Ecologists have long known that there is a linear relationship between the amount of precipitation a location receives and the number of trees that grow in the area. Suppose that the yearly rainfall (X, measured in millimeters) and the amount of ground covered by trees (Y, measured on a scale from 0 to 100) are recorded for 49 different geographic locations. In the sample data, X has a sample mean of 1182.4 and a sample standard deviation of 226.0, while Y has a sample mean of 49.6 and a sample standard deviation of 7.1. The sample correlation between X and Y is 0.673. Please answer the following questions based on this information.
What percentage of the variability in tree cover is not explained by rainfall?
A. 2.1%
B. 21.5%
C. 54.7%
D. 45.3%
E. 67.37%
2. In a statistical test for Ho: β = 0 vs Ha: β ≠0, which of the following statements is the alternative hypothesis (in words)?
A. The population mean of tree cover is not zero.
B. The population mean of tree cover is zero.
C. Tree cover depends on rainfall.
D. Tree cover does not depend on rainfall.
E. The population means of tree cover and rainfall are not equal.
3. In a statistical test for Ho: β = 0 vs Ha: β ≠0, the test statistic of the form t(47) follows which distribution?
A. Standard normal distribution.
B. t-distribution with degree of freedom 47.
C. t-distribution with degree of freedom 48.
D. χ2 distribution with degree of freedom 47.
E. χ2 distribution with degree of freedom 48.
4. Suppose through some calculation via R, it is obtained that the 95% confidence interval for β is (-0.0083, 0.0505). What would the test for Ho: β = 0 vs Ha: β ≠0 conclude?
A. Reject the null hypothesis at the level of significance 0.05 and all smaller α, say 0.01.
B. Fail to reject the null hypothesis at the level of significance 0.05 and all smaller α, say 0.01.
C. Reject the null hypothesis at the level of significance 0.05 and all larger α, say 0.10.
D. Fail to reject the null hypothesis at the level of significance 0.05 and all larger α, say 0.10.
E. Fail to reject the null hypothesis at the level of significance 0.05 and cannot tell for sure when α changes.
5. Suppose now that the 50th observation is recorded and included in the dataset, and the regression equation is recalculated. Which of the following possibilities for the 50th observation would probably change the regression equation the most?
A. X = 900, Y = 45
B. X = 1200, Y = 40
C. X = 1200, Y = 80
D. X = 2400, Y = 75
E. X = 2400, Y = 25