Correlation & Regression (AQA Level 3 Mathematical Studies (Core Maths): Paper 2A: Statistical Techniques): Exam Questions

The scatter diagram opposite gives the length and the wingspan of the twelve most common garden birds in the UK.

Calculate the equation of the regression line of $l$ on $w$ and plot the line on the scatter graph opposite.

The ‘flying dinosaur’ pterodactylus had a wingspan of approximately 1.5 m and a body length of approximately 90cm.

Does the data support the idea that today’s birds are descended from dinosaurs?

You must show working to justify your answer.

In the UK women’s clothing size is usually indicated by numbers.

The British Standards Institute produced a standard set of sizes from 8 to 32

However, stores do not all use the British Standard.

Dresses with the same size number may have different measurements in different stores.

The table below shows some of the actual measurements of size 14 dresses at eight high street stores.

Describe the correlation between the chest size and waist size of the dresses.

You may use the grid below if you wish.

The recommended measurements for a size 14 dress are:

List the stores in order of how closely they follow these recommendations.

You must show your working to justify your answer.

A manufacturer selected nine wooden letters in order to compare their height, width and mass.

Here is the data for eight of the letters.

The letter V has height 1.7 cm and width 3.0 cm but its mass is unknown.

Which value should the manufacturer use to predict the mass of the letter V, its height or its width?

Use a statistical measure to explain your decision.

By comparing the height and width of V with the other letters, comment on the reliability of the predicted mass of V.

For eight workers, their hourly pay, £R, and the average amount they spend per week on food, £F, is recorded.

The data is plotted on the following scatter diagram.

Calculate and plot the mean point.

Give one reason why a regression line of F on R is likely to be a good fit for the data.

(i) Calculate the equation of the regression line of F on R

[2]

(ii) Draw the regression line on the scatter diagram.

[2]

A worker has an hourly pay of £11.00

Estimate the average amount this worker spends per week on food.

A different worker spends on average £115 per week on food.

Give one reason why the regression line should not be used to predict the hourly pay for this worker.

Here are four product moment correlation coefficients.

Choose the value that represents the strongest correlation.

Here are three scatter diagrams, A, B and C.

Choose the letter of the diagram that has a product moment correlation coefficient closest to zero.

This scatter diagram shows the number of ice creams sold and the number of people wearing coats on 12 days at a theme park.

Tom looks at the diagram and says,

“Eating ice cream causes fewer people to wear coats.”

Is Tom correct?

Give a reason for your answer.

Femi does a 5km run each week.

The table shows, for ten weeks,

the time taken for each run
and
her average heart rate during each run.

The scatter diagram shows $h$ against $t$ for the first eight weeks.

Complete the diagram by plotting the points for the final two weeks.

(i) Calculate the equation of the regression line of $h$ on $t$.

[2]

(ii) Draw the regression line on the scatter diagram.

[2]

The next week, Femi takes 20.5 minutes to do the 5km run.

(i) Estimate her average heart rate during this run.

[2]

(ii) Explain why your estimate is likely to be unreliable.

[1]

Year 9 students in the secondary schools in a city all take the same maths test.

Here are some point estimates of the mean score of all these Year 9 students.

No student is in more than one sample.

Using this data, what is the best possible estimate of the population mean?

Explain how the accuracy of the estimate of the population mean could be improved.

Eight of the students who took the maths test take an additional test.

The scores of these students in the two tests are shown in the table.

$M$ is the score in the Maths test.

$A$ is the score in the Additional test.

By calculating the product moment correlation coefficient, describe the correlation between $M$ and $A$.

State the value of perfect negative correlation.

The table gives the time spent running and distance travelled for 8 members of a running club.

Using the product moment correlation coefficient, state the type and strength of the correlation between time and distance.

12 students in a class were due to sit two tests.

One student was absent for test 1

A different student was absent for test 2

The marks are shown in the table.

$x$ is the mark for test 1

$y$ is the mark for test 2

The marks of the 10 students that sat both tests are shown on the scatter diagram.

The class teacher says that one student’s pair of marks could be an outlier. Which student is this?

Calculate the equation of the regression line of $y$ on $x$.

Do not include the outlier.

Use your equation of the regression line to estimate the missing marks.

Test 2 mark for student H ____________________________

Test 1 mark for student L ____________________________

Each student was awarded an overall grade based on the total mark, $t$, of their two tests.

$m$ is the mean of the values of $t$, including the outlier and the two estimated marks.

