Question

Tasks: Review the SPSS data file provided for this lab. The SPSS data file contains data that was collected on 20 children by a researcher. You will analyze the data to determine if any of the following have an effect on a test score (score): hours spent studying (hours), anxiety (anxiety) or A-level entry points (a_points).

Part A: For each regression model indicated, provide the following answers on the hours that follow.

a. What is the ANOVA table?

b. What is the regression equation?

c. Conduct the test for the significance of the Overall Regression Model?

d. What is R² ?

e. What are the 95 % confidence intervals for the estimates of the regression coefficientsthe B/s?

f. Provide an interpretation of the slopes, b 's.

Answer 1

model 1:
a. Analysis of Variance Table

Response: score
Df Sum Sq Mean Sq F value Pr(>F)
hour 1 1539.55 1539.55 37.225 9.166e-06 ***
Residuals 18 744.45 41.36
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

b. regression equation:
score=22.164+0.992hour

Residuals:
Min 1Q Median 3Q Max
-10.6747 -4.1513 0.1568 4.4249 12.2210

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 22.1640 6.5257 3.396 0.00322 **
hour 0.9920 0.1626 6.101 9.17e-06 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.431 on 18 degrees of freedom
Multiple R-squared: 0.6741,   Adjusted R-squared: 0.656
F-statistic: 37.22 on 1 and 18 DF, p-value: 9.166e-06

c.p value of F statistics is very small, the regression model is significant.

d. R-sq=0.6741

e. upper internval= 0.992+2.093*0.1626=1.33

lower interval=0.992-2.093*0.1626=0.6517

f. if a children studies for 1 hr more then the score will increase by 0.992

model 2:
a.Analysis of Variance Table

Response: score
Df Sum Sq Mean Sq F value Pr(>F)
anxiety 1 3.78 3.778 0.0298 0.8648
Residuals 18 2280.22 126.679   

b. regression equation:
score=62.3043-0.0257anxiety
c.Residuals:
Min 1Q Median 3Q Max
-23.0707 -5.1583 0.9036 7.3662 21.0321

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 62.3043 7.9609 7.826 3.35e-07 ***
anxiety -0.0257 0.1488 -0.173 0.865
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 11.26 on 18 degrees of freedom
Multiple R-squared: 0.001654,   Adjusted R-squared: -0.05381
F-statistic: 0.02982 on 1 and 18 DF, p-value: 0.8648

p value of F statistics is very high, the independent variable is not useful.

model 3:
b.Analysis of Variance Table

Response: score
Df Sum Sq Mean Sq F value Pr(>F)
a_point 1 1735.27 1735.27 56.922 5.588e-07 ***
Residuals 18 548.73 30.49
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

b. regression equation:
score=-8.172+2.982 a_points
c.Residuals:
Min 1Q Median 3Q Max
-9.459 -2.644 -1.348 2.874 13.541

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -8.1721 9.2511 -0.883 0.389
a_point 2.9816 0.3952 7.545 5.59e-07 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.521 on 18 degrees of freedom
Multiple R-squared: 0.7597,   Adjusted R-squared: 0.7464
F-statistic: 56.92 on 1 and 18 DF, p-value: 5.588e-07

p value of F statistics is very small, the independent variable is useful.

model 4:
a.Analysis of Variance Table

Response: score
Df Sum Sq Mean Sq F value Pr(>F)
hour 1 1539.55 1539.55 41.218 6.341e-06 ***
anxiety 1 109.48 109.48 2.931 0.1051
Residuals 17 634.97 37.35
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

b. regression equation:
score=11.4949+1.0763hour+0.1452anxiety

c.
Residuals:
Min 1Q Median 3Q Max
-10.2048 -3.3697 -0.2313 3.8562 11.2935

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 11.49493 8.79176 1.307 0.208
hour 1.07627 0.16217 6.637 4.2e-06 ***
anxiety 0.14520 0.08481 1.712 0.105
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.112 on 17 degrees of freedom
Multiple R-squared: 0.722,   Adjusted R-squared: 0.6893
F-statistic: 22.07 on 2 and 17 DF, p-value: 1.881e-05

p value of F statistics is very small, the independent variables are useful.

model 5:
a.Analysis of Variance Table

Response: score
Df Sum Sq Mean Sq F value Pr(>F)
hour 1 1539.55 1539.55 68.200 2.362e-07 ***
a_point 1 360.69 360.69 15.978 0.0009327 ***
Residuals 17 383.76 22.57
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

b. regression equation:
score=-3.9251+0.4765hour+0.1452anxiety

c.Residuals:
Min 1Q Median 3Q Max
-8.258 -2.398 -1.001 2.697 7.595

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -3.9251 8.1143 -0.484 0.634754
hour 0.4765 0.1762 2.703 0.015069 *
a_point 1.9945 0.4990 3.997 0.000933 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.751 on 17 degrees of freedom
Multiple R-squared: 0.832,   Adjusted R-squared: 0.8122
F-statistic: 42.09 on 2 and 17 DF, p-value: 2.604e-07

p value of F statistics is very small, the independent variables are useful.

model 6:
a.Analysis of Variance Table

Response: score
Df Sum Sq Mean Sq F value Pr(>F)
anxiety 1 3.78 3.78 0.120 0.7333
a_point 1 1745.09 1745.09 55.438 9.567e-07 ***
Residuals 17 535.13 31.48
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’

b. regression equation:
score=-11.5+0.04921anxiety+3.01736 a_points
c.Residuals:
Min 1Q Median 3Q Max
-9.7012 -3.3262 -0.8776 3.5782 13.4956

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -11.50001 10.67728 -1.077 0.297
anxiety 0.04921 0.07486 0.657 0.520
a_point 3.01736 0.40525 7.446 9.57e-07 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.611 on 17 degrees of freedom
Multiple R-squared: 0.7657,   Adjusted R-squared: 0.7381
F-statistic: 27.78 on 2 and 17 DF, p-value: 4.396e-06

p value of F statistics is very small, the independent variables are useful.

model 7:
a.Analysis of Variance Table

Response: score
Df Sum Sq Mean Sq F value Pr(>F)
hour 1 1539.55 1539.55 76.9158 1.65e-07 ***
anxiety 1 109.48 109.48 5.4695 0.03266 *
a_point 1 314.71 314.71 15.7231 0.00111 **
Residuals 16 320.26 20.02   
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

b.summaregression equation:
score=-10.6324+0.5708hours+0.1116anxiety+1.8803a_point

c.Residuals:
Min 1Q Median 3Q Max
-8.5698 -2.4992 -0.9947 3.3778 6.7838

Coefficients:
Estimate Std. Error t value Pr(>|t|)   
(Intercept) -10.63243 8.51831 -1.248 0.22992   
hour 0.57076 0.17420 3.276 0.00475 **
anxiety 0.11161 0.06266 1.781 0.09388 .
a_point 1.88030 0.47420 3.965 0.00111 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.474 on 16 degrees of freedom
Multiple R-squared: 0.8598,   Adjusted R-squared: 0.8335
F-statistic: 32.7 on 3 and 16 DF, p-value: 4.667e-07

p value of F statistics is very small, the independent variables are useful.