Question
Consider the following data
(a) Find the correlation coefficient \(r\).
(b) Describe the relation between \(x\) and \(y\).
(c) Find the equation \(y = ax+b\) of the regression line for \(y\) on \(x\).
(d) Find the equation \(x = cy+d\) of the regression line for \(x\) on \(y\).
(e) Find the inverse of the function in question (c); Is it the function in question (d)? [Maximum mark: 10] [with GDC]
Answer/Explanation
Ans.
(a) 0.965
(b) strong positive
(c) y = 2.2x – 0.5
(d) x = 0.423y + 0.385
(e) y = 2.2x – 0.5 ⇔ y + 0.5 = 2.2x ⇔ x = 0.455 y + 0.227. They are different
Question
Statements I, II, III, IV, 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. [Maximum mark: 4] [without GDC]
Answer/Explanation
Question
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.
(a) State which of the diagrams indicate that the pair of variables
(i) is not correlated. (ii) shows strong linear correlation.
(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.
(ii) For the two remaining numbers, state why you reject them for this experiment. [Maximum mark: 7] [without GDC]
Answer/Explanation
Ans.
Question
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. [Maximum mark: 4] [without GDC]
Answer/Explanation
Ans.
(a) (see diagram)
(b)
(c) leaf length and leaf width are positively correlated
Question
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:
(a) Find the correlation coefficient \(r\).
(b) Describe the correlation between the high school and the university grades.
(c) Find the equation of the regression line for \(y\) on \(x\). [Maximum mark: 5] [with GDC]
Answer/Explanation
Ans.
(a) \(r\) = 0.76
(b) There is a fairly strong positive correlation between high school
grades and university grades.
(c) \(y\) = 0.052x – 1.29 (3 s.f.)
Question
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 scores.
(b) On the graph mark the point M which represents the mean of the distribution.
(c) Draw a suitable line of best fit.
(d) Idris scored 32 marks on the French test. Use your graph to estimate the mark
Idris scored on the German test.
Answer/Explanation
Ans.
(a) High positive or high or positive or good correlation etc.
(b) Correct point M(29, 31)
(c) Suitable line should pass through M and have nearly as many crosses (plotted points)
below it as above it.
(d) Accept only value (including non-integers) obtained using line of best fit.
Question
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 point M corresponding to the means has coordinates (56.9, 45.9).
(a) Describe the relationship between the two sets of scores.
(b) On the diagram draw the straight line of best fit given that it passes through the point (0, 69).
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. [Maximum mark: 5] [without GDC]
Answer/Explanation
Ans.
(a) The scores are negatively correlated
(b)
Line must be drawn straight. It must pass through (0, 69).
It must pass through the mean point M = (56.9, 45.9).
(c) 51 is closest. (low 50 or 52)
Question
The following table gives the heights and weights of five sixteen-year-old boys.
(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. [Maximum mark: 6] [with GDC]
Answer/Explanation
Ans.
(a) (i) \(\frac{182+173+162+178+190}{5}\)= 177 cm
(ii) \(\frac{73+68+60+66+75}{5}\)= 68.4 kg
Question
Eight students in Mr. O’Neil’s Physical Education class did pushups and situps. Their
results are shown in the following table.
The graph below shows the results for the first seven students.
(a) Plot the results for the eighth student on the graph.
(b) Find \(\bar{x}\) and \(\bar{y}\) , and draw a line of best fit on the graph.
(c) A student can do 60 pushups. How many situps can the student be expected to do? [Maximum mark: 6] [with GDC]
Answer/Explanation
Ans.
(a) on the graph
(b) \(\bar{x}\) = 34 and \(\bar{y}\)= 40
Notes: a line through (34, 40), a reasonable line.
(c) 50 situps (allow ±2)
Question
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.
(a) Find correct to two decimal places, the correlation coefficient (r).
(b) Use your result from part (a) to comment on the statement:
‘Those who do well in Mathematics also do well in English. [Maximum mark: 4] [with GDC]
Answer/Explanation
Ans.
(a) \(r\) = 0.6399706… ≈ 0.64 (2 d.p.)
(b) There is a degree of positive correlation between scores in Maths and scores in English.
Therefore those who do well in Mathematics are likely to do well in English also. (Or
equivalent statements.)
Question
The following table gives the amount of fuel in a car’s fuel tank, and the number of
kilometres travelled after filling the tank.
(a) On the scatter diagram below, plot the remaining points.
The mean distance travelled is 421 km \((\bar{x})\), and the mean amount of fuel in the tank is
28 litres \((\bar{y})\). This point is plotted on the scatter diagram.
(b) Sketch the line of best fit.
(c) A car travelled 350km. Use your line above to estimate the amount of fuel left in the tank. [Maximum mark: 6] [with GDC]
Answer/Explanation
Ans.
(a)
(b) Straight line with –ve gradient passing through the mean
intercept on y-axis between 50 and 55
(c) 32 (read answer from candidate’s line)
Question
It is decided to take a random sample of 10 students to see if there is any linear
relationship between height and shoe size. The results are given in the table below.
