stats number 8
hotjpka69
mastery_8.odt1.
For a sample of 41 New England cities, a sociologist studies the crime rate in each city (crimes per 100,000 residents) as a function of its poverty rate (in %) and its median income (in $1,000s). The regression results are shown.
ANOVA df SS MS F Significance F
Regression 2 3,594,749 1,797,374.5 12.23270 1.5E-04
Residual 28 4,114,095 146,932.0
Total 30 7,708,844
a.
Calculate the standard error of the estimate. (Round your answer to 2 decimal places.)
formula877.mml
b-1.
What proportion of the variability in crime rate is explained by the variability in the explanatory variables? (Round your answer to 4 decimal places.)
Explained proportion
b-2.
What proportion is unexplained? (Round your answer to 4 decimal places.)
Unexplained proportion
rev: 11_22_2013_QC_41534, 11_29_2013_QC_41646
2.
In a multiple regression with two explanatory variables and 117 observations, it is found that SSR = 4.51 and SST = 8.86.
a. Calculate the standard error of the estimate. (Round your answer to 2 decimal places.)
se
b. Calculate the coefficient of determination R2. (Round your answer to 4 decimal places.)
R2
c. Calculate adjusted R2. (Round your answer to 4 decimal places.)
Adjusted R2
rev: 09_16_2013_QC_34398
3.
The following ANOVA table was obtained when estimating a multiple regression.
ANOVA df SS MS F Significance F
Regression 2 188,875.00 94,437.50 53.98 5.55E-10
Residual 26 45,484.65 1,749.41
Total 28 234,359.65
a. Calculate the standard error of the estimate. (Round your answer to 2 decimal places.)
se
b-1. Calculate the coefficient of determination. (Round your answer to 4 decimal places.)
Coefficient of determination
b-2. Interpret the coefficient of determination.
The proportion of the variation in x that is explained by the regression model.
The proportion of the variation in y that is explained by the regression model.
c. Calculate adjusted R2. (Round your answer to 4 decimal places.)
Adjusted R2
rev: 09_16_2013_QC_34398, 12_06_2013_QC_42096, 12_18_2014_QC_CS-890
4.
The homeownership rate in the United States was 67.4% in 2009. In order to determine if homeownership is linked with income, 2009 state level data on the homeownership rate (Ownership) and median household income (Income) were collected. The data can be found on the text website, labeled Home Ownership.
State
Income
Ownership
Alabama
$38,990
72.3%
Alaska
$60,614
65.7%
Arizona
$44,749
67.4%
Arkansas
$35,548
66.6%
California
$55,144
56.0%
Colorado
$54,940
67.2%
Connecticut
$63,861
69.4%
Delaware
$51,124
75.0%
District of Columbia
$52,151
44.1%
Florida
$44,641
69.4%
Georgia
$42,350
65.9%
Hawaii
$54,659
58.4%
Idaho
$45,788
73.9%
Illinois
$51,880
67.8%
Indiana
$43,315
70.4%
Iowa
$49,731
71.0%
Kansas
$43,727
65.9%
Kentucky
$41,674
69.5%
Louisiana
$44,443
70.3%
Maine
$46,512
72.5%
Maryland
$63,196
68.5%
Massachusetts
$58,383
64.0%
Michigan
$45,004
72.9%
Minnesota
$55,100
71.6%
Mississippi
$34,088
73.4%
Missouri
$47,779
70.5%
Montana
$39,447
69.0%
Nebraska
$48,605
68.8%
Nevada
$50,444
61.2%
New Hampshire
$63,141
74.8%
New Jersey
$63,787
64.9%
New Mexico
$42,552
67.5%
New York
$49,226
53.3%
North Carolina
$40,916
68.4%
North Dakota
$49,085
64.4%
Ohio
$44,889
68.2%
Oklahoma
$44,888
68.1%
Oregon
$48,108
66.8%
Pennsylvania
$47,182
70.7%
Rhode Island
$50,644
61.7%
South Carolina
$40,111
72.6%
South Dakota
$44,836
68.1%
Tennessee
$39,527
69.4%
Texas
$46,485
64.0%
Utah
$57,501
72.8%
Vermont
$51,328
72.9%
Virginia
$59,511
68.6%
Washington
$59,402
64.4%
West Virginia
$39,500
76.8%
Wisconsin
$50,247
69.0%
Wyoming
$51,480
72.4%
SOURCE: www.census.gov
PictureClick here for the Excel Data File
|
State |
Income ($) |
Ownership (%) |
|
Alabama |
38,990 |
72.3 |
|
Alaska |
60,614 |
65.7 |
|
Arizona |
44,749 |
67.4 |
|
Arkansas |
35,548 |
66.6 |
|
California |
55,144 |
56.0 |
|
Colorado |
54,940 |
67.2 |
|
Connecticut |
63,861 |
69.4 |
|
Delaware |
51,124 |
75.0 |
|
District of Columbia |
