- Kaff coffee is sold in packets. A seller measures the masses of the contents of a random sample of 90 packets of Kaff coffee from her stock. The results are shown in the table below.
| Mass \(w ( \mathrm {~g} )\) | Midpoint \(y ( \mathrm {~g} )\) | Frequency f |
| \(240 \leq w < 245\) | 242.5 | 8 |
| \(245 \leq w < 248\) | 246.5 | 15 |
| \(248 \leq w < 252\) | 250.0 | 35 |
| \(252 \leq w < 255\) | 253.5 | 23 |
| \(255 \leq w < 260\) | 257.5 | 9 |
$$\text { (You may use } \sum \mathrm { fy } ^ { 2 } = 5644 \text { 171.75) }$$
A histogram is drawn and the class \(245 \leq w < 248\) is represented by a rectangle of width 1.2 cm and height 10 cm .
- Calculate the width and the height of the rectangle representing the class \(255 \leq w < 260\).
- Use linear interpolation to estimate the median mass of the contents of a packet of Kaff coffee to 1 decimal place.
- Estimate the mean and the standard deviation of the mass of the contents of a packet of Kaff coffee to 1 decimal place.
The seller claims that the mean mass of the contents of the packets is more than the stated mass. Given that the stated mass of the contents of a packet of Kaff coffee is 250 g and the actual standard deviation of the contents of a packet of Kaff coffee is 4 g ,
- test, using a 5\% level of significance, whether or not the seller's claim is justified. State your hypotheses clearly.
(You may assume that the mass of the contents of a packet is normally distributed.) - Using your answers to parts (b) and (c), comment on the assumption that the mass of the contents of a packet is normally distributed.
(Total 14 marks)