{39-1}> 1.873)$[/tex][tex]$=0.0688$[/tex]d). The conclusion :   In this case, the p-value is [tex]$0.688 > \alpha=0.05$[/tex]So, we do not reject [tex]$H_0$[/tex].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.