Beginning Statistics, 2nd Edition
Beginning Statistics, 2nd Edition
2nd Edition
ISBN: 9781932628678
Author: Carolyn Warren; Kimberly Denley; Emily Atchley
Publisher: Hawkes Learning Systems
Question
Book Icon
Chapter 12.3, Problem 9E
To determine

(a)

To find:

The sum of squared errors, SSE.

Expert Solution
Check Mark

Answer to Problem 9E

Solution:

The equation of the least squares regression line is y^=908.07+313.98x and the sum of

Squared error (SSE) is 1148767.96.

Explanation of Solution

Given Information:

The numbers of parking tickets students received during one semester and their monthly

Parking Tickets and Monthly Income
Number of Tickets, Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  1 10 8 3 2 0 5 4 2 1 0
Monthly Income
(in Dollars), Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  2
4000 3800 1500 2000 870 2500 1800 1200 1400

The least Squares regression line is the line for which the average variation from the data is the smallest, also called the line of best fit, given by y^=b0+b1x.

Where Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  3 is the slope of the least-squares regression line for paired data from a sample andBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  4 is the Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  5-intercept for the regression line.

The sum of squared errors (SSE) for a regression line is the sum of the squares of the residuals.

Formula used:

The equation of least-squares regression line is given by

y^=b0+b1x

WhereBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  6, is the slope of the least-squares regression line given as,

b1=nxiyi(xi)(yi)nxi2(xi)2 and b0 is Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  7intercept given as,

b0=yinb1xin

Here n is the number of data pairs in the sample, Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  8 is the Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  9 value of the explanatory variable and Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  10 is the Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  11value of response variable.

The sum of squared errors (SSE) for a regression line is the sum of the squares of the residuals is given by,

SSE=(yiy^)2

Here Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  12 is the Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  13 observed value of the response variable and Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  14 is the predicted variable of Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  15using the list square regression model.

Calculation:

The table of the numbers of parking tickets students received during one semester and

their monthly Incomes is given by,

Number of Tickets,
xi
Monthly Income
(in Dollars), yi
xiyi xi2 yi2
10 4000 40000 100 16000000
8 3800 30400 64 14440000
3 1500 4500 9 2250000
2 2000 4000 4 4000000
0 870 0 0 756900
5 2500 12500 25 6250000
4 1800 7200 16 3240000
2 1000 2000 4 1000000
1 1200 1200 1 1440000
0 1400 0 0 1960000
xi=35 yi=20,070 xiyi=101,800 xi2=223 yi2=51,336,900

Where, xi=x1+x2+x3+...+x10.

Substitute 10 forBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  16, 8 forBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  17, 3 forBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  18… 0 for Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  19 in the above equation,

xi=10+8+3+...+0=35

yi=y1+y2+y3+...+y10

Substitute Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  20 forBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  21, Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  22 forBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  23,Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  24 forBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  25Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  26 for Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  27 in the above equation,

yi=4000+3800+1500+...+1400=20,070

In order to calculate the value of xiyi, substitute 10 for Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  28 and 4000 for Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  29 in xiyi.

xiyi=(10)(4000)=40000

Proceed in the same manner to calculate xiyi for the rest of the data and refer table for the rest of the xiyi values calculated.

xiyi=40000+30400+4500+...+0=101,800

In order to calculate the value ofBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  30, substitute 10 for Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  31 inBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  32.

xi2=(10)(10)=100

Proceed in the same manner to calculateBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  33 for the rest of the data and refer table for the rest of the Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  34 values calculated.

xi2=100+64+9+...+0=223

In order to calculate the value of Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  35substitute 4000 for Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  36 in Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  37.

yi2=(4000)(4000)=16000000

Proceed in the same manner to calculateBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  38 for the rest of the data and refer table for the rest of the Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  39 values calculated.