52,151 |
44.1 |
|
Florida |
44,641 |
69.4 |
|
Georgia |
42,350 |
65.9 |
|
Hawaii |
54,659 |
58.4 |
|
Idaho |
45,788 |
73.9 |
|
Illinois |
51,880 |
67.8 |
|
Indiana |
43,315 |
70.4 |
|
Iowa |
49,731 |
71.0 |
|
Kansas |
43,727 |
65.9 |
|
Kentucky |
41,674 |
69.5 |
|
Louisiana |
44,443 |
70.3 |
|
Maine |
46,512 |
72.5 |
|
Maryland |
63,196 |
68.5 |
|
Massachusetts |
58,383 |
64.0 |
|
Michigan |
45,004 |
72.9 |
|
Minnesota |
55,100 |
71.6 |
|
Mississippi |
34,088 |
73.4 |
|
Missouri |
47,779 |
70.5 |
|
Montana |
39,447 |
69.0 |
|
Nebraska |
48,605 |
68.8 |
|
Nevada |
50,444 |
61.2 |
|
New Hampshire |
63,141 |
74.8 |
|
New Jersey |
63,787 |
64.9 |
|
New Mexico |
42,552 |
67.5 |
|
New York |
49,226 |
53.3 |
|
North Carolina |
40,916 |
68.4 |
|
North Dakota |
49,085 |
64.4 |
|
Ohio |
44,889 |
68.2 |
|
Oklahoma |
44,888 |
68.1 |
|
Oregon |
48,108 |
66.8 |
|
Pennsylvania |
47,182 |
70.7 |
|
Rhode Island |
50,644 |
61.7 |
|
South Carolina |
40,111 |
72.6 |
|
South Dakota |
44,836 |
68.1 |
|
Tennessee |
39,527 |
69.4 |
|
Texas |
46,485 |
64.0 |
|
Utah |
57,501 |
72.8 |
|
Vermont |
51,328 |
72.9 |
|
Virginia |
59,511 |
68.6 |
|
Washington |
59,402 |
64.4 |
|
West Virginia |
39,500 |
76.8 |
|
Wisconsin |
50,247 |
69.0 |
|
Wyoming |
51,480 |
72.4 |
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a-1.
Estimate the model Ownership = β0 + β1Income + ε. (Negative values should be indicated by a minus sign. Round your answers to 4 decimal places.)
y-hat = + Income
a-2. Interpret the model.
For a $1,000 increase in income, homeownership rate is predicted to decrease by 0.02%.
For a $1,000 increase in income, homeownership rate is predicted to increase by 0.02%.
For a $1,000 decrease in income, homeownership rate is predicted to increase by 0.01%.
For a $1,000 decrease in income, homeownership rate is predicted to decrease by 0.01%.
b. What is the standard error of the estimate? (Round your answer to 2 decimal places.)
se
c. Interpret the coefficient of determination.
4.63% of the sample variation in y is explained by the estimated regression equation.
4.63% of the sample variation in x is explained by the estimated regression equation.
3.63% of the sample variation in x is explained by the estimated regression equation.
5.63% of the sample variation in y is explained by the estimated regression equation.
rev: 09_16_2013_QC_34398, 11_01_2013_QC_34398
5.
Consider the following sample data:
x 26 32 16 31 10 30 34 32
y 35 53 40 38 28 47 29 35
PictureClick here for the Excel Data File
|
x |
26 |
32 |
16 |
31 |
10 |
30 |
34 |
32 |
|
y |
35 |
53 |
40 |
38 |
28 |
47 |
29 |
35 |
b.
Calculate b1 and b0. What is the sample regression equation? (Round intermediate calculations to 4 decimal places and final answers to 2 decimal places.)
y-hat = + x
c.
Find the predicted value for y if x equals 18, 23, and 28. (Round intermediate coefficient values and final answers to 2 decimal places.)
y-hat
If x = 18
If x = 23
If x = 28
rev: 09_16_2013_QC_34398, 10_31_2013_QC_34398, 11_21_2013_QC_34398, 12_17_2013_QC_34398
6.
In a simple linear regression based on 28 observations, it is found that b1 = 7.1 and se(bj) = 1.9. Consider the hypotheses (Use Table 2):
H0: β1 ≥ 10 and HA: β1 < 10
a.
At the 5% significance level, find the critical value(s). (Negative value should be indicated by a minus sign. Round your answer to 3 decimal places.)
Critical value
b.
Calculate the value of the appropriate test statistic. (Negative value should be indicated by a minus sign. Round your answer to 2 decimal places.)
Test statistic
c. At the 5% significance level, what is the conclusion to the hypothesis test? Is the slope coefficient less than 10?
Do not reject H0Picture the slope coefficient is not less than 10.
Reject H0Picture the slope coefficient is less than 10.
Do not reject H0Picture the slope coefficient is less than 10.
Reject H0Picture the slope coefficient is not less than 10.
rev: 11_01_2013_QC_34398
7.