Which students were awarded Grade 3?

You must show your working.

You may use the table below

Mrs Hintz gave her A-level class an algebra test, a calculus test and a statistics test.

Each of the three tests had a maximum score of 100

The scores for the 10 students who completed all three tests are shown in the table.

The scatter diagrams show

the calculus scores against the algebra scores

and

the statistics scores against the algebra scores

for the 10 students in the table.

Mrs Hintz expects a stronger positive correlation between the algebra and calculus scores for her class than between the algebra and statistics scores.

(i) In terms of correlation, what does ‘stronger positive’ mean?

[2]

(ii) Use appropriate statistical calculations to check that she is correct.

[3]

(i) Work out the equation of the regression line of $c$ on $a$

[2]

(ii) Draw your regression line on the scatter diagram below.

[2]

Roy scored 51 in the statistics test.

He was absent for the other two tests.

(i) The equation of the regression line of $s$ on $a$ is $s=16.8+0.55a$

Use the equation of the regression line of $s$ on $a$ to estimate Roy’s score in the algebra test

[2]

(ii) Here are two methods to estimate Roy’s score in the calculus test.

Method 1

Use his estimated algebra score with your equation of the regression line of $c$ on $a$

Method 2

Use his statistics score with the equation of the regression line of $c$ on $s$

This equation is $c=0.930+0.934s$

Do these two methods give the same score?

You must show your working.

[4]

James is a decorator.

He has to prepare estimates of the final prices of jobs for potential customers. In the past he based each estimate on the amount of time he thought the job would take.

He wants to base future estimates on the surface area of the walls and ceilings that he has to decorate.

To work out how to do this, James uses data from his last 10 jobs, as shown in the table.

A scatter diagram of $P$ against $A$ is shown below.

One of the jobs required extra time to prepare the walls before decorating.

(i) Using the other 9 points, calculate the equation of the regression line of $P$ on $A$.

[2]

(ii) Draw your regression line on the scatter diagram.

[2]

James decides to use the equation of the regression line as the basis for future estimates.

He also decides to add an extra charge if a lot of preparation is needed.

The extra charge is based on the surface area, and the rates are shown in the table.

James uses this new method to work out an estimate.

The surface area he has to decorate is 84 m²

A large amount of preparation will be needed.

Work out his estimate.

Anna looks after a forest of Scots pine trees.

Anna wants to estimate the mean height of trees in the forest.

She makes point estimates of the mean height of trees in different areas of the forest.

The table shows three of Anna’s point estimates.

They were made on the same day.

Work out the best estimate of the population mean.

Each tree in the forest is numbered.

Trees numbered from 001 to 225 are between 20 and 40 years old.

Anna wants to choose a sample of 10 of these trees at random.

To do this she uses 3-digit random numbers.

Complete the table below to show the number of each tree in Anna’s sample.

The table shows the diameter, height and age of each tree in Anna’s sample.

Anna wants to use the diameter of a tree to estimate

its height
its age.

Which of these estimates is likely to be more reliable?

Use product moment correlation coefficients to help you decide.

Give an example of two variables which have both of the following features.

The correlation between the variables is strong.

One of the variables causes the other to change.

State the variable that causes the other to change.

Give an example of two variables which have both of the following features.

The correlation between the variables is strong.

The variables do not cause a change in each other.

Explain why the variables do not cause a change in each other.

Which of the following cannot be a correct value for a product moment correlation coefficient?

Here are four scatter diagrams, A, B, C and D.

Complete the table by matching the coefficient to the letter of the correct diagram. You do not need to calculate the coefficient.

The scatter diagram shows the correlation between the speed of the blades of a windmill and wind speed.

Bill looks at the diagram and says,

“Increasing the speed of the blades of the windmill causes the wind speed to increase.”

Is Bill correct?

Explain your answer.

Every weekday, Alex drives from home to work.

He notices that his journey time changes depending upon the time he leaves home.

He collects this data over a 2-week period.

Complete the scatter diagram of $J$ against $T$ by plotting the last two points from the table above.

It is appropriate to exclude two of the points when calculating the equation of the regression line of $J$ on $T$.

Identify the two points.

Give a reason for your answer.

The points are ( ____ , ____) and ( ____ , ____)

Excluding the two points you identified in question (b) (i), calculate the equation of the regression line of $J$ on $T$.

Draw your regression line on the scatter diagram.

Work out the time that Alex should leave home in order to arrive at 8.30 am