2 It is claimed that the heights of a particular age group of boys follow a normal distribution with mean 125 cm and standard deviation 12 cm . Observations for a randomly chosen group of 60 boys in this age group are summarised in the following table. The table also gives the expected frequencies, correct to 2 decimal places, based on the normal distribution with mean 125 cm and standard deviation 12 cm .
| Height, \(x \mathrm {~cm}\) | \(x < 100\) | \(100 \leqslant x < 110\) | \(110 \leqslant x < 120\) | \(120 \leqslant x < 130\) | \(130 \leqslant x < 140\) | \(x \geqslant 140\) |
| Observed frequency | 0 | 3 | 15 | 23 | 11 | 8 |
| Expected frequency | 1.12 | 5.22 | 13.97 | 19.38 | 13.97 | 6.34 |
- Show how the expected frequency for \(130 \leqslant x < 140\) is obtained.
- Carry out a goodness of fit test, at the \(5 \%\) significance level, to determine whether the claim is supported by the data.