yi2=16000000+14400000+2250000+...+1960000=51,336,900

The slope of the least-squares regression line is given as,

b1=nxiyi(xi)(yi)nxi2(xi2)2

Substitute 35 for xi, 20, 070 for yi, 101, 800 for xiyi, 223 for xi2 and 51, 336, 900 for Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  40 in the above equation ofBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  41,

b1=10(101,800)(35)(20,070)10(223)(35)2=3155501005=313.98

The Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  42-intercept of regression line is given as,

b0=yinb1xin

Substitute 35 for xi, 20, 070 for yi and 313.98 for b0

b0=20,07010(313.987)3510=20071098.93=908.07

The equation of least-squares regression line is,

y^=908.07+313.98x

The table of the numbers of parking tickets students received during one semester and

their monthly incomes is given by,

Number of Tickets,
xi
Monthly Income
(in Dollars), yi
Predicted value, yi Residual, (yiyi^) Squared
error, (yiyi^)2
10 4000 4047.87 -47.87 2291.5369
8 3800 3419.91 380.09 144468.4081
3 1500 1850.01 -350.01 122507.0001
2 2000 1536.03 463.97 215268.1609
0 870 908.07 -38.07 1449.3249
5 2500 2477.97 22.03 485.3209
4 1800 2163.99 -363.99 132488.7201
2 1000 1536.03 -536.03 287328.1609
1 1200 1222.05 -22.05 486.2025
0 1400 908.07 491.93 241995.1249
xi=35 yi=20,070 yi^=20,070 (yiyi^)=0 (yiyi^)2=1148767.96

The predicted values Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  43 are obtained by substituting the values of Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  44 in the fitted regression line,

y^=908.07+313.98x

ForBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  45, substitute 10 for Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  46in the above equation,

y^=908.07+313.98x=908.07+313.98(10)=4047.87

Similarly the other values of Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  47 are obtained and are shown in the third column of the table:

The residuals (yiyi^) in column four of the table are obtained by the difference of valued of Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  48and Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  49.

Substitute 4000 for Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  50and 4047.87 for Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  51in above equation,

(yiyi^)=40004047.87=47.87

Similarly the other values of (yiyi^) are obtained and are shown in the fourth column of the table.

The squared errors are obtained as, (yiyi^)2

Substitute Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  52for Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  53 in (yiyi^),

Thus, (yiyi^)=2291.5369

Similarly the other values of (yiyi^)2 are obtained and are shown in the last column of the table.

The sum of squared errors (SSE) for a regression line is,

SSE=(yiyi^)2

From the above table SSE is given by,

(yiyi^)2=2291.5369+144468.4081+...+241995.12489=1148767.96

Thus, the sum of squared error (SSE) is 114767.96.

Conclusion:

The equation of least-squares regression line is,

y^=908.07+313.98x

The sum of squared errors (SSE) is 1148767.96.

To determine

(b)

To find:

The Standard error of estimate, Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  54.

Expert Solution
Check Mark

Answer to Problem 9E

Solution:

The Standard error of estimate Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  55is 378.94.

Explanation of Solution

Given Information:

The numbers of parking tickets students received during one semester and their monthly

Parking Tickets and Monthly Income
Number of Tickets, Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  56 10 8 3 2 0 5 4 2 1 0
Monthly Income
(in Dollars), Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  57
4000 3800 1500 2000 870 2500 1800 1000 1200 1400

Incomes is,

Formula used:

The standard error of estimate is a measure of the deviation of the sample data points from the regression line and is given by:

Se=(yiyi^)2n2=SSEn2

Here SSE is the sum of squared error and Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  58 is the number of paired data set in the sample, Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  59 is the Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  60 observed value of the response variable,

Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  61 is the predicted variable of Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  62using the list square regression model.

Calculation:

The formula of Standard error of estimate is,

Se=(yiyi^)2n2=SSEn2

From part a substitute Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  63 for SSE andBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  64 for Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  65 in the above equation of Standard error of estimate.

