1. the Length and Width of 10 Leaves Are Shown on the Scatter Diagram Below

1. 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.

Working:
Answer:
(c) …………………………………………..
…………………………………………...

(Total 4 marks)

2. The following table gives the heights and weights of five sixteen-year-old boys.

Name / Height / Weight
Blake / 182 cm / 73 kg
Jorge / 173 cm / 68 kg
Chin / 162 cm / 60 kg
Ravi / 178 cm / 66 kg
Derek / 190 cm / 75 kg

(a) Find

(i) the mean height;

(ii) the mean weight.

(b) Plot the above data on the grid below and draw the line of best fit.

Working:
Answers:
(a) (i) ……………………………………..
(ii) ……………………………………..

(Total 4 marks)

3. The diagram below shows the marks scored by pupils in a French test and a German test. The mean score on the French test is 29 marks and on the German test is 31 marks.

(a) Describe the relationship between the marks scored in the two tests.

(b) On the graph mark the point M which represents the mean of the distribution.

(d) Idris scored 32 marks on the French test. Use your graph to estimate the mark Idris scored on the German test.

Working:
Answers:
(a) …………………………………………..
(d) …………………………………………..

(Total 4 marks)

4. Statements I, II, III, IV and V represent descriptions of the correlation between two variables.

I High positive linear correlation
II Low positive linear correlation
III No correlation
IV Low negative linear correlation
V High negative linear correlation

Which statement best represents the relationship between the two variables shown in each of the scatter diagrams below.

Answers:

(a) …………………………………………

(b) …………………………………………

(d) …………………………………………

(Total 4 marks)

5. The heights and weights of 10 students selected at random are shown in the table below.

Student / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10
Height
x cm / 155 / 161 / 173 / 150 / 182 / 165 / 170 / 185 / 175 / 145
Weight
y kg / 50 / 75 / 80 / 46 / 81 / 79 / 64 / 92 / 74 / 108

(a) Plot this information on a scatter graph. Use a scale of 1 cm to represent 20 cm on the
x-axis and 1 cm to represent 10 kg on the y-axis.

(4)

(b) Calculate the mean height.

(1)

(d) It is given that Sxy = 44.31.

(i) By first calculating the standard deviation of the heights, correct to two decimal places, show that the gradient of the line of regression of y on x is 0.276.

(ii) Calculate the equation of the line of best fit.

(iii) Draw the line of best fit on your graph.

(6)

(e) Use your line to estimate

(i) the weight of a student of height 190 cm;

(ii) the height of a student of weight 72 kg.

(2)

(f) It is decided to remove the data for student number 10 from all calculations. Explain briefly what effect this will have on the line of best fit.

(1)

(Total 15 marks)

6. Eight students in Mr. O’Neil’s Physical Education class did pushups and situps. Their results are shown in the following table.

Student / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8
number of pushups (x) / 24 / 18 / 32 / 51 / 35 / 42 / 45 / 25
number of situps (y) / 32 / 28 / 38 / 40 / 30 / 52 / 48 / 52

The graph below shows the results for the first seven students.

(a) Plot the results for the eighth student on the graph.

(b) If = 34 and = 40 , draw a line of best fit on the graph.

Working:
Answers:
(c) ......

(Total 8 marks)

7. The scatter diagram below shows the relationship between the number of vehicles per thousand of population and the number of people killed in road accidents over an eight year period in Calmville.

Let x be the number of vehicles per thousand and y be the number of people killed. The following information is known.

= 270, = 650 sx = 22.3 sy = 96.2, sxy = 2077.75

(a) (i) Calculate the product–moment correlation coefficient (r).

(ii) Explain clearly the statistical relationship between the variables x and y

(4)

(b) Write the equation of the regression line of y on x, expressing it in the form y = mx + c (where m and c are given correct to 3 significant figures).

(4)

(i) There were 250 vehicles per 1000 of population. Find the number of people killed.

(ii) Explain why it is not a good idea to use the regression line to estimate the number of people killed when the number of vehicles is 150 per thousand.

(3)

(Total 11 marks)

8. A group of 15 students was given a test on mathematics. The students then played a computer game. The diagram below shows the scores on the test and the game.

The mean score on the mathematics test was 56.9 and the mean score for the computer game was 45.9. The point M has coordinates (56.9, 45.9).

(a) Describe the relationship between the two sets of scores.

A straight line of best fit passes through the point (0, 69).

(b) On the diagram draw this straight line of best fit.

Jane took the tests late and scored 45 at mathematics.

(c) Using your graph or otherwise, estimate the score Jane expects on the computer game, giving your answer to the nearest whole number.

Working:
Answers:
(a) ......
(c) ......

(Total 8 marks)

9. The following are the results of a survey of the scores of 10 people on both a mathematics (x) and a science (y) aptitude test:

Student / Mathematics (x) / Science (y)
1 / 90 / 85
2 / 38 / 60
3 / 58 / 78 / = 73
4 / 85 / 70 / = 78
5 / 73 / 65
6 / 82 / 71
7 / 56 / 80 / Sx = 16.7
8 / 73 / 90 / Sy =10.8
9 / 95 / 96 / Sxy = 100.1
10 / 80 / 85

(a) Copy the graph below on graph paper and fill in the missing points for students 7–10 on the graph.

