Question

a real - estate agent conducted an experiment to test the effect of selling a staged home vs. selling an empty home. to do so, the agent obtained a list of 10 comparable homes just listed for sale that were currently empty. he randomly assigned 5 of the homes to be \staged,\ meaning filled with nice furniture and decorated. the owners of the 5 homes all agreed to have their homes staged by professional decorators. the other 5 homes remained empty. the hypothesis is that empty homes are not as appealing to buyers as staged homes and, therefore, sell for lower prices than staged homes. the mean selling price of the 5 empty homes was $150,000 with a standard deviation of $22,000. the mean selling price of the 5 staged homes was $175,000 with a standard deviation of 35,000. a dotplot of each sample shows no strong skewness and no outliers. the agent tests ( h_0:mu_1-mu_2 = 0,h_a:mu_1-mu_2lt0 ), where ( mu_1= ) the true mean selling price of all comparable empty homes and ( mu_2= ) the true mean selling price of all comparable staged homes. the conditions for inference are met. what are the values of the test statistic and ( p ) - value for a ( t ) - test about a difference in means? find the ( t ) - table here. ( r=\frac{150,000 - 175,000}{sqrt{\frac{22,000^2}{5}+\frac{35,000^2}{5}}} ). the ( p ) - value is between 0.10 and 0.15. ( r=\frac{150,000 - 175,000}{sqrt{\frac{22,000^2}{5}+\frac{35,000^2}{5}}} ). the ( p ) - value is between 0.20 and 0.30.

Explanation:

Step1: Calculate the test statistic

The formula for the test statistic \(t\) for a two - sample \(t\) - test is \(t=\frac{\bar{x}_{1}-\bar{x}_{2}}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}\), where \(\bar{x}_{1} = 150000\), \(\bar{x}_{2}=175000\), \(s_{1} = 22000\), \(s_{2}=35000\), \(n_{1}=n_{2}=5\).

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Step2: Determine the degrees of freedom

For a two - sample \(t\) - test with \(n_{1}=n_{2}=n\), the degrees of freedom \(df=\min(n_{1}-1,n_{2}-1)\) (using the conservative approach). Here \(n_{1}=n_{2}=5\), so \(df = 4\)

Step3: Find the \(P\) - value

Using a \(t\) - table with \(df = 4\) and a one - tailed test (since \(H_{a}:\mu_{1}-\mu_{2}<0\)), for \(t=-1.35\).
Looking at the row \(df = 4\) in the \(t\) - table:
\(t_{0.10,4}=1.533\), \(t_{0.15,4}=1.190\)
Since \(1.190<|t| = 1.35<1.533\), the \(P\) - value is between \(0.10\) and \(0.15\)

Answer:

\(t=\frac{150000 - 175000}{\sqrt{\frac{22000^{2}}{5}+\frac{35000^{2}}{5}}}\). The \(P\) - value is between \(0.10\) and \(0.15\)