S4 June 2008 Q5
5. A machine is filling bottles of milk. A random sample of 16 bottles was taken and the volume of milk in each bottle was measured and recorded. The volume of milk in a bottle is normally distributed and the unbiased estimate of the variance, \(s^2\), of the volume of milk in a bottle is 0.003
(a) Find a 95% confidence interval for the variance of the population of volumes of milk from which the sample was taken. (5)
The machine should fill bottles so that the standard deviation of the volumes is equal to 0.07
(b) Comment on this with reference to your 95% confidence interval. (3)
| Scheme | Marks |
|---|---|
| Confidence interval \(= \left(\dfrac{15 \times 0.003}{27.488},\ \dfrac{15 \times 0.003}{6.262}\right)\) | M1 B1 B1 |
| \(= (0.00164,\ 0.00719)\) | A1 A1 |
| (5) |
| Scheme | Marks |
|---|---|
| \(0.07^2 = 0.0049\) | M1 |
| 0.0049 is within the 95% confidence interval. | A1 |
| There is no evidence to reject the idea that the standard deviation of the volumes is 0.07 or The machine is working well. | A1 |
| (3) |
Notes
(corrected from the printed mark scheme: the scheme prints “… the standard deviation of the volumes is not 0.07”; since 0.0049 lies inside the interval the conclusion is that it is 0.07)