Using data from 50 workers, a researcher estimates Wage = β0 + β1 Education + β2 Experience +β3 Age + ε, where Wage is the hourly wage rate and Education, Experience, and Age are the years of higher education, the years of experience, and the age of the worker, respectively. A portion of the regression results is shown in the following table.
Coefficients Standard Error t Stat p-value
Intercept 7.95 4.29 1.19 0.0641
Education 1.87 0.40 3.51 0.0002
Experience 0.48 0.19 3.34 0.0027
Age −0.01 0.06 −0.19 0.7170
a-1. What is the point estimate for β1?
1.87
0.48
a-2. Interpret this value.
As Education increases by 1 unit, Wage is predicted to increase by 1.87 units.
As Education increases by 1 unit, Wage is predicted to increase by 0.48 units, holding Age and Experience constant.
As Education increases by 1 unit, Wage is predicted to increase by 0.48 units.
As Education increases by 1 unit, Wage is predicted to increase by 1.87 units, holding Age and Experience constant.
a-3. What is the point estimate for β2?
0.48
1.87
a-4. Interpret this value.
Same interpretation by using 1.87 or -0.01
As Experience increases by 1 unit, Wage is predicted to increase by 0.48 units, holding Age and Education constant.
b.
What is the sample regression equation? (Negative value should be indicated by a minus sign. Round your answers to 2 decimal places.)
y-hat = + Education + Experience + Age
c.
What is the predicted value for Age = 23, Education = 4 and Experience = 2. (Do not round intermediate calculations. Round your answer to 2 decimal places.)
y-hat
rev: 09_16_2013_QC_34398
8.
A social scientist would like to analyze the relationship between educational attainment and salary. He collects the following sample data, where Education refers to years of higher education and Salary is the individual’s annual salary in thousands of dollars:
Education 3 4 6 2 5 4 8 0
Salary $40 36 56 35 72 47 107 52
PictureClick here for the Excel Data File
|
Education |
3 |
4 |
6 |
2 |
5 |
4 |
8 |
0 |
|
Salary |
40 |
36 |
56 |
35 |
72 |
47 |
107 |
52 |
a.
Find the sample regression equation for the model: Salary = β0 + β1Education + ε. (Round intermediate calculations to 4 decimal places. Enter your answers in thousands rounded to 2 decimal places.)
formula537.mml + Education
b. Interpret the coefficient for education.
As Education increases by 1 unit, an individual’s annual salary is predicted to decrease by $7,000.
As Education increases by 1 unit, an individual’s annual salary is predicted to increase by $8,000.
As Education increases by 1 unit, an individual’s annual salary is predicted to increase by $7,000.
As Education inceases by 1 unit, an individual’s annual salary is predicted to decrease by $8,000.
c.
What is the predicted salary for an individual who completed 7 years of higher education? (Round intermediate coefficient values to 2 decimal places and final answer, in dollars, to the nearest whole number.)
formula723.mml $
rev: 09_16_2013_QC_34398, 10_31_2013_QC_34398
9.
Consider the following simple linear regression results based on 20 observations. Use Table 2.
Coefficients Standard Error t Stat p-value Lower 95% Upper 95%
Intercept 30.7705 4.6589 6.6047 0.0000 20.98 40.56
x1 0.1071 0.1879 0.5700 0.5757 −0.29 0.50
a-1. Choose the hypotheses to determine if the intercept differs from zero.
H0: β0 ≥ 0; HA: β0 < 0
H0: β0 ≤ 0; HA: β0 > 0
H0: β0 = 0; HA: β0 ≠ 0
a-2. At the 5% significance level, what is the conclusion to the hypothesis test? Does the intercept differ from zero?
Do not reject H0Picture the intercept differs from zero.
Do not reject H0Picture the intercept is greater than zero.
Reject H0Picture the intercept differs from zero.
Reject H0Picture the intercept is greater than zero.
b-1.
Construct the 95% confidence interval for the slope coefficient. (Negative values should be indicated by a minus sign. Round your intermediate calculations to 4 decimal places,"tα/2,df" value to 3 decimal places and final answers to 2 decimal places.)
Confidence interval to
b-2.
At the 5% significance level, does the slope differ from zero?
No, since the interval contains zero.
Yes, since the interval does not contain zero.
Yes, since the interval contains zero.
No, since the interval does not contain zero.
rev: 11_01_2013_QC_34398, 11_30_2013_QC_41780
10.
In a simple linear regression, the following information is given:
formula321.mml = − 29; formula726.mml= 48;
formula323.mml
formula324.mml
a.
Calculate b1. (Negative value should be indicated by a minus sign. Round your answer to 2 decimal places.)
b1
b.
Calculate b0. (Round intermediate calculations to 4 decimal places and final answer to 2 decimal places.)
b0
c-1.
What is the sample regression equation? (Negative value should be indicated by a minus sign. Round your answers to 2 decimal places.)
y-hat = + x
c-2.
Predict y if x equals −21.(Round intermediate coefficient values and final answer to 2 decimal places.)
y-hat
rev: 09_17_2013_QC_34398, 10_31_2013_QC_34398
