A2 October 2020 Q1
1. Gina receives a large number of packages from two companies, \(A\) and \(B\). She believes that the variance of the weights of packages from company \(A\) is greater than the variance of the weights of packages from company \(B\).
Gina takes a random sample of 7 packages from company \(A\) and an independent random sample of 10 packages from company \(B\). Her results are summarised below
\[\bar{a} = 300 \qquad \mathrm{S}_{aa} = 145\,496 \qquad \bar{b} = 233.4 \qquad \mathrm{S}_{bb} = 56\,364.4\][You may assume that the weights of packages from the two companies are normally distributed.]
Test Gina’s belief. Use a 5% level of significance and state your hypotheses clearly. (6)
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{H}_0 : \sigma_a^{\,2} = \sigma_b^{\,2} \qquad \mathrm{H}_1 : \sigma_a^{\,2} \gt \sigma_b^{\,2}\) | B1 | 2.5 |
| \(s_a^{\,2} = \dfrac{145\,496}{6} = [24\,249.33\ldots] \qquad s_b^{\,2} = \dfrac{56\,364.4}{9} = [6262.711\ldots]\) | M1 | 1.1b |
| \(F_{6,9} = \dfrac{\text{“}24\,249.33\ldots\text{”}}{\text{“}6262.711\ldots\text{”}}\) | M1 | 2.1 |
| \(= 3.872\ldots\) | A1 | 1.1b |
| \(F_{6,9}\) (5% one-tail) c.v. = 3.37 | B1 | 1.1b |
| Significant, there is evidence to support Gina’s belief | A1 | 2.2b |
| (6) | ||
| (6 marks) |
Notes
1st B1 for both hypotheses in terms of \(\sigma\).
1st M1 for at least one of the \(s^2\) calculations correctly attempted
NB \(s_a = 155.72\ldots\) accept awrt 156 and \(s_b = 79.137\ldots\) accept awrt 79.1
2nd M1 for a correct calculation of the test statistic (ft their \(s^2\))
1st A1 for awrt 3.87
2nd B1 for correct cv awrt 3.37
2nd A1 for a correct conclusion mentioning Gina’s belief (o.e.)