Statistics & Probability

A survey records the lengths, x cm, of 100 fish. Grouped frequency table (class width 10 cm; upper bounds 10, 20, 30, 40, 50, 60): 0 < x ≤ 10: 20; 10 < x ≤ 20: 19; 20 < x ≤ 30: 13; 30 < x ≤ 40: 15; 40 < x ≤ 50: 15; 50 < x ≤ 60: 18. (a) Copy and complete the cumulative frequency table. [1] (b) On the grid, draw a cumulative frequency curve. [2] (c) Use your curve to estimate the median, the interquartile range, and the 20th percentile. [4] (d) A fish is considered "oversized" if their length exceeds 30 cm. Find the number of oversized fish. [2] (e) In a second survey of 100 fish at a different site, the values had the same median but a larger interquartile range. Describe how the cumulative frequency curve for this second survey would differ from the curve drawn in (b). [2]

Box-and-whisker plots show the length (cm) of fish kept in two tanks. Tank A: min 42, Q1 49, median 53, Q3 60, max 66. Tank B: min 44, Q1 48, median 49, Q3 53, max 58. There are 7 fish in Tank A with length greater than 60 cm. (a) Find the total number of fish kept in Tank A. [1] (b) Make one comment comparing the averages and one comment comparing the spread of the lengths in the two tanks. Use figures to support your answers. [4] (c) A third tank Tank C has fish with the same median as Tank B but a smaller interquartile range than both Tank A and Tank B. Sketch a possible box plot for Tank C on the same scale. [2]

A box contains 6 white balls and 7 black balls, making 13 balls in total. Two balls are drawn from the box at random, one after the other, without replacement. (a) Complete the probability tree diagram showing the two balls drawn and the corresponding probabilities. [2] (b) Find the probability that the two balls drawn are of the same type. [2] (c) Find the probability that at least one of the two balls drawn is white. [2] (d) The two balls are returned to the box and a third ball is drawn at random. Given that the first two balls drawn were both black, find the probability that the third ball drawn is white. [1]