Se=1148767.96102=143595.995=378.94

Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  66

Conclusion:

Thus, the Standard error of estimate Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  67 is 378.94.

To determine

(c)

To Construct:

A 95% prediction interval for the given value of explanatory variable, Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  68.

Expert Solution
Check Mark

Answer to Problem 9E

Solution:

The prediction interval is (2032.14, 3551.76).

Explanation of Solution

The prediction interval is a confidence interval for an individual value of the response

Variable y, at a given fixed value of the explanatory variable, x.

Formula used:

The formula to calculate the margin of error of a prediction interval for an individual value of the response variable, y, is given by,

E=tα/2Se1+1n+n(x0x¯)2n(xi2)(xi)2.

Se is the Standard error of estimate,.

n is the number of data pairs in the sample.

x0 is the fixed value of the explanatory variable, x.

x¯ is the mean of the x-values for the data points in the sample.

xi is the ith value of the explanatory variable.

The formula to calculate the Prediction interval is given by,

y^E<Y<y^+E

or

(y^E,y^+E).

Where y^ is the predicted value of the response variable, y, when x=x0 and E is the

margin of error

The formula to calculate arithmetic mean is given by,

x¯=x1+x2+.....+xnn

Where x1, x2,…. xn are explanatory variables and n is the total number of explanatory

Variables.

Calculation:

Arithmetic Mean is given as,

x¯=x1+x2+.....+xnn

From part a substitute 10 forBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  69, 10 forBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  70, 8 for Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  71…..0 for Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  72 in the above equation of arithmetic mean,

x¯=10+8+..+010=3510=3.5

Since the level of confidence is 95%,

α=10.95=0.05

Then using the t-distribution table, the critical value for this test

tα/2=t0.05/2=t0.025

For Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  73-distribution with 8 degrees of freedom,

t0.025=1.86

The formula to calculate the margin of error of a prediction interval is,

E=tα/2Se1+1n+n(x0x¯)2n(xi2)(xi)2

Substitute 1.86 forBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  74, 3.5 forBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  75, 10 forBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  76, 35 forBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  77, 223 forBeginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  78, 6 for Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  79 and Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  80for Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  81in the above equation of margin of error,

E=(1.86)(378.94)1+110+10(63.5)210(223)(35)2=759.81

Predicted value Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  82 for Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  83 is

y^=908.07+313.98x=908.07+313.98(6)=2791.95Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  84

The prediction interval is:

(y^E,y^+E)

Substitute 2791.95for Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  85and 759.81 for Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  86in the above formula,

prediction interval=(2791.95-759.81,2791.95+759.81)=(2032.14,3551.76)

The prediction interval is (2032.14,3551.76).

Conclusion:

Thus, the prediction interval is (2032.14,3551.76).

To determine

(d)

To Construct:

A 95% confidence interval for the Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  87intercept of regression line.

Expert Solution
Check Mark

Answer to Problem 9E

Solution:

The 95% confidence interval for the Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  88intercept Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  89 is (496.45, 1319.69).

Explanation of Solution

Given Information:

The numbers of parking tickets students received during one semester and their monthly

Incomes is,

Parking Tickets and Monthly Income
Number of Tickets, x 10 8 3 2 0 5 4 2 1 0
Monthly Income 4000 3800 1500 2000 870 2500 1800 1000 1200 1400
(in Dollars), y

Calculation:

To generate the regression table in excel follow the given steps:

1. Under data tab, choose data analytics and then select regression.

2. Select the input Y range and enter the range of the given Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  90 and select the input X range and enter the range of the given Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  91 data.

3.Choose 95% confidence interval and click OK.

The following table will appear.