(a) Write down the equation of the regression line of shoe size \((y)\) on height \((x)\),
giving your answer in the form \(y = mx + c\).
(b) Use your equation in part (a) to predict the shoe size of a student who is 162 cm in height.
(c) Write down the correlation coefficient.
(d) Describe the correlation between height and shoe size. [Maximum mark: 7] [with GDC]
Answer/Explanation
Ans.
(a) \(y\) = 0.070x – 3.22
(b) \(y\) = 0.070 × 162 – 3.22 = 8.12 Therefore shoe size 8 or 9 (8.12).
(c) \(r\) = 0.681
(d) Moderately strong, positive correlation.
Question
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.
(a) (i) Find the correlation coefficient \(r\) for this data.
(ii) What does the value of the correlation coefficient suggest about the
relationship between the two variables?
(b) Write down the equation of the regression line for \(y\) on \(x\) in the form \(y = ax + b\).
(c) Use your equation for the regression line to predict
(i) the time that it would take a 30 year old person to reach proficiency, giving
your answer correct to the nearest hour;
(ii) the age of a person who would take 8 hours to reach proficiency, giving your answer correct to the nearest year. [Maximum mark: 8] [with GDC]
Answer/Explanation
Ans.
(a) (i) \(r\) = 0.935 (3 s.f.)
(ii) it suggests a strong positive correlation between the two variables.
(b) \(y\) = 0.291x + 1.56
(c) (i) \(y\) = 0.291 × 30 + 1.56 = 10.29 = 10 hours
(ii) 8 = 0.291x + 1.56
\(x\) = 22.13 = 22 years
Question
The heights and weights of 10 students selected at random are shown in the table
below.
(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.
(b) Calculate the mean height.
(c) Calculate the mean weight.
(d) (i) Find the equation of the line of best fit.
(ii) Draw the line of best fit on your graph.
(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.
(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. [Maximum mark: 12] [with GDC]
Answer/Explanation
Ans.
(a)
(b) Mean height = 166.1 = 166 (3 s.f.)
(c) Mean weight = 74.9 (3 s.f.)
(d) (i) \(y\) = 0.276x + 29.1
(ii) Line on graph. Note: y-intercept at 29.1, straight line through (166, 74.9).
(e) (i) \(y\) = 0.276 × 190 + 29.1= 81.5 kg
(ii) 72 = 0.276x + 29.1 ⇒ x=\(\frac{72-29.1}{0.276}\)= 155 cm.
OR From the graph (i) \(y\) = 81 (±1) (ii) \(x\) = 155 (±1)
(f) The ‘line of best fit’ becomes closer to the remaining points.
OR Gradient becomes steeper and the line is more accurate ‘best fit’.
OR Any reasonable explanation. (Line becomes \(y\) = 1.10x – 113)
Question
A shopkeeper wanted to investigate whether or not there was a correlation between the prices of food 10 years ago in 1992, with their prices today. He chose 8 everyday items and the prices are given in the table below.
(a) Calculate the mean and the standard deviation of the prices
(i) in 1992;
(ii) in 2002.
(b) (i) Find the correlation coefficient.
(ii) Comment on the relationship between the prices.
(c) Find the equation of the line of the best fit in the form \(y = mx + c\).
(d) What would you expect to pay now for an item costing $2.60 in 1992?
(e) Which item would you omit to increase the correlation coefficient? [Maximum mark: 12] [without GDC]
Answer/Explanation
Ans.
(a) (i) 1992 mean = \(\$\)1.59, Sd = \(\$\)0.727 (or 0.73) (accept 0.777 or 0.78)
(ii) 2002 mean = \(\$\)1.98, Sd = \(\$\)0.635 (or 0.64) (accept 0.679 or 0.68)
(b) (i) r = 0.672
(ii) There is a weak positive correlation
(c) \(y\) = 0.588x + 1.05
(d) \(y\) = 0.582 × 2.60 + 1.05 = $2.56
OR \(y\) = 0.587 × 2.60 + 1.05 = $2.58
OR \(y\) = 0.588 × 2.60 + 1.05 = $2.58
(e) Coffee – because it is the only item to go down in price.
OR
Rolls – because the price increased significantly.
Question
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:
(a) Plot this information on a scatter graph. Use a scale of 1 cm to represent 10 units
on the x-axis and 1 cm to represent 10 units on the y-axis.
(b) Find and plot the point M \(( \bar{x} , \bar{y} )\) on the graph.
(c) Find the equation of the regression line of y on x in the form \(y = ax + b\).
(d) Graph this line on the above graph.
(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. [Maximum mark: 12] [without GDC]
Answer/Explanation
Ans.
(a)
(b) Point M(73,78) plotted correctly.
(c) \(y\) = 0.359x + 51.8
(d) Reasonable line of best fit. Note: going through M, y intercept anywhere from 50 to 54
(e) \(y\) = 0.359 × 88 + 51.8 = 83
OR
\(y\) = 83 (±2) if read from the graph and method is shown.