S4 June 2006 Q2
2. The weights, in grams, of apples are assumed to follow a normal distribution.
The weights of apples sold by a supermarket have variance \(\sigma_s^2\). A random sample of 4 apples from the supermarket had weights
114, 110, 119, 123.
(a) Find a 95% confidence interval for \(\sigma_s^2\). (7)
The weights of apples sold on a market stall have variance \(\sigma_M^2\). A second random sample of 7 apples was taken from the market stall. The sample variance \(s_M^2\) of the apples was 318.8.
(b) Stating your hypotheses clearly test, at the 1% level of significance, whether or not there is evidence that \(\sigma_M^2 \gt \sigma_s^2\). (5)
| Scheme | Marks |
|---|---|
| \(\left(\bar{x} = \dfrac{466}{4} = 116.5\right) \qquad s_x^2 = \dfrac{54386 - 4\bar{x}^2}{3},\ = 32.\dot{3}\) or \(\dfrac{97}{3}\) or awrt 32.3 | M1, A1 |
| \(0.216 \lt \dfrac{3s_x^2}{\sigma^2} \lt 9.348\) | B1 M1 B1 |
| \(10.376\ldots \lt \sigma^2 \lt 449.07\ldots\) awrt 10.4, 449 | A1, A1 |
| (7) |
Notes
The question paper file gives the second weight as 100; the mark scheme’s working (\(\Sigma x = 466\), \(\Sigma x^2 = 54\,386\)) uses 110, so the question is shown here with 110.
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \sigma_M^2 = \sigma_s^2 \qquad \mathrm{H}_1 : \sigma_M^2 \gt \sigma_s^2\) both (\(\sigma_M = \sigma_s\), \(\sigma_M \gt \sigma_s\) are OK) | B1 |
| \(\dfrac{s_M^2}{s_s^2} = \dfrac{318.8}{32.\dot{3}} = 9.859\ldots\) awrt 9.86 | M1 A1 |
| \(F_{6,3}\) (1% c.v.) \(= 27.91\) | B1 |
| \(9.86 \lt 27.91\), insufficient evidence of an increase in variance … to say \(\sigma_M^2 \gt \sigma_s^2\) is OK. … variance can be assumed to be the same is OK | A1ft |
| (5) | |
| (12 marks) |
Notes
NB \(\dfrac{32.\dot{3}}{318.8} = 0.101\ldots\) only gets M1 A1 if appropriate F value attempted