IB Questionbank Mathematics Higher Level 3rd edition11.The cumulative frequency graph below represents the weight in grams of 80 apples picked from a particular tree. (a)Estimate the(i)median weight of the apples;(ii)30th percentile of the weight of the apples.(2) (b)Estimate the number of apples that weigh more than 110 grams.(2)(Total 4 marks)

2.The heights in metres of a random sample of 80 boys in a certain age group were measured and the following cumulative frequency graph obtained. (a)(i)Estimate the median of these data.(ii)Estimate the interquartile range for these data.(3) (b)(i)Produce a frequency table for these data, using a class width of 0.05 metres.(ii)Calculate unbiased estimates of the mean and variance of the heights of the population of boys in this age group.(5)

(c)A boy is selected at random from these 80 boys.(i)Find the probability that his height is less than or equal to 1.15 metres.(ii)Given that his height is less than or equal to 1.15 metres, find the probability that his height is less than or equal to 1.12 metres.(5)(Total 13 marks)

3.The company Fresh Water produces one-litre bottles of mineral water. The company wants to determine the amount of magnesium, in milligrams, in these bottles.A random sample of ten bottles is analysed and the results are as follows:6.7, 7.2, 6.7, 6.8, 6.9, 7.0, 6.8, 6.6, 7.1, 7.3.Find unbiased estimates of the mean and variance of the amount of magnesium in the one-litre bottles.(Total 4 marks)

4.(a)Consider the set of numbers a, 2a, 3a, ..., na where a and n are positive integers.(i)Show that the expression for the mean of this set is .(ii)Let a = 4. Find the minimum value of n for which the sum of these numbers exceeds its mean by more than 100.(6) (b)Consider now the set of numbers x1, ... , xm, y1, ... , y1, ... , yn where xi = 0 for i = 1, ... , m and yi = 1 for i = 1, ... , n.(i)Show that the mean M of this set is given by and the standard deviationS by .(ii)Given that M = S, find the value of the median.(11)(Total 17 marks)

5.The test scores of a group of students are shown on the cumulative frequency graph below. (a)Estimate the median test score.(1) (b)The top 10 % of students receive a grade A and the next best 20 % of students receive a grade B. Estimate(i)the minimum score required to obtain a grade A;(ii)the minimum score required to obtain a grade B.(4)(Total 5 marks)

6.Consider the data set {k 2, k, k +1, k + 4}, where k.(a)Find the mean of this data set in terms of k.(3) Each number in the above data set is now decreased by 3.(b)Find the mean of this new data set in terms of k.(2)(Total 5 marks)

7.A recruitment company tests the aptitude of 100 applicants applying for jobs in engineering. Each applicant does a puzzle and the time taken, t, is recorded.The cumulative frequency curve for these data is shown below. Using the cumulative frequency curve,(a)write down the value of the median;(1) (b)determine the interquartile range;(2) (c)complete the frequency table below.

Time to complete puzzle in secondsNumber of applicants20 < t 30

30 < t 35

35 < t 40

40 < t 45

45 < t 50

50 < t 60

60 < t 80

(2)(Total 5 marks)

8.A test marked out of 100 is written by 800 students. The cumulative frequency graph for the marks is given below.

(a)Write down the number of students who scored 40 marks or less on the test.(2) (b)The middle 50 % of test results lie between marks a and b, where a < b.Find a and b.(4)(Total 6 marks)