2.4 Correlation & Regression

The table below shows a comparison of the average house price, H (£100 000), and the average yearly income, I (£10 000), for different areas around the UK in 2021.

The equation of the regression line of I on H is calculated to be I = 0.06H+15.92. 
A particularly unscrupulous politician uses this to claim that if you want a salary of £35 000, all you need to do is buy a house that costs £583 000.

Comment on the validity of the politician’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.

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?

The table below shows data from the large data set on the daily mean temperature in Heathrow, T_H℃, and the daily mean temperature in Beijing, T_B℃, on the same day for a random sample of 10 days during May 2015.

The equation for the regression line of T_B on T_H is T_B = 4.55+1.46T_H. 

The table below shows T_H and T_B on the same day for a random sample of 3 other days during May 2015.

Considering this second sample, use the regression line equation and the values of T_H to predict values of T_B and find the average percentage difference of these estimated values of T_B from the true values of T_B. Hence, comment on how accurately this regression line equation can predict values.

Using your knowledge of the large data set, explain whether there is likely to be a causal relationship between T_H and T_B.

A researcher claims that this correlation between T_H and T_B is a coincidence.  How could you use data from the large data set to check this claim?

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.

Myfanwy 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.  

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

Myfanwy’s lab partner suggests that the value 0.056 in the regression equation represents the current in the circuit when the voltage applied is zero.  Explain why he might suggest this, but also suggest a reason why his interpretation is most likely incorrect.

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.

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

Based on your diagram, along with the regression equation, to what extent would you say that a person’s height may be used as an accurate predictor of his or her weight?

The table below shows data from the large data set on the daily mean pressure, p (hPa), and daily total sunshine, s (hrs), in Camborne for a random sample of 12 days in 2015.

The equation of the regression line of s on p is  s = 270.5 + 0.271 p.

Give an interpretation of the value of the gradient of the regression line.

Use your knowledge of the large data set to explain whether there is likely to be a causal relationship between daily mean pressure and daily total sunshine.

Explain why it would not be reliable to use this regression equation to predict:

Use the regression equation to predict the daily total sunshine for a day with a daily mean pressure of 1017 hPa.  How does this compare with the actual data from the 1017 hPa day in the table?

The table includes data for the 1^st and 15^th days of each of the months covered by the large data set for 2015.

Suggest another factor that may have had an effect on the daily total sunshine data recorded in the table.  Explain how taking account of that factor might allow a better model to be produced describing the relationship between daily mean pressure and daily total sunshine.

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.

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 app uses this regression equation to predict that a journey of distance 7 km would take 39.5 minutes.  Explain why this is incorrect.

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?

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

=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.

Use the regression equation to find an estimate for the mean value and range of S. State any assumptions that are needed.

Comment on the reliability of using the regression equation to:

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 =  and  y = . The values for the first ten days are plotted on the scatter diagram below.

An ice cream shop owner in Camborne is trying to use data from the large data set alongside their own past sales data to help them estimate future sales.  The mean daily temperature per month, T ℃, is shown with the mean daily number of ice creams sold per month, I, from 2015 in the table below.

The equation for the regression line of I on T is  I = 429.5 + 42.5T.

Find an estimate for the expected total number of ice creams sold in the month of July if the average daily temperature for that month is 14.9° C. 

Suggest which other data from the large data set could be used to improve this model.

The ice cream shop owner claims that there is a causal link between I and T, and so if the shop sells more ice cream, the month will be hotter. 

Comment on this claim.