S1 January 2013 Q5

EdexcelOld spec15 marksData ProcessingNormal Distribution

5. A survey of 100 households gave the following results for weekly income £\(y\).

Income \(y\) (£)Mid-pointFrequency \(f\)
\(0 \leqslant y \lt 200\)10012
\(200 \leqslant y \lt 240\)22028
\(240 \leqslant y \lt 320\)28022
\(320 \leqslant y \lt 400\)36018
\(400 \leqslant y \lt 600\)50012
\(600 \leqslant y \lt 800\)7008

(You may use \(\sum fy^2 = 12\,452\,800\))

A histogram was drawn and the class \(200 \leqslant y \lt 240\) was represented by a rectangle of width 2 cm and height 7 cm.

(a) Calculate the width and the height of the rectangle representing the class \(320 \leqslant y \lt 400\) (3)
(b) Use linear interpolation to estimate the median weekly income to the nearest pound. (2)
(c) Estimate the mean and the standard deviation of the weekly income for these data. (4)

One measure of skewness is \(\dfrac{3(\text{mean} - \text{median})}{\text{standard deviation}}\).

(d) Use this measure to calculate the skewness for these data and describe its value. (2)

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.

(e) Find \(\mathrm{P}(240 \lt X \lt 400)\). (2)
(f) With reference to your calculations in parts (d) and (e) and the data in the table, comment on Katie’s suggestion. (2)
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