Mathematical Studies

Mathematical Studies

IB Standard Level

Year 11

TEST

Statistics

2016

Instructions

Calculators are permitted for the test.
Unless otherwise stated in the question, all numerical answers must be given exactly or correct to three significant figures.

Name:______

Total marks/55

Answer all the questions in the spaces provided. Unless otherwise stated in the question, all numerical answers must be given exactly or correct to three significant figures. Where an answer is wrong, some marks may be given for correct method, provided this is shown by written working. Working may be continued below the box if necessary. Solutions found from a graphic display calculator should be supported by suitable working, e.g. if graphs are used to find a solution, you should sketch these as part of your answer.

Question 1

The table shows the number of children in 50 families.

Number of
children / Frequency / Cumulative
frequency
1 / 3 / 3
2 / m / 22
3 / 12 / 34
4 / p / q
5 / 5 / 48
6 / 2 / 50
T

(a)Write down the value of T.

(b)Find the values of m, p and q.

(Total 4 marks)

Question 2

The following table shows the age distribution of teachers who smoke at LaughlinHigh School.

Ages / Number of smokers
20 ≤ x < 30 / 5
30 ≤ x < 40 / 4
40 ≤ x < 50 / 3
50 ≤ x < 60 / 2
60 ≤ x < 70 / 3

(a)Calculate an estimate of the mean smoking age.

(b)On the following grid, construct a histogram to represent this data.

(Total 4 marks)

Question 3

The graph below shows the cumulative frequency for the yearly incomes of 200 people.

Use the graph to estimate

(a)the number of people who earn less than 5000 British pounds per year;

(b)the median salary of the group of 200 people;

(c)the lowest income of the richest 20% of this group.

(Total 4 marks)

Question 4

The bar chart below shows the number of people in a selection of families.

(a)How many families are represented?

(b)Write down the mode of the distribution.

(c)Find, correct to the nearest whole number, the mean number of people in a family.

(Total 4 marks)

Question 5(5 marks)

The weight in kilograms of 12 students in a class are as follows.

63 76 99 65 63 51 52 95 63 71 65 83

(a)State the mode.

(1)

(b)Calculate

(i)the mean weight;

(ii)the standard deviation of the weights.

(2)

When one student leaves the class, the mean weight of the remaining 11 students becomes 70kg.

(2)

Question 6(5 marks)

The heights of 200 students are recorded in the following table.

Height (h) in cm / Frequency
140 ≤ h < 150 / 2
150 ≤ h < 160 / 28
160 ≤ h < 170 / 63
170 ≤ h < 180 / 74
180 ≤ h < 190 / 20
190 ≤ h < 200 / 11
200 ≤ h < 210 / 2

(a)Write down the modal group.

(1)

(b)Calculate an estimate of the mean and standard deviation of the heights.

(4)

The cumulative frequency curve for this data is drawn below.

(c)Write down the median height.

(1)

(d)The upper quartile is 177.3 cm. Calculate the interquartile range.

(2)

(e)Find the percentage of students with heights less than 165 cm.

(2)

Question 7(4 marks)

The length and width of 10 leaves are shown on the scatter diagram below.

(a)Plot the point M (97, 43) which represents the mean length and the mean width.

(b)Draw a suitable line of best fit.

(c)Write a sentence describing the relationship between leaf length and leaf width for this sample.

Question 8(4 marks)

Ten students were given two tests, one on Mathematics and one on English.
The table shows the results of the tests for each of the ten students.

Student / A / B / C / D / E / F / G / H / I / J
Mathematics (x) / 8.6 / 13.4 / 12.8 / 9.3 / 1.3 / 9.4 / 13.1 / 4.9 / 13.5 / 9.6
English (y) / 33 / 51 / 30 / 48 / 12 / 23 / 46 / 18 / 36 / 50

(a)Find, correct to two decimal places, the product moment correlation coefficient (r).

(2)

(b)Use your result from part (a) to comment on the statement:

‘Those who do well in Mathematics also do well in English.’

(2)

Question 9(6 marks)

Ten students were asked for their average grade at the end of their last year of high school and their average grade at the end of their last year at university. The results were put into a table as follows:

Student / High School grade, x / University grade, y
1
2
3
4
5
6
7
8
9
10 / 90
75
80
70
95
85
90
70
95
85 / 3.2
2.6
3.0
1.6
3.8
3.1
3.8
2.8
3.0
3.5
Total / 835 / 30.4

(a)Find the correlation coefficient r, giving your answer to two decimal places.

(2)

(b)Describe the correlation between the high school grades and the university grades.

(2)

(c)Find the equation of the regression line for y on x in the form y = ax + b.

(2)

Question 10 (10 marks)

The Type Fast secretarial training agency has a new computer software spreadsheet package. The agency investigates the number of hours it takes people of varying ages to reach a level of proficiency using this package. Fifteen individuals are tested and the results are summarised in the table below.

Age
(x) / 32 / 40 / 21 / 45 / 24 / 19 / 17 / 21 / 27 / 54 / 33 / 37 / 23 / 45 / 18
Time
(in hours)
(y) / 10 / 12 / 8 / 15 / 7 / 8 / 6 / 9 / 11 / 16 / t / 13 / 9 / 17 / 5

(a)(i)Find the product-moment correlation coefficient r for this data.

(2)

(ii)What does the value of the correlation coefficient suggest about the relationship between the two variables?

(1)

(b)Given that the mean time taken was 10.6 hours, write the equation of the regression line for y on x in the form y = ax + b.

(3)

(c)Use your equation for the regression line to predict

(i)the time that it would take a 33 year old person to reach proficiency, giving your answer correct to the nearest hour;

(2)

(ii)the age of a person who would take 8 hours to reach proficiency, giving your answer correct to the nearest year.

(2)

END OF TEST