9231 P21 - Nov 2018 - Q11O - 12 marks
Question 11 OR alternative.
A machine is used to produce metal rods. When the machine is working efficiently, the lengths, \(x\ \mathrm{cm}\), of the rods have a normal distribution with mean \(150\ \mathrm{cm}\) and standard deviation \(1.2\ \mathrm{cm}\). The machine is checked regularly by taking random samples of \(200\) rods. The latest results are shown in the following table.
| Interval | \(146 \leq x < 147\) | \(147 \leq x < 148\) | \(148 \leq x < 149\) | \(149 \leq x < 150\) | \(150 \leq x < 151\) | \(151 \leq x < 152\) | \(152 \leq x < 153\) | \(153 \leq x < 154\) |
|---|---|---|---|---|---|---|---|---|
| Observed frequency | 1 | 2 | 23 | 52 | 69 | 36 | 15 | 2 |
As a first check, the sample is used to calculate an estimate for the mean.
(i) Show that an estimate for the mean from this sample is close to \(150\ \mathrm{cm}\).
As a second check, the results are tested for goodness of fit of the normal distribution with mean \(150\ \mathrm{cm}\) and standard deviation \(1.2\ \mathrm{cm}\). The relevant expected frequencies are shown in the following table.
| Interval | \(x < 147\) | \(147 \leq x < 148\) | \(148 \leq x < 149\) | \(149 \leq x < 150\) | \(150 \leq x < 151\) | \(151 \leq x < 152\) | \(152 \leq x < 153\) | \(153 \leq x\) |
|---|---|---|---|---|---|---|---|---|
| Observed frequency | 1 | 2 | 23 | 52 | 69 | 36 | 15 | 2 |
| Expected frequency | 1.24 | 8.32 | 30.94 | 59.50 | 59.50 | 30.94 | 8.32 | 1.24 |
(ii) Show how the expected frequency for \(151 \leq x < 152\) is obtained.
(iii) Test, at the \(5\%\) significance level, the goodness of fit of the normal distribution to the results.