Suppose that we want to test whether the mean selling price of houses with is the same as the mean selling price of houses without a garage. We collect two samples and obtain the following information:
Houses with garage
Sample size: 71
Mean selling price (in thousands of dollars): 238.2
Sample variance: 2,013.9
Houses with garage
Sample size: 34
Mean selling price (in thousands of dollars): 185.5
Sample variance: 784.3
Using the above information and an alpha of 0.05, carry out a hypothesis test (with unequal variances) to test the null hypothesis H0: µ1 = µ2.
To keep things simple, you may use the standard normal distribution (rather than the t-distribution).
Part (a)
Please open the “Assignment 2 Real Estate.xls” file in Excel and test the null hypothesis H0: µ1 = µ2. Consistent with Question 1, please use the “t-Test: Two-Sample Assuming Unequal Variance”.
You must submit your Excel file as part of the assignment.
Part (b)
What is your decision regarding the null hypothesis (“reject” or “do not reject”)? Explain on which cell in your Excel output you base your decision.
Note: Whenever you see a number such as 4.02583E-10 in Excel, that is scientific notation. Written in decimal form, that number is 0.000000000402583, i.e., it is very small.
Part (c)
Interpret your result from part (b) at a level that would be understandable to a high school graduate who has not taken a college-level statistics course.
Part (a):
To test the null hypothesis H0: µ1 = µ2, we can perform a hypothesis test using the "t-Test: Two-Sample Assuming Unequal Variance" function in Excel.
Based on the provided information, we have two samples:
Houses with garage:
Sample size (n1) = 71
Mean selling price (in thousands of dollars) = 238.2
Sample variance (s1^2) = 2,013.9
Houses without garage:
Sample size (n2) = 34
Mean selling price (in thousands of dollars) = 185.5
Sample variance (s2^2) = 784.3
Using an alpha level (significance level) of 0.05, we want to determine whether there is enough evidence to reject the null hypothesis.
1. Open the "Assignment 2 Real Estate.xls" file in Excel.
2. Select an empty cell where you want the results of the test to appear.
3. In the Excel menu, go to "Data" > "Data Analysis" (if you don't see "Data Analysis," you may need to enable it first).
4. Choose "t-Test: Two-Sample Assuming Unequal Variances" from the list and click "OK."
5. In the "t-Test: Two-Sample Assuming Unequal Variances" dialog box, enter the range for the first sample (Houses with garage) and the range for the second sample (Houses without garage).
6. Enter the alpha level (0.05) in the "Alpha" field.
7. Make sure the "Labels" box is checked if your data has headers.
8. Click "OK" to perform the hypothesis test.
Part (b):
The decision regarding the null hypothesis ("reject" or "do not reject") can be made based on the p-value provided in the Excel output.
In the Excel output, there should be a field labeled "P(T<=t) one-tail" or similar, which represents the p-value of the test. Compare this p-value to your selected alpha level (0.05).
If the p-value is less than the alpha level (p-value < 0.05), we reject the null hypothesis. If the p-value is greater than or equal to the alpha level (p-value >= 0.05), we fail to reject the null hypothesis.
Part (c):
Based on the decision from part (b), we interpret the result as follows:
Since the p-value (0.000000000402583) is much smaller than the chosen alpha level (0.05), we reject the null hypothesis. This means that there is enough evidence to conclude that the mean selling price of houses with a garage is statistically significantly different from the mean selling price of houses without a garage.
In simpler terms, the data suggests that houses with a garage tend to have a different mean selling price than houses without a garage.