Question

A real estate agent would like to know if the number of bedrooms in a house can be used to predict the selling price of the house. More specifically, she wants to know whether a larger number of bedrooms leads to a higher selling price. Records for 25 houses that recently sold in the area were selected at random, and data on the number of bedrooms (X) and the selling price Y (in $000s) for each house were used to fit the model E(y) = β0 + β1x.

The value of the test statistic for testing β1 is 11.3383 and the corresponding standard error is 1.29384. Which of the following statements is true?

A. There is sufficient evidence (at   α = 0.05) to conclude that number of bedrooms is positively linearly related to selling price.    

B. At any reasonable α, there is no relationship between the number of bedrooms and selling price.    

C. There is insufficient evidence (at α= 0.05) to conclude that number of bedrooms is positively linearly related to selling price.    

D. There is insufficient evidence (at α = 0.10) to conclude that number of bedrooms is a useful linear predictor of selling price.

Answer 1

Solution: We are given the test statistic for testing is 11.3383 and standard error of $\beta_{1}$ is 1.29384.

From the given information, we can find the estimate of $\beta_{1}$

$\beta_{1}= 1.29384 \times 11.3383$

rounded to two decimal places.

14.67

The positive sign of $\beta_{1}$ suggests that there is a positive linear relationship between bedrooms and the selling price.

Now we need to find the t critical value at 0.05 significance level for df = 23 in order to find whether there is a significant linear relationship between the bedrooms and the selling price.

The t critical value is:

$t_{critical}=2.069$

Since the test statistic of $\beta_{1}$ is greater than the t critical value, we, therefore, reject the null hypothesis and conclude there exists a positive linear relationship between the bedrooms and the selling price.