7. A shoe manufacturer sees a report from another country stating that the length of adult male feet is normally distributed with a mean of 22.4 cm and a standard deviation of 2.8 cm . The manufacturer wishes to see if this model is appropriate for his customers and collects data on the length, correct to the nearest cm, of the right foot of a random sample of 200 males giving the following results:
| Length (cm) | \(\leq 18\) | \(19 - 21\) | \(22 - 24\) | \(25 - 27\) | \(\geq 28\) |
| No. of Men | 24 | 48 | 69 | 41 | 18 |
The expected frequencies for the \(\leq 18\) and \(19 - 21\) groups are calculated as 16.46 and 58.44 respectively, correct to 2 decimal places.
- Calculate expected frequencies for the other three classes.
- Stating your hypotheses clearly, test at the \(10 \%\) level of significance whether or not this data can be modelled by the distribution \(\mathrm { N } \left( 22.4,2.8 ^ { 2 } \right)\).
(7 marks)
The manufacturer wishes to refine the model by not assuming a mean and standard deviation. - Explain briefly how the manufacturer should proceed.
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