The table below represents the weights, \(W\), in grams, of 80 packets of roasted peanuts.
| Weight \(W\) | \(80<W\le85\) | \(85<W\le90\) | \(90<W\le95\) | \(95<W\le100\) | \(100<W\le105\) | \(105<W\le110\) | \(110<W\le115\) |
|---|
| Number of packets | 5 | 10 | 15 | 26 | 13 | 7 | 4 |
Use the midpoint of each interval to find an estimate for the standard deviation of the weights.[3]
Copy and complete the following cumulative frequency table for the above data.[1]
| Weight \(W\) | \(W\le85\) | \(W\le90\) | \(W\le95\) | \(W\le100\) | \(W\le105\) | \(W\le110\) | \(W\le115\) |
|---|
| Number of packets | 5 | 15 | | | | | 80 |
A cumulative frequency graph of the distribution is shown below, with a scale of 2 cm for 10 packets on the vertical axis and 2 cm for 5 grams on the horizontal axis. Use the graph to estimate (i) the median and (ii) the upper quartile. Give your answers to the nearest gram.[4]
the median;
the upper quartile (that is, the third quartile). Give your answers to the nearest gram.
Let \(W_1,W_2,\ldots,W_{80}\) be the individual weights of the packets, and let \(\overline W\) be their mean. Find \(\sum_{i=1}^{80}(W_i-\overline W)\).[2]
One of the 80 packets is selected at random. Given that its weight satisfies 85 < W ≤ 110, find the probability that its weight is greater than 100 grams.[4]