Correlation & Regression (AQA A Level Maths: Statistics): Exam Questions

The table below shows data from the United States regarding annual per capita cheese consumption (in pounds) and the divorce rate (number of divorces per 1000 people) for ten years between 2000 and 2018:

Draw a scatter diagram to represent this data, with per capita cheese consumption on the horizontal axis and divorce rate on the vertical axis.

(i) Describe the correlation between per capita cheese consumption and divorce rate.

(ii) A newspaper reports the following statement:

"Eating more cheese reduces the chances of getting a divorce."

Comment on the validity of the statement.

Priya has been applying different voltages ($v$, measured in volts) to an electrical circuit in her lab and recording the resulting currents ($i$, measured in amps). The smallest voltage she applied was 0.5 volts, and the largest voltage she applied was 120 volts.

She found the equation of the regression line of $i$ on $v$ to be $i=0.056+0.332v$.

(i) Interpret the value 0.332 in this context.

(ii) Use the equation to predict the current for a voltage of 70 volts.

Explain why it would not be sensible to use the regression equation to work out:

(i) the current resulting from a voltage of 2000 volts

(ii) the voltage corresponding to a current of 20 amps.

A ride-sharing app collected data on the time, $t$ minutes, taken to complete a journey of distance, $d$ miles. Data from a random sample of 8 journeys is detailed in the table below.

By plotting a scatter diagram of $t$ on $d$ for this data, explain whether or not it is appropriate to use a linear regression model on this data.

Using a new random sample of thousands of journeys, the ride-sharing app calculated the regression line of time on distance to be $t=-1.8+5.9d$.

The regression equation predicts that for journeys less than 0.3 miles the time taken will be less than zero minutes.  What is the most likely reason that the regression equation gives this false prediction?

The following table shows the height, $h$ cm, and weight, $w$ kg, for each of eleven students at a sixth form college.

The following statistics were calculated for the data on height:

mean $=159.5$ cm, standard deviation $=45.3$ cm

An outlier is an observation which lies more than 2 standard deviations from the mean.

(i) Show that $h=17$ is an outlier.

(ii) Explain why this outlier should be omitted from the data.

With the outlier data excluded, the equation of the regression line of $w$ on $h$ is $w=-87.6+0.845h$.

(i) Exclude the outlier from the recorded measurements and draw a scatter diagram to represent the data for the remaining ten students.

(ii) Draw the regression line on your diagram.

A maths teacher randomly selects 10 students from a class of 30 to answer a survey. The survey asks students how many practice questions they completed when revising for a recent test, $Q$, and their percentage score in that test, $S$ %. Summary statistics for $Q$ are shown below.

$\overline{Q}=21$, range of $Q=20$

The equation of the regression line of $S$ on $Q$ is $S=34+2Q$.

Explain which variable is the response variable.

Comment on the reliability of using the regression equation to:

(i) estimate the scores of the other students in the maths class,

(ii) estimate the scores of this cohort of students in a science class.

Ella measures how the extension, $x$ mm, of a thin piece of metal wire varies with the force applied to it, $F$ kN. She records her results in the table below.

Ella calculates the regression line of $F$ on $x$ to be $F=0.004-69.3x$.

Explain why this equation must be wrong.

The correct equation for the regression line of $F$ on $x$ is $F=6.16+67.6x$.

Interpret the value of 67.6 in this context.

Using the correct regression line, Ella estimates that if she applies a force of 1000 kN then the wire will show an extension of 14.7 mm. 

Give two reasons why Ella’s estimate may not be accurate.

The table below shows a comparison of the average house price, $H$ (£1 000), and the average yearly income, $I$ (£1 000), for different areas around the UK in 2021.

(i) Plot a scatter diagram of $I$ against $H$, and

(ii) describe the correlation shown.

The equation of the regression line of $I$ on $H$ is calculated to be $I=0.06H+15.92$.

A news reporter uses this to claim that if you want a salary of £50 000, all you need to do is buy a house that costs £568 000.

Comment on the validity of the news reporter's claim.

Two researchers, Alwyn and Beth, are working on a project collecting data about the self-reported happiness of students on a scale from 0 to 10, $H$, and the number of exams sat by those students, $n$. After collecting data from 1000 students, they construct a scatter diagram and find the equation of the regression line of $H$ on $n$ to be $H=7.63-0.82n$.

Explain what correlation the data is likely to show in the scatter diagram.

What information about the original data set would need to be checked before using the regression line equation to estimate the self-reported happiness of a student sitting 8 exams?

After calculating the equation of the line of regression, Alwyn accidentally deletes all the data collected about the self-reported happiness scores.  Alwyn says it’s not a problem since he can use the regression line and the number of exams sat to recalculate all the values. Beth says that Alwyn is wrong and the original data is lost forever.

Explain which researcher is correct.

A consultant is trying to improve the efficiency of how a factory making chewing gum operates.  To help them do this, they collect many types of data about the factory workers.  One such type of data is the number of chewing gum packets made per shift.  The list below shows the number of chewing gum packets made by a particular worker (Worker 1) during the last 10 shifts worked.

Calculate the mean number of chewing gum packets made per shift by Worker 1 to the nearest whole number of packets.

The table below shows the mean number of chewing gum packets, $N$, made by various workers along with how many hours of training, $T$ hours, they have received.

(i) Including your answer from (a), plot a scatter diagram of the data in the table above.

(ii) Given that the equation of the regression line of $N$ on $T$ is $N=18T+95$, add the regression line to your scatter diagram.

The consultant then goes on to collect even more data on other factory workers and records some of it in the table below.

Without adding this new data to your scatter diagram, what advice could the consultant give to the factory to improve the efficiency of their workers?

Paige takes a sample of 9 cities throughout the UK to compare the percentage of people living in a city who identify as vegan, $V$ %, and the percentage of restaurants offering vegan options in that same city, $R$ %.

The regression line of $R$ on $V$ is calculated, and it is used to predict values of $R$ for $V=1.35$ and $V=1.03$. The values returned are $R=70.73$ and $R=50.314$ respectively.

Find the equation of the regression line of $R$ on $V$.

In one of the cities, 1.16% of people were vegan and 55.9% of restaurants offered vegan options.

Use the equation of the regression line of $R$ on $V$ to estimate the percentage of restaurants offering vegan options in a city in which 1.16% of people are vegan. Give your estimated value of $R$ to 3 significant figures. Compare this to the information above.

Paige discovers that in one city every restaurant offers vegan options. Paige suggests that the equation of the regression line of $R$ on $V$ can be used to find the percentage of people in this city who identify as vegan. Explain why Paige is likely wrong.

An owner of a beach resort is comparing parasol sales, £$p$, and sun cream sales, £$s$, at the resort over a period of eleven days. The data is standardised by coding the variables using $x=\frac{s-153}{103}$ and $y=\frac{p-32}{37}$. The values for the first ten days are plotted on the scatter diagram below.

On the eleventh day, the resort sold £246 worth of sun cream and £69 worth of parasols. Use this information to complete the scatter diagram.

The equation for the regression line of $y$ on $x$ is $y=0.19+0.83x$.

(i) Show that by using the regression line of $y$ on $x$ and the coding equations above, the regression line of $p$ on $s$ can be written in the form $p=a+bs$, where $a$ and $b$ are constants to be found to 3 significant figures.

(ii) Hence, or otherwise, find an estimate for the amount of parasol sales on a day where there are £170 of sun cream sales.