4.6 We have five observations on x and y. They are x₁ = 3,2,1,-1,0 with corresponding y values y = 4,2,3,1,0. The fitted least squares line is y, = 1.2 +0.8x, the sum of squared least squares residuals is = 3.6 and Σ(-y)² = 10. Carry out this exercise with a hand calculator. a. Calculate the fitted values y, and their sample mean y. Compare this value to the sample mean of the y values. b. Calculate Σ, (; – 5) and Σ1 ( − 5)/Σ (-5). - i=1 c. The least squares residuals are ê; = 0.4, -0.8, 1, 0.6, and -1.2. Calculate Σ( - ) ê i=1 2 d. Calculate 1 -/-(-)² and compare it to the results in part (b). i=1 e. Show, algebraically, that Σ(-y) (y; - y) = i=1 y; - Ny². Calculate this value. f. Using Σ(x₁ - x)² = 10, and previous results, calculate r= [Σº (9; − 3)(x - 1)] / [√Σ (91 − 5)² √Σ²2, (1; − 5) ²
4.6 We have five observations on x and y. They are x₁ = 3,2,1,-1,0 with corresponding y values y = 4,2,3,1,0. The fitted least squares line is y, = 1.2 +0.8x, the sum of squared least squares residuals is = 3.6 and Σ(-y)² = 10. Carry out this exercise with a hand calculator. a. Calculate the fitted values y, and their sample mean y. Compare this value to the sample mean of the y values. b. Calculate Σ, (; – 5) and Σ1 ( − 5)/Σ (-5). - i=1 c. The least squares residuals are ê; = 0.4, -0.8, 1, 0.6, and -1.2. Calculate Σ( - ) ê i=1 2 d. Calculate 1 -/-(-)² and compare it to the results in part (b). i=1 e. Show, algebraically, that Σ(-y) (y; - y) = i=1 y; - Ny². Calculate this value. f. Using Σ(x₁ - x)² = 10, and previous results, calculate r= [Σº (9; − 3)(x - 1)] / [√Σ (91 − 5)² √Σ²2, (1; − 5) ²
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.4: Distributions Of Data
Problem 19PFA
Related questions
Question
Please answer 4.6
![sion above is 199
b. Using the answer to (a), what is the correlation between WAGE and EDUC? (Hint: Y
correlation between WAGE and the fitted value WAGE?]
c. The sample mean and variance of EDUC are 14.315 and 8.555, respectively. Calculate the leverage
of observations with EDUC= 5, 16, and 21. Should any of the values be considered large?
d. Omitting the ninth observation, a person with 21 years of education and wage rate $30.76, and
reestimating the model we find ô= 14.25 and an estimated slope of 2.470095. Calculate DFBETAS
for this observation. Should it be considered large?
e. For the ninth observation, used in part (d), DFFITS = -0.0571607. Is this value large? The leverage
value for this observation was found in part (c). How much does the fitted value for this observation
change when this observation is deleted from the sample?
f.
For the ninth observation, used in parts (d) and (e), the least squares residual is -10.18368. Calcu-
late the studentized residual. Should it be considered large?
4.6 We have five observations on x and y. They are x = 3,2,1,-1,0 with corresponding y values
y = 4,2,3,1,0. The fitted least squares line is ŷ, = 1.2 +0.8x,, the sum of squared least squares
residuals is = 3.6 and (y₁ - y)² = 10. Carry out this exercise with a hand calculator.
a. Calculate the fitted values y; and their sample meanỹ. Compare this value to the sample mean of
the y values.
b. Calculate Σ ( – y) and Σ1 ( – 5)/Σ(-5).
c. The least squares residuals are ê; = 0.4,-0.8, 1, 0.6, and -1.2. Calculate
d. Calculate 1 -/-(-y)² and compare it to the results in part (b).
(-) ê,.
e. Show, algebraically, that Σ(-) (y;-) = Lii-Ny². Calculate this value.
f. Using Σ(x₁ - x)² = 10, and previous results, calculate
r=
r = [Σ²2 (0 - 3)(x - 5)] / [√ Σ (9; – 5)²³√Σ², (3; – 5) ²
-
What statistic is r? Calculate ² and compare this value to the values in parts (d) and (b).
4.7 We have data on 2323 randomly selected households consisting of three persons in 2013. Let ENTERT
denote the monthly entertainment expenditure ($) per person per month and let INCOME ($100) be
monthly household income. Consider the regression model
ENTERT, =B₁ + B₂INCOME; + e,, i = 1,..., 2323
Assume that assumptions SR1-SR6 hold. The OLS estimated equation is ENTERT, = 9.820 +
0.503INCOME,. The standard error of the slope coefficient estimator is se (b₂) = 0.029, the standard](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b1ec9e1-71c5-41a7-88e5-1ec0b0e62692%2Fc5e98b98-9a99-4d32-a7be-b9406adef221%2F3a7zvhm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:sion above is 199
b. Using the answer to (a), what is the correlation between WAGE and EDUC? (Hint: Y
correlation between WAGE and the fitted value WAGE?]
c. The sample mean and variance of EDUC are 14.315 and 8.555, respectively. Calculate the leverage
of observations with EDUC= 5, 16, and 21. Should any of the values be considered large?
d. Omitting the ninth observation, a person with 21 years of education and wage rate $30.76, and
reestimating the model we find ô= 14.25 and an estimated slope of 2.470095. Calculate DFBETAS
for this observation. Should it be considered large?
e. For the ninth observation, used in part (d), DFFITS = -0.0571607. Is this value large? The leverage
value for this observation was found in part (c). How much does the fitted value for this observation
change when this observation is deleted from the sample?
f.
For the ninth observation, used in parts (d) and (e), the least squares residual is -10.18368. Calcu-
late the studentized residual. Should it be considered large?
4.6 We have five observations on x and y. They are x = 3,2,1,-1,0 with corresponding y values
y = 4,2,3,1,0. The fitted least squares line is ŷ, = 1.2 +0.8x,, the sum of squared least squares
residuals is = 3.6 and (y₁ - y)² = 10. Carry out this exercise with a hand calculator.
a. Calculate the fitted values y; and their sample meanỹ. Compare this value to the sample mean of
the y values.
b. Calculate Σ ( – y) and Σ1 ( – 5)/Σ(-5).
c. The least squares residuals are ê; = 0.4,-0.8, 1, 0.6, and -1.2. Calculate
d. Calculate 1 -/-(-y)² and compare it to the results in part (b).
(-) ê,.
e. Show, algebraically, that Σ(-) (y;-) = Lii-Ny². Calculate this value.
f. Using Σ(x₁ - x)² = 10, and previous results, calculate
r=
r = [Σ²2 (0 - 3)(x - 5)] / [√ Σ (9; – 5)²³√Σ², (3; – 5) ²
-
What statistic is r? Calculate ² and compare this value to the values in parts (d) and (b).
4.7 We have data on 2323 randomly selected households consisting of three persons in 2013. Let ENTERT
denote the monthly entertainment expenditure ($) per person per month and let INCOME ($100) be
monthly household income. Consider the regression model
ENTERT, =B₁ + B₂INCOME; + e,, i = 1,..., 2323
Assume that assumptions SR1-SR6 hold. The OLS estimated equation is ENTERT, = 9.820 +
0.503INCOME,. The standard error of the slope coefficient estimator is se (b₂) = 0.029, the standard
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