S4 June 2007 Q6
6. A butter packing machine cuts butter into blocks. The weight of a block of butter is normally distributed with a mean weight of 250 g and a standard deviation of 4 g. A random sample of 15 blocks is taken to monitor any change in the mean weight of the blocks of butter.
(a) Find the critical region of a suitable test using a 2% level of significance. (3)
(b) Assuming the mean weight of a block of butter has increased to 254 g, find the probability of a Type II error. (5)
| Scheme | Marks |
|---|---|
| \(\dfrac{\overline{X} - 250}{\frac{4}{\sqrt{15}}} \gt 2.3263\) or \(\dfrac{\overline{X} - 250}{\frac{4}{\sqrt{15}}} \lt -2.3263\) \(\pm\) 2.3262 | B1 M1 |
| \(\overline{X} \gt 252.40\ldots\) or \(\overline{X} \lt 247.6\ldots\) awrt 252 and 248 | A1 |
| (3) |
Notes
Only needs to try and find one side for M1
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(\overline{X} \lt 252.4 / \mu = 254) - \mathrm{P}(\overline{X} \lt 247.6 / \mu = 254)\) using their ‘252.4’ and ‘247.6’ | M1 |
| \(= \mathrm{P}\left(Z \lt \dfrac{252.4 - 254}{\frac{4}{\sqrt{15}}}\right) - \mathrm{P}\left(Z \lt \dfrac{247.6 - 254}{\frac{4}{\sqrt{15}}}\right)\) stand using \(4/\sqrt{15}\), 254 their ‘252.4’ or ‘247.6’ | M1 |
| \(= \mathrm{P}(Z \lt -1.5492) - \mathrm{P}(Z \lt -6.20)\) \(-1.5492\) and \(-6.20\) o.e. | A1 |
| \(= (1 - 0.9394) - (1 - 1)\) | M1 |
| \(= 0.0606\) | A1 |
| (5) |
Notes
Only need to see one of the standardisation for second M1
If consider only 252.4 and get 0.0606 they get M0 M1 A0 M1 A1 ie they can get 3/5