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Standard Deviation and Cumulative Frequency

Statistics-1 1. One thousand candidates sit an examination. The distribution of marks is shown in the following grouped frequency table.

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Marks|1–10|11–20|21–30|31–40|41–50|51–60|61–70|71–80|81–90|91–100| Number of candidates|15|50|100|170|260|220|90|45|30|20| (a)Copy and complete the following table, which presents the above data as a cumulative frequency distribution. (3) Mark|? 10|? 20|? 30|? 40|? 50|? 60|? 70|? 80|? 90|? 100| Number of candidates|15|65|||||905|||| b)Draw a cumulative frequency graph of the distribution, using a scale of 1 cm for 100 candidates on the vertical axis and 1 cm for 10 marks on the horizontal axis. (5) (c)Use your graph to answer parts (i)–(iii) below, (i)Find an estimate for the median score. (2) (ii)Candidates who scored less than 35 were required to retake the examination. How many candidates had to retake? (3) (iii)The highest-scoring 15% of candidates were awarded a distinction. Find the mark above which a distinction was awarded. (3) Total 16 marks) 2. At a conference of 100 mathematicians there are 72 men and 28 women. The men have a mean height of 1. 79 m and the women have a mean height of 1. 62 m. Find the mean height of the 100 mathematicians. (Total 4 marks) 3. The mean of the population x1, x2, …….. , x25 is m. Given that = 300 and = 625, find (a)the value of m; (b)the standard deviation of the population. (Total 4 marks) 4. A supermarket records the amount of money d spent by customers in their store during a busy period.

The results are as follows: Money in $ (d)|0–20|20–40|40–60|60–80|80–100|100–120|120–140| Number of customers (n)|24|16|22|40|18|10|4| (a)Find an estimate for the mean amount of money spent by the customers, giving your answer to the nearest dollar ($). (2) (b)Copy and complete the following cumulative frequency table and use it to draw a cumulative frequency graph. Use a scale of 2 cm to represent $20 on the horizontal axis, and 2 cm to represent 20 customers on the vertical axis. (5) Money in $ (d)|