Standard deviation of grouped parcel masses, quartiles and a percentile from a cumulative frequency curve, and the expected postage charge
Standard deviation of grouped parcel masses, quartiles and a percentile from a cumulative frequency curve, and the expected postage charge
15 marks
A courier depot weighs \(160\) parcels on one morning. Their masses, \(m\) kg, are summarised in the table.
| Mass (\(m\) kg) | \(0\le m<2\) | \(2\le m<4\) | \(4\le m<6\) | \(6\le m<8\) | \(8\le m<10\) | \(10\le m<14\) | \(14\le m<20\) |
|---|---|---|---|---|---|---|---|
| Number of parcels | \(16\) | \(24\) | \(30\) | \(34\) | \(16\) | \(28\) | \(12\) |
- Use the table to calculate an estimate of the standard deviation of the masses of these parcels.[3]
The data are shown on the cumulative frequency curve below.
- Use the curve to estimate the interquartile range of the masses.
- The heaviest \(35\%\) of the parcels each have a mass greater than \(k\) kg. Use the curve to find the value of \(k\).[6]
The depot charges by weight band. A parcel lighter than \(4\) kg costs \(\$5\) to send, a parcel of at least \(4\) kg but less than \(14\) kg costs \(\$9\), and a parcel of \(14\) kg or more costs \(\$15\).
- Write down the probability that a parcel chosen at random from the \(160\) costs \(\$5\) to send.[2]
The random variable \(X\) is the charge, in dollars, for a parcel chosen at random from the \(160\).
| \(x\) | \(5\) | \(9\) | \(15\) |
|---|---|---|---|
| \(\mathrm{P}(X=x)\) | \(0.675\) |
- Complete the table above.[2]
- Find \(\mathrm{E}(X)\).[2]
Solution:
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Answer:(a) \(4.35\text{ kg}\) (b)(i) \(6\text{ kg}\) (b)(ii) \(k=8\) (c) \(0.25\) (d) \(0.25,\ 0.675,\ 0.075\) (e) \(\mathrm{E}(X)=8.45\) dollars
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