S3 June 2011 Q3
3. A factory manufactures batches of an electronic component. Each component is manufactured in one of three shifts. A component may have one of two types of defect, \(D_1\) or \(D_2\), at the end of the manufacturing process. A production manager believes that the type of defect is dependent upon the shift that manufactured the component. He examines 200 randomly selected defective components and classifies them by defect type and shift. The results are shown in the table below.
| Defect type Shift | \(D_1\) | \(D_2\) |
|---|---|---|
| First shift | 45 | 18 |
| Second shift | 55 | 20 |
| Third shift | 50 | 12 |
Stating your hypotheses, test, at the 10% level of significance, whether or not there is evidence to support the manager’s belief. Show your working clearly. (10)
| Scheme | Marks | ||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| M1A1 | ||||||||||||||||||||||||||||
| \(\mathrm{H}_0\) : Type of defect is independent of Shift (no association) \(\mathrm{H}_1\) : Type of defect is not independent of Shift (association) | B1 | ||||||||||||||||||||||||||||
| M1A1 | ||||||||||||||||||||||||||||
| \(\dfrac{(O - E)^2}{E} = 1.5934..\) or \(\dfrac{O_i^2}{E_i} - 200 = 201.5934 - 200 = 1.5934..\) awrt 1.59 | A1 | ||||||||||||||||||||||||||||
| \(\nu = (3 - 1)(2 - 1) = 2\) | B1 | ||||||||||||||||||||||||||||
| \(\chi^2_2(0.10) = 4.605\) | B1ft | ||||||||||||||||||||||||||||
| \(1.59 \lt 4.605\) so insufficient evidence to reject \(\mathrm{H}_0\) | M1 | ||||||||||||||||||||||||||||
| Insufficient evidence to support manager’s belief /claim. | A1 | ||||||||||||||||||||||||||||
| (10 marks) |
Notes
1st M1 for some use of \(\dfrac{\text{Row Total} \times \text{Col.Total}}{\text{Grand Total}}\) May be implied by correct \(E_i\)
1st A1 for all expected frequencies correct
B1 for both hypotheses. Must mention “defect” and “shift” at least once
Use of “relationship” or “correlation” or “connection” is B0
2nd M1 for at least two correct terms (as in 3rd or 4th column) or correct expressions with their \(E_i\)
2nd A1 for all correct terms. May be implied by a correct answer.(2 dp or better-allow eg 0.10…)
3rd M1 for a correct statement linking their test statistic and their cv . Must be \(\chi^2\) not normal.
4th A1 for a correct comment in context - must mention “manager’s belief” or “shift” and “defect type” - condone “relationship” or “connection” here but not “correlation”. No follow through e.g. “There is evidence of a relationship between shift and type of defect”