Section 11.1 Day 2

Recall Linear model is appropriate for set of data if:

Recall Linear model is appropriate for set of data if: 1) Conditional means fall near a line

Linear model is appropriate for set of data if: 1) Conditional means fall near a line

Linear model is appropriate for set of data if: 1) Conditional means fall near a line 2) Variability is about the same for each conditional distribution

Linear model is appropriate for set of data if: 1) Conditional means fall near a line 2) Variability is about the same for each conditional distribution

Linear model is appropriate for set of data if: 1) Conditional means fall near a line 2) Variability is about the same for each conditional distribution

Spread in values of x is measured by:

Recall, most of the time the theoretical slope, β1, is _________.

Recall, most of the time the theoretical slope, β1, is unknown.

Recall, most of the time the theoretical slope, β1, is unknown. So, we use ___ to estimate β1.

Recall, most of the time the theoretical slope, β1, is unknown. So, we use b1 to estimate β1.

The slope b1 _____ from sample to sample.

The slope b1 varies from sample to sample. Is this variation a good thing or not so good?

The slope b1 varies from sample to sample. Is this variation a good thing or not so good? Not so good as the variation affects our predictions.

Recall The slope b1 varies from sample to sample. Bold line is true regression line for the population.

Formula for estimating the standard error of the slope b1.

The smaller the standard error of the slope, the better for us. Formula for estimating the standard error of the slope b1. The smaller the standard error of the slope, the better for us.

Formula for estimating the standard error of the slope b1.

Formula for estimating the standard error of the slope b1. residuals

Formula for estimating the standard error of the slope b1.

Formula for estimating the standard error of the slope b1. SSE

Formula for estimating the standard error of the slope b1.

Formula for estimating the standard error of the slope b1. Spread in the values of x or Variability in the values of x

Key Concept The slope b1 of the regression line varies less from sample to sample when: Sample size is ______________ Residuals are _______________ Values of x are ______________

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

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 ______________

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 750, P5 For each quantity, decide which value will give you the larger variability in b1 (assuming all other things remain the same).

Page 750, P5 a. The standard deviation, σ, of the individual response values of y at each value of x: 3 or 5

Page 750, P5 a) 3 or 5, a greater variability in the conditional distributions of y results in a larger value in the numerator of Note: s is estimate of

Page 750, P5 b. The spread of the x-values: 3 or 10

Page 750, P5 b) 3 or 10, a smaller spread in the values of x results in smaller value in the denominator of

Page 750, P5 c. The number of observations, n: 10 or 20

Page 750, P5 c) 10 or 20, because all else being equal, a larger (random) sample size tends to result in less variability in the estimates of parameters

Page 750, P5 d. the true slope, β1: 1 or 3

Page 750, P5 d) The true, or theoretical, slope β1 does not matter, everything else being equal.

Page 750, P5 e. the true intercept, βo: 1 or 7

Page 750, P5 e) The true, or theoretical, intercept βo does not matter because, everything else being equal, all the intercept does is indicate whether one cloud of points is higher or lower than another.

Page 744, D3

Page 744, D3 Plot II will produce regression lines with the smallest variation in slopes because: the conditional distributions of responses (residuals) have smaller variation than in Plot I and the x-values have greater spread than in Plot III.

Page 744, D3 Plot I has about twice the variability in x as Plot III but also twice the variability in y. Consequently, the two have roughly the same variation in slopes of the regression line.

If you want a regression equation with the smallest estimated standard error of the slope, which of these lists would you use for the values of the explanatory variable? You may assume that each conditional distribution of y has the same variability.

If you want a regression equation with the smallest estimated standard error of the slope, which of these lists would you use for the values of the explanatory variable? You may assume that each conditional distribution of y has the same variability. A. 5, 10, 15, 20, 25 B. 5, 5, 5, 15, 15, 15, 25, 25, 25 C. 5, 5, 5, 5, 5, 25, 25, 25, 25, 25 D. 5, 5, 10, 10, 15, 15, 20, 20, 25, 25 E. 10, 10, 10, 15, 15, 15, 20, 20, 20

If you want a regression equation with the smallest estimated standard error of the slope, which of these lists would you use for the values of the explanatory variable? You may assume that each conditional distribution of y has the same variability.

x = 15 If you want a regression equation with the smallest estimated standard error of the slope, which of these lists would you use for the values of the explanatory variable? You may assume that each conditional distribution of y has the same variability. A. 5, 10, 15, 20, 25 B. 5, 5, 5, 15, 15, 15, 25, 25, 25 C. 5, 5, 5, 5, 5, 25, 25, 25, 25, 25 D. 5, 5, 10, 10, 15, 15, 20, 20, 25, 25 E. 10, 10, 10, 15, 15, 15, 20, 20, 20

x = 15 If you want a regression equation with the smallest estimated standard error of the slope, which of these lists would you use for the values of the explanatory variable? You may assume that each conditional distribution of y has the same variability. A. 5, 10, 15, 20, 25 B. 5, 5, 5, 15, 15, 15, 25, 25, 25 C. 5, 5, 5, 5, 5, 25, 25, 25, 25, 25 D. 5, 5, 10, 10, 15, 15, 20, 20, 25, 25 E. 10, 10, 10, 15, 15, 15, 20, 20, 20

Page 750, P4

Page 750, P4

Page 750, P4 To find SSE: Enter Sulfate in L1 and Redness in L2. Do LinReg L3 = LResid2 Sum(L3)

Page 750, P4 s is estimate of the common variability of y at each x.

Page 751, P8

Page 751, P8 Want smallest spread in x values to largest spread to get largest to smallest standard error of the slope

Page 751, P8 Largest to smallest standard error of the slope, : V, II, I, IV, III

Questions?

Page 749, P3

Page 749, P3

Page 749, P3

Page 749, P3

Page 749, P3

Page 752, E2

Page 752, E2

Page 752, E2

Page 752, E2

Page 752, E2

Page 752, E2

Page 752, E2 c. The estimated slope is - 0.23643.

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

Page 752, E2 b1

Page 752, E2 b0 b1

Page 752, E2 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.

Section 11.1 Extra Problems

Page 754, E3

Page 754, E3 In this case, a linear model is known to fit the situation perfectly:

Page 754, E3 In this case, a linear model is known to fit the situation perfectly: What is the theoretical slope?

Page 754, E3 In this case, a linear model is known to fit the situation perfectly: What is the theoretical slope? How do you interpret this slope in context?

Page 754, E3 In this case, a linear model is known to fit the situation perfectly: What is the theoretical slope? How do you interpret this slope in context? For every 1-cm increase in the distance across, the circumference will increase by 3.1416 cm.

Page 754, E3 b) All of the points may not fall exactly on a line with this slope because it is difficult to measure C and d with much accuracy.

Page 754, E4

Page 754, E4

Page 754, E4

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?