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 cm and standard deviation 1.2 cm . The machine is checked regularly by taking random samples of 200 rods. The latest results are shown in the following table.
| Interval | \(146 \leqslant x < 147\) | \(147 \leqslant x < 148\) | \(148 \leqslant x < 149\) | \(149 \leqslant x < 150\) |
| Observed frequency | 1 | 2 | 23 | 52 |
| \(150 \leqslant x < 151\) | \(151 \leqslant x < 152\) | \(152 \leqslant x < 153\) | \(153 \leqslant x < 154\) |
| 69 | 36 | 15 | 2 |
As a first check, the sample is used to calculate an estimate for the mean.
- Show that an estimate for the mean from this sample is close to 150 cm .
As a second check, the results are tested for goodness of fit of the normal distribution with mean 150 cm and standard deviation 1.2 cm . The relevant expected frequencies, found using the normal distribution function given in the List of Formulae (MF10), are shown in the following table.
| Interval | \(x < 147\) | \(147 \leqslant x < 148\) | \(148 \leqslant x < 149\) | \(149 \leqslant x < 150\) |
| Observed frequency | 1 | 2 | 23 | 52 |
| Expected frequency | 1.24 | 8.32 | 30.94 | 59.50 |
| \(150 \leqslant x < 151\) | \(151 \leqslant x < 152\) | \(152 \leqslant x < 153\) | \(153 \leqslant x\) |
| 69 | 36 | 15 | 2 |
| 59.50 | 30.94 | 8.32 | 1.24 |
- Show how the expected frequency for \(151 \leqslant x < 152\) is obtained.
- Test, at the \(5 \%\) significance level, the goodness of fit of the normal distribution to the results.
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