- A survey of 100 households gave the following results for weekly income \(\pounds y\).
| Income \(y\) (£) | Mid-point | Frequency \(f\) |
| \(0 \leqslant y < 200\) | 100 | 12 |
| \(200 \leqslant y < 240\) | 220 | 28 |
| \(240 \leqslant y < 320\) | 280 | 22 |
| \(320 \leqslant y < 400\) | 360 | 18 |
| \(400 \leqslant y < 600\) | 500 | 12 |
| \(600 \leqslant y < 800\) | 700 | 8 |
(You may use \(\sum f y ^ { 2 } = 12452\) 800)
A histogram was drawn and the class \(200 \leqslant y < 240\) was represented by a rectangle of width 2 cm and height 7 cm .
- Calculate the width and the height of the rectangle representing the class
$$320 \leqslant y < 400$$
- Use linear interpolation to estimate the median weekly income to the nearest pound.
- Estimate the mean and the standard deviation of the weekly income for these data.
One measure of skewness is \(\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }\).
- Use this measure to calculate the skewness for these data and describe its value.
Katie suggests using the random variable \(X\) which has a normal distribution with mean 320 and standard deviation 150 to model the weekly income for these data.
- Find \(\mathrm { P } ( 240 < X < 400 )\).
- With reference to your calculations in parts (d) and (e) and the data in the table, comment on Katie's suggestion.