The heights, h metres, of the 120 boys in an athletics club are recorded.
The table shows information about the heights of the boys.
(a) (i) Write down the modal class.
……………… \(\leq h < ………………\)
(ii) Calculate an estimate of the mean height.
(b) (i) One boy is chosen at random from the club.
Find the probability that this boy has a height greater than 1.8m.
(ii) Three boys are chosen at random from the club.
Calculate the probability that one of the boys has a height greater than 1.8m and the other two boys each have a height of 1.4m or less.
(c) (i) Use the frequency table on page 4 to complete the cumulative frequency table.
(ii) On the grid, draw a cumulative frequency diagram to show this information.
(d) Use your diagram to find an estimate for
(i) the median height,
(ii) the 40th percentile.
▶️ Answer/Explanation
(a)(i) Ans: \(1.5 < h \leq 1.6\)
The modal class is the interval with the highest frequency (34 boys).
(a)(ii) Ans: 1.62 or 1.623… m
Midpoints: 1.35, 1.45, 1.55, 1.65, 1.75, 1.85. Multiply by frequencies: 8.1, 36.8, 52.7, 39.6, 24.5, 12.95. Sum (174.65) ÷ 120 ≈ 1.623 m.
(b)(i) Ans: \(\frac{14}{120}\)
Boys >1.8m: 14. Probability = \(\frac{14}{120}\).
(b)(ii) Ans: \(\frac{21}{20060}\)
Combinations: \(\frac{14 \times 8 \times 7 \times 3}{120 \times 119 \times 118}\) simplifies to \(\frac{21}{20060}\).
(c)(i) Ans: 55, 79, 106, 120
Cumulative frequencies are calculated by adding each subsequent frequency.
(d)(i) Ans: 1.62 to 1.63 m
Median (60th boy) falls in the 1.6-1.7 range, estimated from the cumulative frequency diagram.
(d)(ii) Ans: 1.57 to 1.58 m
40th percentile (48th boy) falls in the 1.5-1.6 range, estimated from the diagram.
The probability that it rains on Monday \(\frac{3}{5}\)
If it rains on Monday, the probability that it rains on Tuesday is \(\frac{4}{7}\)
If it does not rain on Monday, the probability that it rains on Tuesday is \(\frac{5}{7}\)
(a) Complete the tree diagram.
(b) Find the probability that it rains
(i) on both days,
(ii) on Monday but not on Tuesday,
(iii) on only one of the two days.
(c) If it does not rain on Monday and it does not rain on Tuesday, the probability that it does not rain on Wednesday is \(\frac{1}{4}\)
Calculate the probability that it rains on at least one of the three days.
▶️ Answer/Explanation
(a)
(b)(i) Ans: \(\frac{12}{35}\)
Multiply the probability of rain on Monday (\(\frac{3}{5}\)) by the probability of rain on Tuesday given rain on Monday (\(\frac{4}{7}\)).
(b)(ii) Ans: \(\frac{9}{35}\)
Multiply the probability of rain on Monday (\(\frac{3}{5}\)) by the probability of no rain on Tuesday given rain on Monday (\(\frac{3}{7}\)).
(b)(iii) Ans: \(\frac{19}{35}\)
Add the probability of rain only on Monday (\(\frac{9}{35}\)) and rain only on Tuesday (\(\frac{2}{5} \times \frac{5}{7} = \frac{10}{35}\)).
(c) Ans: \(\frac{34}{35}\)
First, find the probability of no rain on all three days (\(\frac{2}{5} \times \frac{2}{7} \times \frac{1}{4} = \frac{1}{35}\)). Subtract this from 1 to get the probability of rain on at least one day.