# Mathematics

## Case Answers

1 | ||||||

Using ANOVA, we reject the null hypothesis that all ratings are the same; so at least one differs from the rest. | ||||||

Anova: Single Factor | ||||||

SUMMARY | ||||||

Groups | Count | Sum | Average | Variance | ||

Quality | 200 | 879 | 4.395 | 0.5818844221 | ||

Ease of Use | 200 | 833 | 4.165 | 0.6108291457 | ||

Price | 200 | 734 | 3.67 | 1.1367839196 | ||

Service | 200 | 828 | 4.14 | 0.7943718593 | ||

ANOVA | ||||||

Source of Variation | SS | df | MS | F | P-value | F crit |

Between Groups | 55.505 | 3 | 18.5016666667 | 23.6907048446 | 0 | 2.6160889565 |

Within Groups | 621.65 | 796 | 0.7809673367 | |||

Total | 677.155 | 799 | ||||

2 | ||||||

The proportion of on time deliveries in 2010 was 0.9850. | ||||||

We may test the null hypothesis that the proportion of on time deliveries in 2012 is < 0.985 to determine if it has improved (the alternate hypothesis is p > 0.985) | ||||||

The sample proportion for 2014 is 0.9907 | ||||||

z = (0.9907 – 0.985)/SQRT(.985*(1-0.985)) = | 0.046893334 | |||||

Critical value = 1.645 | ||||||

p-value = | 0.4812991205 | |||||

Therefore, we cannot conclude a significant improvement | ||||||

3 | A chart of the average number of defects by year shows a declining trend. | |||||

We may test for differences between 2010 and 2014 (assuming the samples are the monthly data since we don’t know the actual number of shipments) | ||||||

t-Test: Two-Sample Assuming Unequal Variances | ||||||

2008 | 2012 | |||||

Mean | 826.3333333333 | 496.25 | ||||

Variance | 135.3333333333 | 2940.0227272727 | ||||

Observations | 12 | 12 | ||||

Hypothesized Mean Difference | 0 | |||||

df | 12 | |||||

t Stat | 20.6189486406 | |||||

P(T<=t) one-tail | 0 | |||||

t Critical one-tail | 1.7822875556 | |||||

P(T<=t) two-tail | 0.0000000001 | |||||

t Critical two-tail | 2.1788128297 | |||||

The test clearly shows a significant difference in the mean defect rates. | ||||||

4 | Testing hypotheses that the mean cost has improved for one of the new processes, we cannot conclude a signficant improvement. | |||||

t-Test: Two-Sample Assuming Unequal Variances | ||||||

Current | Process A | |||||

Mean | 289.6 | 285.5 | ||||

Variance | 2061.1448275862 | 4217.6379310345 | ||||

Observations | 30 | 30 | ||||

Hypothesized Mean Difference | 0 | |||||

df | 52 | |||||

t Stat | 0.2834045089 | |||||

P(T<=t) one-tail | 0.3889960249 | |||||

t Critical one-tail | 1.6746891537 | |||||

P(T<=t) two-tail | 0.7779920499 | |||||

t Critical two-tail | 2.0066468051 | |||||

t-Test: Two-Sample Assuming Unequal Variances | ||||||

Current | Process B | |||||

Mean | 289.6 | 298.4333333333 | ||||

Variance | 2061.1448275862 | 435.3574712644 | ||||

Observations | 30 | 30 | ||||

Hypothesized Mean Difference | 0 | |||||

df | 41 | |||||

t Stat | -0.9683208014 | |||||

P(T<=t) one-tail | 0.1692809435 | |||||

t Critical one-tail | 1.6828780021 | |||||

P(T<=t) two-tail | 0.338561887 | |||||

t Critical two-tail | 2.0195409704 | |||||

5 | Conduct two sample tests on mean years at PLE for each factor. | |||||

t-Test: Two-Sample Assuming Unequal Variances | ||||||

Female | Male | |||||

Mean | 5.5307692308 | 5.5407407407 | ||||

Variance | 12.2506410256 | 6.4494301994 | ||||

Observations | 13 | 27 | ||||

Hypothesized Mean Difference | 0 | |||||

df | 18 | |||||

t Stat | -0.0091747587 | |||||

P(T<=t) one-tail | 0.4963903133 | |||||

t Critical one-tail | 1.7340636066 | |||||

P(T<=t) two-tail | 0.9927806266 | |||||

t Critical two-tail | 2.1009220402 | |||||

CONCLUSION: NO SIGNIFICANT DIFFERENCE BY GENDER | ||||||

t-Test: Two-Sample Assuming Unequal Variances | ||||||

College Grad N | College Grad Y | |||||

Mean | 4.8923076923 | 5.8481481481 | ||||

Variance | 5.8191025641 | 9.1095156695 | ||||

Observations | 13 | 27 | ||||

Hypothesized Mean Difference | 0 | |||||

df | 29 | |||||

t Stat | -1.0788151945 | |||||

P(T<=t) one-tail | 0.1447808862 | |||||

t Critical one-tail | 1.6991270265 | |||||

P(T<=t) two-tail | 0.2895617724 | |||||

t Critical two-tail | 2.0452296421 | |||||

CONCLUSION: NO SIGNIFICANT DIFFERENCE BY COLLEGE GRAD STATUS | ||||||

t-Test: Two-Sample Assuming Unequal Variances | ||||||

Local N | Local Y | |||||

Mean | 3.6294117647 | 6.947826087 | ||||

Variance | 5.6909558824 | 5.2726086957 | ||||

Observations | 17 | 23 | ||||

Hypothesized Mean Difference | 0 | |||||

df | 34 | |||||

t Stat | -4.4186414248 | |||||

P(T<=t) one-tail | 0.0000480512 | |||||

t Critical one-tail | 1.6909242552 | |||||

P(T<=t) two-tail | 0.0000961023 | |||||

t Critical two-tail | 2.0322445093 | |||||

CONCLUSION: SIGNIFICANT DIFFERENCE BY LOCAL AREA | ||||||

EMPLOYEES FROM THE LOCAL AREA HAVE GREATER RETENTION |

826.33333333333337 837.41666666666663 785.91666666666663 669.08333333333337 496.25

## Defects After Delivery

Defects After Delivery | |||||

Defects per million items received from suppliers | |||||

Month | 2010 | 2011 | 2012 | 2013 | 2014 |

January | 812 | 828 | 824 | 682 | 571 |

February | 810 | 832 | 836 | 695 | 575 |

March | 813 | 847 | 818 | 692 | 547 |

April | 823 | 839 | 825 | 686 | 542 |

May | 832 | 832 | 804 | 673 | 532 |

June | 848 | 840 | 812 | 681 | 496 |

July | 837 | 849 | 806 | 696 | 472 |

August | 831 | 857 | 798 | 688 | 460 |

September | 827 | 839 | 804 | 671 | 441 |

October | 838 | 842 | 713 | 645 | 445 |

November | 826 | 828 | 705 | 617 | 438 |

December | 819 | 816 | 686 | 603 | 436 |

Average | 826.3333333333 | 837.4166666667 | 785.9166666667 | 669.0833333333 | 496.25 |