(4)

(b) Plot the point M (, ) on the graph.

(1)

(2)

(d) Graph this line on the above graph.

(2)

(e) Given that a student receives an 88 on the mathematics test, what would you expect this student's science score to be? Show how you arrived at your result.

(2)

(Total 11 marks)

10. A study was carried out to investigate possible links between the weights of baby rabbits and their mothers. A sample of 20 pairs of mother rabbits (x) and baby rabbits (y) was chosen at random and their weights noted. This information was plotted on a scatter diagram and various statistical calculations were made. These appear below.

mean of x / mean of y / sx / sy / sxy / sum of x / sum of y
3.78 / 3.46 / 0.850 / 0.689 / 0.442 / 75.6 / 69.2

(a) Show that the product-moment correlation coefficient r for this data is 0.755.

(2)

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

(3)

(ii) Use your equation for the regression line to estimate the weight of a rabbit given that its mother weighs 3.71 kg.

(2)

(Total 7 marks)

11. The sketches below represent scatter diagrams for the way in which variables x, y and z change over time, t, in a given chemical experiment. They are labelled , and .

(a) State which of the diagrams indicate that the pair of variables

(i) is not correlated;

(1)

(ii) shows strong linear correlation.

(1)

(b) A student is given a piece of paper with five numbers written on it. She is told that three of these numbers are the product moment correlation coefficients for the three pairs of variables shown above. The five numbers are

0.9, –0.85, –0.20, 0.04, 1.60

(i) For each sketch above state which of these five numbers is the most appropriate value for the correlation coefficient.

(3)

(ii) For the two remaining numbers, state why you reject them for this experiment.

(2)

∑t = 124 ∑w = 250 st = 6.08 sw = 10.50 stw = 55.00

for 20 data points.

Calculate

(i) the product moment correlation coefficient for this data;

(2)

(ii) the equation of the regression line of w on t in the form w = at + b.

(5)

(Total 14 marks)

12. The following table gives the amount of fuel in a car’s fuel tank, and the number of kilometres travelled after filling the tank.

Distance travelled (km) / 0 / 220 / 276 / 500 / 680 / 850
Amount of fuel in tank (litres) / 55 / 43 / 30 / 24 / 10 / 6

(a) On the scatter diagram below, plot the remaining points.

The mean distance travelled is 421 km (), and the mean amount of fuel in the tank is 28 litres (). This point is plotted on the scatter diagram.

(b) Sketch the line of best fit.

A car travelled 350 km.

(Total 6 marks)

13. A number of employees at a factory were given x additional training sessions each. They were then timed on how long (y seconds) it took them to complete a task. The results are shown in the scatter diagram below. A list of descriptive statistics is also given.

n = 9,

sum of x values: S x = 54,

sum of y values: S y = 81,

mean of x values: = 6,

mean of y values: = 9,

standard deviation of x: sx = 1.94,

standard deviation of y: sy = 2.35,

covariance: sxy = –3.77.

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

(2)

(b) What is the nature of the relationship between the amount of additional training and the time taken to complete the task?

(2)

(1)

(d) (i) Determine the equation of the linear regression line for y on x.

(ii) Find the expected time to complete the task for an employee who only attended three additional training sessions.

(4)

(Total 9 marks)

14. In an experiment a vertical spring was fixed at its upper end. It was stretched by hanging different weights on its lower end. The length of the spring was then measured. The following readings were obtained.

Load (kg) x / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8
Length (cm) y / 23.5 / 25 / 26.5 / 27 / 28.5 / 31.5 / 34.5 / 36 / 37.5

(a) Plot these pairs of values on a scatter diagram taking 1 cm to represent 1 kg on the horizontal axis and 1 cm to represent 2 cm on the vertical axis.

(4)

(b) (i) Write down the mean value of the load ().

(1)

(ii) Write down the standard deviation of the load.

(1)

(iii) Write down the mean value of the length ().

(1)

(iv) Write down the standard deviation of the length.

(1)

It is given that the covariance Sxy is 12.17.

(d) (i) Write down the correlation coefficient, r, for these readings.

(1)

(ii) Comment on this result.

(2)

(e) Find the equation of the regression line of y on x.

(2)

(f) Draw the line of regression on the scatter diagram.

(2)

(g) (i) Using your diagram or otherwise, estimate the length of the spring when a load of 5.4 kg is applied.

(1)

(ii) Malcolm uses the equation to claim that a weight of 30 kg would result in a length of 62.8 cm. Comment on his claim.

(1)

(Total 18 marks)

15. The figure below shows the lengths in centimetres of fish found in the net of a small trawler.

(a) Find the total number of fish in the net.

(2)

(b) Find (i) the modal length interval;

(ii) the interval containing the median length;

(iii) an estimate of the mean length.

(5)

(ii) How many fish (if any) have length greater than three standard deviations above the mean?

(3)

The fishing company must pay a fine if more than 10% of the catch have lengths less than 40cm.

(d) Do a calculation to decide whether the company is fined.