SUMMARY OUTPUT

Regression Statistics
Multiple R 0.94662523
R Square 0.89609937
Adjusted R Square 0.88311172
Standard Error 378.940622
Observations 10

ANOVA
  df SS MS F Significance F
Regression 1 9907642.04 9907642 68.99665 3.32835E-05
Residual 8 1148767.96 143596
Total 9 11056410      
Coefficients Standard Error t Stat P-value Lower 95.0% Upper 95.0%
Intercept 908.0696517 178.5009634 5.087197 0.000945 496.4456924 1319.69361
X 313.9800995 37.79968083 8.306422 3.33E-05 226.8138793 401.14632

From the above result,

1. Multiple R is the absolute value of the Correlation Coefficient, Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  92.

2. R2 is the Coefficient of determination, r2.

3. Standard Error is the standard error of the estimate, Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  93.

4. The intersection of the Residual row and the SS column is the sum of squared errors, SSE.

5. The Lower 95.0% and the Upper 95.0% columns give the lower and upper endpoints of the 95% confidence intervals for the Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  94intercept and slope.

6. The coefficient columns gives the values for the coefficients, that is, the

Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  95intercept and slope, of the regression line.

The row labeled Intercept is the row for the values corresponding to the Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  96intercept. The last two values in this row are the lower and upper endpoints for a 95% confidence for the Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  97intercept of the regression line, β0.

Thus, the 95% confidence interval for β0 is,

496.45<β0<1319.69

Confidence Interval=(496.45, 1319.69).

Conclusion:

Thus, the 95% confidence interval for the Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  98intercept Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  99 is(496.45, 1319.69).

To determine

(e)

To Construct:

A 95% confidence interval for the slope of regression line.

Expert Solution
Check Mark

Answer to Problem 9E

Solution:

Thus, the 95% confidence interval for the slope β1 is(226.81, 401.15).

Explanation of Solution

Given Information:

The numbers of parking tickets students received during one semester and their monthly

Incomes is,

Parking Tickets and Monthly Income
Number of Tickets, x 10 8 3 2 0 5 4 2 1 0
Monthly Income 4000 3800 1500 2000 870 2500 1800 1000 1200 1400
(in Dollars), y

Calculation:

To generate the regression table in excel follow the given steps:

1. Under data tab, choose data analytics and then select regression.

2. Select the input Y range and enter the range of the given Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  100 and select the input X range and enter the range of the given Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  101 data.

3.Choose 95% confidence interval and click OK.

The following table will appear.

SUMMARY OUTPUT

Regression Statistics
Multiple R 0.946625253
R Square 0.89609937
Adjusted R Square 0.883111792
Standard Error 378.940622
Observations 10
ANOVA
  df SS MS F Significance F
Regression 1 9907642.04 9907642 68.99665 3.32835E-05
Residual 8 1148767.96 143596
Total 9 11056410      
Coefficients Standard Error t Stat P-value Lower 95.0% Upper 95.0%
Intercept 908.0696517 178.5009634 5.087197 0.000945 496.4456924 1319.69361
X 313.9800995 37.79968083 8.306422 3.33E-05 226.8138793 401.14632

From the above result,

1. Multiple R is the absolute value of the Correlation Coefficient, Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  102.

2. R2 is the Coefficient of determination, r2.

3. Standard Error is the standard error of the estimate, Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  103.

4. The intersection of the Residual row and the SS column is the sum of squared errors, SSE.

5. The Lower 95.0% and the Upper 95.0% columns give the lower and upper endpoints of the 95% confidence intervals for the Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  104intercept and slope.

6. The coefficient columns gives the values for the coefficients, that is, the

Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  105intercept and slope, of the regression line.

The row labeled Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  106 is the row for the values corresponding to the slope of regression line. The last two values in this row are the lower and upper endpoints for a 95% confidence for the slope of the regression line, β1.

Thus, the 95% confidence interval for β1 is,

226.81<β1<401.15

Confidence Interval=(226.81, 401.15).

Conclusion:

Thus, the 95% confidence interval for the slope Beginning Statistics, 2nd Edition, Chapter 12.3, Problem 9E , additional homework tip  107 is(226.81, 401.15).

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