Solutions

Please note that the steps show rounded numbers, but that the final answers to the problems are calculated without rounding.

Problem Part Solution
1 - Estimated linear regression equation:
\[\hat{Y} = b_0 + b_1X\]

True linear regression equation:
\[Y = \beta_0 + \beta_1X + \epsilon\]
2 - See the wiki for a review of this important concept.
3 A A
The appropriate graphs to check for a linear relationship are a scatterplot and a residual plot. The scatterplot seems to show a linear relationship and there is no pattern in the residual plot, so we can conclude that there is a linear relationship in the data.
3 B b
The appropriate graph to check for constant variance is a residual plot. There is no pattern in the residual plot, so we can conclude that there is a constant variance in the data.
3 C c
The appropriate graph to check for a normal error term is a Q-Q plot of the residuals. The points in the plot are close to the line, so we can conclude that there is a normal error term in the data.
4 - \(r = 0.704\)
5 - \(\hat{Y} = -29.859 + 37.72X\)
6 - \(Y = 49.73\)
7 - (22.999, 52.441) We are 95% confident that the slope of the true true linear regression line of Lactic with Taste is between 22.999 and 52.441.
8 - \(H_0: \beta_1 = 0\)
\(H_a: \beta_1 \neq 0\)
9 - \(t = 5.249\)
10 - \(\text{P-value} = 0.00001405\)
11 - reject the null hypothesis
12 - There is sufficient evidence to suggest that the slope of the true linear regression line does not equal zero. We conclude that there is a linear relationship between the concentration of lactic acid in cheese and the quality of its taste.
13 A a
The appropriate graphs to check for a linear relationship are a scatterplot and a residual plot. The scatterplot does not seem to show a significant linear relationship, so we cannot conclude that there is a linear relationship in the data.
13 B b
The appropriate graph to check for constant variance is a residual plot. There is no pattern in the residual plot, so we can conclude that there is a constant variance in the data.
13 C c
The appropriate graph to check for a normal error term is a Q-Q plot of the residuals. The points in the plot are close to the line, so we can conclude that there is a normal error term in the data.
14 - \(\hat{Y} = 25,838.626 + -0.034X\)
15 - \(Y = 22,401.192\)
16 - (-0.073, 0.004) We are 90% confident that the slope of the true true linear regression line of Lactic with Taste is between -0.073 and 0.004.
17 - \(H_0: \beta_1 = 0\)
\(H_a: \beta_1 \neq 0\)
18 - \(t = -1.476\)
19 - \(\text{P-value} = 0.144\)
20 - fail to reject the null hypothesis
21 - There is insufficient evidence to suggest that the slope of the true linear regression line does not equal zero. We conclude that there is not a linear relationship between the mileage of a Prius listed for sale and its price.
22 - \(r = -0.181\)
23 - \(\hat{Y} = 62.825 + -18.236X\)
24 - \(Y = 49.148\)
25 - (-41.855, 5.383) We are 95% confident that the slope of the true true linear regression line of Lead with BRS is between -41.855 and 5.383.
26 - \(H_0: \beta_1 = 0\)
\(H_a: \beta_1 \neq 0\)
27 - \(t = -1.54\)
28 - \(\text{P-value} = 0.128\)
29 - fail to reject the null hypothesis
30 - There is insufficient evidence to suggest that the slope of the true linear regression line does not equal zero. We conclude that there is not a linear relationship between a child’s level of lead exposure and his or her behavioral rating.
31 - d. The actual Y value was 4.5 units higher than the predicted Y value