Section 11.1 Day 3

Page 752, E2

Page 752, E2 a. For each pair of variables, tell whether you think a line gives a suitable summary of the relationship.

Mean Gas versus Mean Temp

Mean Gas versus Mean Temp A line is appropriate at least for values of x of 65°F or less. There is no obvious curvature, and the variation around the line is fairly uniform. The points around 70 degrees vary less than the points at the other temperatures and appear to be approaching the horizontal limit of 0.

Mean KWH versus Mean Temp

Mean KWH versus Mean Temp There appears to be no curvature. A line can be used as a summary of the relationship, but there is more variation in the responses at the lower values of x than at the higher (i.e., lack of homogeneity).

Mean KWH versus Heat DD

Mean KWH versus Heat DD No, this plot shows that this is not a good data set for supporting a linear regression model. Although the data show a slight positive trend, the variation in responses is too great at the larger values of x (Heat DD) for inferential techniques to be correct or useful.

Heat DD versus Mean Temp

Heat DD versus Mean Temp This plot appears to be quite linear, but . . . .

Heat DD versus Mean Temp Residuals vs Mean Temp

Heat DD versus Mean Temp This plot appears to be quite linear, but a residual plot shows pronounced curvature. It appears that the relationship between these variables is not linear. Again, the problem appears to be with points with values of x greater than 65°F.

Page 752, E2 b. By looking at the scatterplots, estimate which of the four pairs of variables has the largest standard error of the slope and which has the smallest.

Key Concept The slope b1 of the regression line varies less from sample to sample when: Sample size is larger Residuals are smaller Values of x are further apart

Page 752, E2 b. By looking at the scatterplots, estimate which of the four pairs of variables has the largest standard error of the slope and which has the smallest.

Page 752, E2 b. Note: Look at the units to determine the approximate residuals and spread in x-values and, therefore, the approximate standard error.

largest standard error of the slope

smallest standard error of the slope largest standard error of the slope

Page 752, E2 c. Compute the slope of the LSRL for the relationship between y = Mean Gas and x = Mean Temp. Interpret the slope in context. Compute the estimated standard error of the slope.

Page 752, E2 c. The estimated slope is - 0.23643. Interpret? Look at descriptions of variables

Page 752, E2 c. The estimated slope is - 0.23643. If one month has a mean daily temperature that is 1°F higher than another month, its mean daily gas usage tends to be 0.236 therms less.

Page 752, E2 The standard error of the slope is:

Page 752, E2 The standard error of the slope is:

Page 752, E2

Page 752, E2 b1?

Page 752, E2 b1

Page 752, E2 b0? b1

Page 752, E2 b0 b1

Page 752, E2 sb1? b0 b1

Page 752, E2 b0 b1 sb1

Page 752, E2 s? b0 b1 sb1

Page 752, E2 b0 b1 sb1

Page 752, E2 Here, s is the estimate of the common variability in the mean natural gas usage daily for the month for each fixed temperature.

Page 749, P3

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Page 754, E5

Page 754, E5 a. The soil samples should have the larger variability in the slope because the distance of y from the regression line tends to be larger compared to the spread in x.

Page 754, E5 The standard error for the soil samples is 0.36165. From P6, the standard error for the rock samples was 0.1472. As predicted, the standard error for the soil samples is much larger.

Page 754, E6

Page 754, E6 a. After the outlying point is removed, the remaining points all fall close to a line, so the variation in the residuals will decrease. Because the variation in the x-values will not change much, this implies that the standard error of the slope will also decrease. So, you expect the estimated standard error of the slope to be larger for the ______ data.

Page 754, E6 a. After the outlying point is removed, the remaining points all fall close to a line, so the variation in the residuals will decrease. Because the variation in the x-values will not change much, this implies that the standard error of the slope will also decrease. So, you expect the estimated standard error of the slope to be larger for the original data.

Page 754, E6

Questions?