S4 June 2009 Q1
1. A company manufactures bolts with a mean diameter of 5 mm. The company wishes to check that the diameter of the bolts has not decreased. A random sample of 10 bolts is taken and the diameters, \(x\) mm, of the bolts are measured. The results are summarised below.
\[\sum x = 49.1 \qquad \sum x^2 = 241.2\]Using a 1% level of significance, test whether or not the mean diameter of the bolts is less than 5 mm.
(You may assume that the diameter of the bolts follows a normal distribution.) (8)
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \mu = 5;\ \mathrm{H}_1 : \mu \lt 5\) both | B1 |
| CR: \(t_9(0.01) \gt 2.821\) | B1 |
| \(\bar{x} = 4.91\) | B1 |
| \(s^2 = \dfrac{1}{9}\left(241.2 - \dfrac{49.1^2}{10}\right) = 0.0132222\) \(s\) = awrt 0.115 | M1 A1 |
| \(t = \dfrac{|4.91 - 5|}{\dfrac{\sqrt{0.013222}}{\sqrt{10}}} = \pm 2.475\) 2.47 – 2.48 | M1 A1 |
| Since 2.475 is not in the critical region there is insufficient evidence to reject \(\mathrm{H}_0\) and conclude that the mean diameter of the bolts is not less than (not equal to) 5 mm. | A1ft |
| (8 marks) |