# Applied Statistics

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Introduction

NAPIER UNIVERSITY

SCHOOL OF MATHEMATICS AND STATISTICS

MODULE MA32808

APPLIED STATISTICS

MULTIPLE REGRESSION COURSEWORK

Student: Nicolas LEGRAIS 07007619

Assessor: Phillip DARBY

Moderator: Dr Sandra BONELLIE

1/ In order to obtain an equation to predict the quality of the product, we used a model with all the variables.

Equation to predict the quality of the product, ignoring the variable shift:

Qualprod= -10,354+0,041*Temp1+0,002*Temp2+0,671*Recycle+0,620*Qualraw

This model has an R-Square value of 0,952 (95,2% of the variations are explained by the variables) but this equation isn’t the good one because with have too much variables with the high sig. (Appendix Q1)

Prediction of the mean quality of the product (with a 95% confidence interval) if the following settings were used:

a/ Temp1=200 Temp2=300 Recycle=4% Qualraw=15

Prediction of the mean quality: PRE_1=10,4458

Mean confidence interval: [LMCI_1 ; UNCI_1]=[6,6514 ; 14,2402]

b/ Temp1=200 Temp2=300 Recycle=14% Qualraw=15

Prediction of the mean quality: PRE_1=17,1519

Mean confidence interval: [LMCI_1 ; UNCI_1]=[6,2760 ; 28,0278]

- We saw in 1/ that we can’t accept the simple model because there were too much variables with a high sig. Thus we have used different approaches to variable selection in order to obtain the final equation.

We used Stepwise regression, Backward elimination and Forward selection. In each approach we can see that we obtained the same R-Square value of 0,952 but we also obtained a better equation than with the simple model. We can also see that in each approach the variable Temp2 has been dropped.

Middle

,042

3

(Constant)

-9,726

2,796

-3,478

,002

Qualraw

,620

,088

,665

7,053

,000

Temp1

,041

,013

,278

3,100

,005

Recycle

,642

,210

,145

3,056

,005

a Dependent Variable: Qualprod

Variables Entered/Removed(b)

Model | Variables Entered | Variables Removed | Method |

1 | Qualraw, Temp2, Recycle, Temp1(a) | . | Enter |

2 | . | Temp2 | Backward (criterion: Probability of F-to-remove >= ,100). |

a All requested variables entered.

b Dependent Variable: Qualprod

Model Summary(c)

Model | R | R Square | Adjusted R Square | Std. Error of the Estimate |

1 | ,976(a) | ,952 | ,945 | 2,5823 |

2 | ,976(b) | ,952 | ,947 | 2,5326 |

a Predictors: (Constant), Qualraw, Temp2, Recycle, Temp1

b Predictors: (Constant), Qualraw, Recycle, Temp1

c Dependent Variable: Qualprod

ANOVA(c)

Model | Sum of Squares | Df | Mean Square | F | Sig. | |

1 | Regression | 3322,691 | 4 | 830,673 | 124,570 | ,000(a) |

Residual | 166,708 | 25 | 6,668 | |||

Total | 3489,399 | 29 | ||||

2 | Regression | 3322,639 | 3 | 1107,546 | 172,681 | ,000(b) |

Residual | 166,760 | 26 | 6,414 | |||

Total | 3489,399 | 29 |

a Predictors: (Constant), Qualraw, Temp2, Recycle, Temp1

b Predictors: (Constant), Qualraw, Recycle, Temp1

c Dependent Variable: Qualprod

Coefficients(a)

Model | Unstandardized Coefficients | Standardized Coefficients | t | Sig. | ||

B | Std. Error | Beta | ||||

1 | (Constant) | -10,354 | 7,686 | -1,347 | ,190 | |

Temp1 | ,041 | ,014 | ,277 | 3,007 | ,006 | |

Temp2 | ,002 | ,025 | ,008 | ,088 | ,931 | |

Recycle | ,671 | ,390 | ,152 | 1,720 | ,098 | |

Qualraw | ,620 | ,090 | ,666 | 6,911 | ,000 | |

2 | (Constant) | -9,726 | 2,796 | -3,478 | ,002 | |

Temp1 | ,041 | ,013 | ,278 | 3,100 | ,005 | |

Recycle | ,642 | ,210 | ,145 | 3,056 | ,005 | |

Qualraw | ,620 | ,088 | ,665 | 7,053 | ,000 |

a Dependent Variable: Qualprod

Variables Entered/Removed(a)

Model | Variables Entered | Variables Removed | Method |

1 | Qualraw | . | Forward (Criterion: Probability-of-F-to-enter <= ,050) |

2 | Temp1 | . | Forward (Criterion: Probability-of-F-to-enter <= ,050) |

3 | Recycle | . | Forward (Criterion: Probability-of-F-to-enter <= ,050) |

a Dependent Variable: Qualprod

Model Summary(d)

