1.3 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  on is calculated to be 

Find the value of  correct to 2 decimal places.

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, , and the number of exams sat by those students, .  After collecting data from 1000 students, they construct a scatter diagram and find the equation of the regression line of  on  to be .

Given the following summary statistics:

Find the values of  and  correct to 4 significant figures and hence, 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.

392     414     536     474     212     396     427     545     459      234

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, , made by various workers along with how many hours of training,  hours, they have received.

Worker	1	2	3	4	5	6	7	8	9
		512	499	359	393	432	456	520	475
	18	24	22.5	15	16	20	21	22	21

Given that the equation of the regression line of  on  is ,  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.

Worker	10	11	12	13	14	15	16	17	18
	600	598	584	602	593	585	591	601	605
	29	28.5	32	29	34.5	30.5	37	31	30

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?

A snack shop owner has noticed that the sale of energy drinks seems to increase later in the school term.  He decides to collect data over the final ten days of a school term to see if the sale of the energy drinks per day, , increases as the number of days until the school holidays, , decreases.

Over the ten days the snack shop owner collects the following summary statistics:

Find the value of the product moment correlation coefficient between  and  correct to 4 decimal places.

The snack shop owner uses this data to calculate the regression line of  on  and uses it to predict the number of energy drinks he will sell on the first day of the new term, when there are still 90 days until the holidays.  State two reasons why this is unlikely to give a reliable prediction.

Hatter has noticed that over the past 50 years there seems to be fewer hatmakers in London. He also knows that global temperatures have been rising over the same time period. He decides to see if there could be any correlation, so he collects data on the number of hatmakers each year in London , and the yearly global mean temperatures,  from the past 50 years and records the information in the graph below.

Explain why the product moment correlation coefficient between  and  can not be .

M Hatter calculates the following statistics for his data:

Find the value of the product moment correlation coefficient, , correct to 4 decimal places.

Hatter concludes that the rise in mean global temperature is what is causing hatmakers in London to go out of business.

Explain whether M. Hatter’s conclusion is fully justified.

On 21^st January 2020, doctors in China started recording and reporting the number of new daily cases of an unknown virus.  Over the first five days there were 916 new cases.

The table below shows the number of new cases, , of the virus in a town in China and the number of days, , after 21^st January 2020. The number of new cases were not available for the 6^th and 11^th days for this town.

Given that for days 1 to 5 the value of , use the data you have for the 10 days when cases were reported to calculate the values of  and .

The value of the product moment correlation coefficient between the number of days after 21^st January 2020 and the number of new cases was calculated as  . 

Use this value of  and your answers from part (a) to find the value of .

The equation for the regression line of  on  is found to be  .

Find the value of  correct to 1 decimal place.     

A restaurant owner, Mr Capazio, suspects that there is positive correlation between the number of alcoholic beverages a person has with their meal and the amount of time it takes them to pay their bill at the end of the evening.  He decides to collect some data to test his theory.

The table below shows the number of alcoholic beverages consumed, , and the amount of time taken to pay the bill,  seconds, for a sample of 10 visitors to the restaurant on a particular night.

Number of drinks,
Time taken,  seconds

Using the coding  find the values of and  .

Calculate the product moment correlation coefficient between  and .

Mr Capazio calculates the regression line of  on  to be  . 

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 (, measured in volts) to an electrical circuit in her lab and recording the resulting currents (, 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  on v to be  = 0.056+0.332.  

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?

An A level music teacher is collecting data on the number of hours his students spend rehearsing their  final piece, , and the number of mistakes made in their exam, .  He calculates the following summary data of ten of his students.

Show that the product moment correlation coefficient for these data is , correct to 3 decimal places. 

The music teacher calculates the equation of the regression line of  on  to be  .

Show that  correct to 3 significant figures and find the value of .

An estate agent, Terry, claims that there is a correlation between the value of a house, (£1000) and the distance between that house and the nearest nightclub,  (miles).

Terry has a database containing over 100 houses and he takes a random sample of seven houses to investigate his claim.  The scatter graph below shows the results:

Terry calculates the product moment correlation coefficient as . Using the scatter graph, explain how you know Terry’s PMCC value is incorrect.

Terry’s results are recorded below:

	1.8	2.1	2.5	3.7	4.9	5.2	7.2
	500	560	330	250	260	180	190

Given that and 

find the values of  ,

The table below shows some data on daily mean air pressure,  (hPa), and daily total sunshine,  (mins), in a certain area over a random sample of 7 days.

	1017	1023	1020	1022	1011	1019	1017
	328	380	260	372	304	316	288

Fill in the table below using the coding

	6
	7

Find the values of

Use your answers to part (b) to find the product moment correlation coefficient for the daily mean air pressure and daily total sunshine. Comment on the relationship between the two variables.

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

The regression line of  on  is calculated, and it is used to predict values of  for  and , the values returned are  and  respectively.

Find the equation of the regression line of  on .

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  on  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 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  on  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.

Effie loves watching the turtles play in the lake near her house, she thinks that there is a relationship between the number of turtles and the number of ducks that live on different parts of the lake.  She decides to investigate this further and gathers data on duck and turtle populations from six wildlife centres.  Effie records the data in the table below.

Effie codes the results using the codes  and  .  Some of the values for  and  are recorded in the table below.

Centre	A	B	C	D	E	F
			2			1.5
	1					1.2

Find the values of  and hence, complete the table.

Use your answers to parts (a) and (b) to find the regression line of  on , show your working clearly.

Charlie is interested to find out if there is positive correlation between the number of letters in someone’s name, , and the time,  rounded to the nearest five seconds, it takes her six-year-old sister to correctly guess the spelling of the name.  She decides to test this by looking at a random sample of different names and timing how long it takes her sister to guess their spelling.

Letters,
Time,
Frequency

Given that , find the value of  and hence find the number of names in Charlie’s sample.

Charlie calculates the equation of the regression line of  on  to be  and the product moment correlation coefficient to be 0.84428 correct to 5 decimal places.

Find the value of and  each to 5 significant figures. 

Two variables,  and , are thought to be connected in the form , where  and  are constants.  A random sample of 100 pairs of data are taken from data sets  and and are coded such that   and .  The data from the coded records are summarised below.

Given that the product moment correlation coefficient between  and  is , find the value of  correct to 3 significant figures.

Find the least squares regression line of  on 

Find the least squares regression line of  on