S4 June 2007 Q4
4. The length \(X\) mm of a spring made by a machine is normally distributed \(\mathrm{N}(\mu, \sigma^2)\). A random sample of 20 springs is selected and their lengths measured in mm. Using this sample the unbiased estimates of \(\mu\) and \(\sigma^2\) are
\[\bar{x} = 100.6, \qquad s^2 = 1.5.\]Stating your hypotheses clearly test, at the 10% level of significance,
(a) whether or not the variance of the lengths of springs is different from 0.9, (6)
(b) whether or not the mean length of the springs is greater than 100 mm. (6)
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \sigma^2 = 0.9 \qquad \mathrm{H}_1 : \sigma^2 \neq 0.9\) | B1 |
| \(\nu = 19\) | |
| CR (Lower tail 10.117) | B1 |
| Upper tail 30.144 | B1 |
| Test statistic \(= \dfrac{19 \times 1.5}{0.9} = 31.6666\), significant | M1 A1 |
| There is sufficient evidence that the variance of the length of spring is different to 0.9 | A1 |
| (6) |
Notes
Only need to see 30.144
Need variance in conclusion
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \mu = 100 \qquad \mathrm{H}_1 : \mu \gt 100\) | B1 |
| \(t_{19} = 1.328\) | B1 |
| \(t = \dfrac{100.6 - 100}{\sqrt{\dfrac{1.5}{20}}} = 2.19\) | M1 A1 A1 |
| Significant. The mean length of spring is greater than 100 | B1 |
| (6) |
Notes
Conclusion must be in context. Length of spring needed