Model | R | R Square | Adjusted R Square | Std. Error of the Estimate |

1 | ,961(a) | ,924 | ,921 | 3,0767 |

2 | ,967(b) | ,935 | ,930 | 2,8974 |

3 | ,976(c) | ,952 | ,947 | 2,5326 |

a Predictors: (Constant), Qualraw

b Predictors: (Constant), Qualraw, Temp1

c Predictors: (Constant), Qualraw, Temp1, Recycle

d Dependent Variable: Qualprod

ANOVA(d)

Model | Sum of Squares | Df | Mean Square | F | Sig. | |

1 |

Conclusion

,002

Qualraw

,620

,088

,665

7,053

,000

Temp1

,041

,013

,278

3,100

,005

Recycle

,642

,210

,145

3,056

,005

a Dependent Variable: Qualprod

APPENDIX Q1b

Coefficients(a)

Model | Unstandardized Coefficients | Standardized Coefficients | t | Sig. | Collinearity Statistics | |||

B | Std. Error | Beta | Tolerance | VIF | ||||

1 | (Constant) | -10,354 | 7,686 | -1,347 | ,190 | |||

Temp1 | ,041 | ,014 | ,277 | 3,007 | ,006 | ,225 | 4,442 | |

Temp2 | ,002 | ,025 | ,008 | ,088 | ,931 | ,260 | 3,839 | |

Recycle | ,671 | ,390 | ,152 | 1,720 | ,098 | ,246 | 4,067 | |

Qualraw | ,620 | ,090 | ,666 | 6,911 | ,000 | ,206 | 4,858 | |

2 | (Constant) | -9,726 | 2,796 | -3,478 | ,002 | |||

Temp1 | ,041 | ,013 | ,278 | 3,100 | ,005 | ,228 | 4,378 | |

Recycle | ,642 | ,210 | ,145 | 3,056 | ,005 | ,814 | 1,228 | |

Qualraw | ,620 | ,088 | ,665 | 7,053 | ,000 | ,207 | 4,841 |

a Dependent Variable: Qualprod

APPENDIX Q2a

Means

Case Processing Summary

Cases | ||||||

Included | Excluded | Total | ||||

N | Percent | N | Percent | N | Percent | |

Qualprod * Shift | 30 | 100,0% | 0 | ,0% | 30 | 100,0% |

Report

Qualprod

Shift | N | Mean | Median | Std. Deviation | Minimum | Maximum | Range |

Dayshift | 16 | 11,206 | 11,550 | 4,8480 | 2,6 | 18,3 | 15,7 |

Nightshift | 14 | 30,036 | 29,000 | 6,1366 | 22,4 | 45,8 | 23,4 |

Total | 30 | 19,993 | 18,200 | 10,9692 | 2,6 | 45,8 | 43,2 |

T-Test

Group Statistics

Shift | N | Mean | Std. Deviation | Std. Error Mean | |

Qualprod | Dayshift | 16 | 11,206 | 4,8480 | 1,2120 |

Nightshift | 14 | 30,036 | 6,1366 | 1,6401 |

Independent Samples Test

Levene's Test for Equality of Variances | t-test for Equality of Means | |||||||||

F | Sig. | t | df | Sig. (2-tailed) | Mean Difference | Std. Error Difference | 95% Confidence Interval of the Difference | |||

Lower | Upper | |||||||||

Qual prod | Equal variances assumed | ,141 | ,711 | -9,382 | 28 | ,000 | -18,8295 | 2,0070 | -22,9405 | -14,7184 |

Equal variances not assumed | -9,233 | 24,693 | ,000 | -18,8295 | 2,0393 | -23,0322 | -14,6268 |

APPENDIX Q2b

Variables Entered/Removed(b)

Model | Variables Entered | Variables Removed | Method |

1 | Shift, Temp2, Qualraw, Recycle, Temp1(a) | . | Enter |

a All requested variables entered.

b Dependent Variable: Qualprod

Model Summary

Model | R | R Square | Adjusted R Square | Std. Error of the Estimate |

1 | ,976(a) | ,953 | ,944 | 2,6033 |

a Predictors: (Constant), Shift, Temp2, Qualraw, Recycle, Temp1

ANOVA(b)

Model | Sum of Squares | Df | Mean Square | F | Sig. | |

1 | Regression | 3326,751 | 5 | 665,350 | 98,178 | ,000(a) |

Residual | 162,648 | 24 | 6,777 | |||

Total | 3489,399 | 29 |

a Predictors: (Constant), Shift, Temp2, Qualraw, Recycle, Temp1

b Dependent Variable: Qualprod

Coefficients(a)

Model | Unstandardized Coefficients | Standardized Coefficients | t | Sig. | ||

B | Std. Error | Beta | ||||

1 | (Constant) | -9,677 | 7,798 | -1,241 | ,227 | |

Temp1 | ,035 | ,015 | ,241 | 2,318 | ,029 | |

Temp2 | ,006 | ,026 | ,019 | ,220 | ,827 | |

Recycle | ,666 | ,393 | ,151 | 1,695 | ,103 | |

Qualraw | ,594 | ,096 | ,638 | 6,163 | ,000 | |

Shift | 1,596 | 2,062 | ,074 | ,774 | ,446 |

a Dependent Variable: Qualprod

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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