9 The finance department of a retail firm recorded the daily income each day for 300 days. The results are summarised in the histogram.
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- Find the number of days on which the daily income was between \(\pounds 4000\) and \(\pounds 6000\).
- Calculate an estimate of the number of days on which the daily income was between \(\pounds 2700\) and \(\pounds 3600\).
- Use the midpoints of the classes to show that an estimate of the mean daily income is \(\pounds 3275\).
An estimate of the standard deviation of the daily income is \(\pounds 1060\). The finance department uses the distribution \(\mathrm { N } \left( 3275,1060 ^ { 2 } \right)\) to model the daily income, in pounds.
- Calculate the number of days on which, according to this model, the daily income would be between \(\pounds 4000\) and \(\pounds 6000\).
- It is given that approximately \(95 \%\) of values of the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\) lie within the range \(\mu \pm 2 \sigma\). Without further calculation, use this fact to comment briefly on whether the proposed model is a good fit to the data illustrated in the histogram.