1. Section 11.1
Day 2

2. Recall
Linear model is appropriate for set of data if:

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

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

5. 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

6. 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

7. 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

8. Spread in values of x is measured by:

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

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

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

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

13. The slope b1 _____ from sample to sample.

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

15. 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.

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

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

18. 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.

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

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

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

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

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

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

25. 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 ______________

26. 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 ______________

27. 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 ______________

28. 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

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

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

31. 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

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

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

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

35. 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

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

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

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

39. 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.

40. Page 744, D3

41. 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.

42. 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.

43. 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.

44. 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

45. 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.

46. 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

47. 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

48. Page 750, P4

49. Page 750, P4

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

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

53. Page 751, P8

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

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

56. Questions?

57. Page 749, P3

58. Page 749, P3

59. Page 749, P3

60. Page 749, P3

61. Page 749, P3

62. Page 752, E2

63. Page 752, E2

64. Page 752, E2

65. Page 752, E2

66. Page 752, E2

67. Page 752, E2

68. Page 752, E2
c. The estimated slope is

69. 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 therms less.

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

71. Page 752, E2

72. Page 752, E2 b1

73. Page 752, E2 b0 b1

74. Page 752, E2 b0 b1
sb1

75. Page 752, E2 b0 b1
sb1

76. 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.

77. Section 11.1
Extra Problems

78. Page 754, E3

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

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

81. 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?

82. 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
cm.

83. 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.

84. Page 754, E4

85. Page 754, E4

86. Page 754, E4

87. Page 754, E5

88. 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.

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

90. Page 754, E6

91. 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.

92. 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.

93. Page 754, E6

94. Questions?