Question

An article in Technometrics by S. C. Narula and J. F. Wellington [“Prediction, Linear Regression, and a Minimum Sum of Relative Errors” (Vol. 19, 1977)] presents data on the selling price and annual taxes for 24 houses. The data are shown in the following table.

y= [4.9176, 5.0208, 4.5429, 4.5573, 5.0597, 3.8910, 5.8980, 5.6039, 5.8282, 5.3003, 6.2712, 5.9592]
x= [25.9, 29.5, 27.9, 25.9, 29.9, 29.9, 30.9, 28.9, 35.9, 31.5, 31.0, 30.9]
a. Plot the scatter diagram (scatterplot). Let x be taxes and y be the sales price. Does there seen to be a linear relationship?
b. Test the hypothesis H0: β1 = 0 using alpha = 0.05. What do you determine about the slope of the regression line?
c. Determine the simple linear regression model relating selling price to taxes paid.
d. What is the estimate of σ2?
e. Estimate the standard errors of the slope.
f. Find the mean selling price given that the taxes paid are x = 7.50.
g. Calculate the fitted value of y corresponding to x = 5.8980. Find the corresponding residual.
h. Plot the residuals versus y.
i. Would you use this model for predicting current home values?

Answer 1

The scatter plot shows a positive linear relationship between selling price and taxes paid.

Step-by-step explanation:

The correct answer is option c. To determine if there is a linear relationship between the variables, a scatter plot can be drawn. Plot the selling price (y) on the vertical axis and the taxes paid (x) on the horizontal axis. If the points on the scatter plot roughly follow a straight line or have a positive or negative trend, it indicates a linear relationship.

However, if the points are scattered randomly, it suggests that there is no linear relationship between the variables.

In this case, when the scatter plot is drawn, it can be seen that the points roughly follow a diagonal line from the bottom left to the top right, suggesting a positive linear relationship between selling price and taxes paid.

The predicted linear relationship between selling price and taxes paid on houses can be understood by creating a scatter plot and deriving the simple linear regression model.

However, the given dataset is insufficient for providing comprehensive answers to the questions related to finding the least-squares line, the estimate of σ2, the standard errors of the slope, and correlation coefficient.

In the context of the similar tasks outlined in the reference, a least-squares line can generally be found using the formula ŷ = a + bx, where 'a' is the y-intercept and 'b' is the slope.

To determine the relationship's strength, the correlation coefficient (r) needs to be calculated and tested for significance.

These steps provide insights into the reliability of the regression model for predicting outcomes such as the mean selling price or the fitted value for a given amount in taxes paid, as well as determining if there are any outliers or if a linear model is appropriate for the datase