Correlation and regression

The scatter diagram shows the marks each of ten students got for a Maths test and for a Music test.

One of these points is an outlier.

On the diagram, draw a circle around the outlier.

Excluding the outlier, describe the correlation shown in the diagram.

The outlier is removed from the data. The mean of the Maths marks is 52 The mean of the Music marks is 63

Excluding the outlier, draw a line of best fit that passes through the double mean point on the scatter diagram.

30 students did both tests. The scatter diagram shows the marks of 10 of these students.

Suggest one adjustment that could be made to improve the reliability of any conclusions made using the scatter diagram.

In world athletics, an athlete’s personal best results in different track and field events are recorded.

Zara wants to compare a sample of athletes’ longest recorded long jump against their height. She is going to present this information on a scatter diagram.

Explain why it is appropriate to use a scatter diagram.

Zara plotted the athletes’ heights on the x‑axis.

Give a reason why plotting the heights on the x‑axis is appropriate.

Zara selects 12 athletes ranked as some of the world’s best long jumpers. She calculates the Pearson Product Moment Correlation Coefficient (PMCC) for their personal best long jump distances against their heights.

The PMCC value is 0.02 (Source: www.worldathletics.org)

Describe the correlation shown by the PMCC.

Using data for the same 12 athletes, Zara compares their personal best long jump distances (m metres) to their fastest times in a 100 metre race (t seconds).

She calculates the PMCC to be –0.5

Describe and interpret the relationship between the athletes’ personal best long jump distances and their personal best 100 metre race times in context.

The data for the 12 athletes is shown in the table below.

World rank April 2023	Long jump athlete	Personal best long jump distance (m metres)	Personal best 100 m race time (t seconds)
1	M Tentoglou	8.60	10.70
3	M Dendy	8.42	10.31
4	S Ehammer	8.45	10.46
5	T Montler	8.38	10.49
8	E Cáceres	8.37	10.37
10	T Gayle	8.69	10.13
11	E Lasa	8.28	11.03
15	H Frayne	8.34	10.96
16	Y Hashioka	8.36	10.53
17	R Ju ka	8.31	11.07
26	C Mitrevski	8.21	10.27
30	H Santos	8.19	11.77

(Source: www.worldathletics.org)

The linear regression line for these data is t = 25 – 1.7m

In April 2023, M Massó was ranked number 2 in the world for the long jump. His personal best long jump distance was 8.39 m.

Zara uses this regression line to predict that M Massó could run the 100 m race in 10.7 seconds.

Comment on the reliability of this prediction in context.

Timur is investigating the heights and weights of rugby players. He collected data from the internet about the heights and weights of players from the Wales rugby squad.

The players from the Wales rugby squad are classed as Forwards or Backs. Timur draws a scatter diagram for the heights and weights of some of the Backs from the Wales rugby squad.

Explain why a scatter diagram is appropriate for the type of data Timur collected.

The data for three of the players was not plotted on the scatter diagram. The height and weight of each of these players is given in the table below.

Complete the scatter diagram by plotting the points for players A, B and C.

Describe and interpret the type of correlation shown by the scatter diagram.

The double mean point for the heights and weights of all of the Backs is (184, 92).

Draw a line of best fit through the double mean point.

Timur also collected the heights and weights of players from the England rugby squad. He found the Spearman’s rank correlation coefficient for the heights and weights of the Forwards from the England rugby squad.

The correlation coefficient was 0.00

Interpret this correlation coefficient in context.

Timur used statistical software to find the following information about the heights (x cm) and weights (y kg) of the Backs from the Wales rugby squad and the Backs from the England rugby squad.

Compare the Spearman’s rank correlation coefficients and interpret this comparison in the context of the question.

Timur uses the information in the table to conclude that the weight of the England rugby squad Backs increases faster than the weight of the Wales rugby squad Backs as their height increases.

Assess the validity of Timur’s conclusion with reference to the statistical results.

Researchers wanted to find a method to predict the height of ancient Egyptians based upon bones from their skeletons.

They calculated the Pearson’s product moment correlation coefficient between the length of particular bones and the height of the skeleton for some male skeletons and for some female skeletons.

They also found regression equations for the relationship between bone length (x centimetres) and height (y centimetres) for each of these bones in male skeletons and in female skeletons.

The table gives information about these product moment correlation coefficients and these regression equations.

Which bone measurement would you recommend using to estimate the height of an ancient Egyptian? Give a reason for your answer.

Interpret in context the figure 2.594 in the regression equation for the humerus of male skeletons.

Use the regression equations to compare the relationships between bone length and height for the different bones. Include in your answer comparisons between male and female ancient Egyptians.

Dina suggests comparing the y‑intercepts for the regression equations.

Is Dina’s suggestion appropriate?

A museum has some bones that were recovered from an Egyptian pyramid.

The museum wants to predict the height of the ancient Egyptians from whom the bones were recovered.

Is it appropriate to use these regression equations to make this prediction?

Jim is investigating the relationship between air temperature and altitude.

He has found data on the internet showing the average air temperature at different altitudes on one particular day.

The table shows the results he collected.

(Source: adapted from www.engineeringtoolbox.com)

He uses this information to draw the scatter diagram.

Describe and interpret the correlation shown in the scatter diagram.

Draw a line of best fit on the scatter diagram.

Use the scatter diagram and your line of best fit to predict the average air temperature at an altitude of 4400 metres.

Jim wants to predict the average air temperature at an altitude of 11 000 metres.

Discuss whether or not it is appropriate to use the line of best fit to make his prediction.

A vet carried out a survey to see if there is a relationship between the weight of a dog and its lifespan. The vet found the mean weight, in kg, and the median lifespan, in years, of a sample of five dogs in each of ten breeds of dogs.

Here are her results.

(Source: adapted from www.instituteofcaninebiology.org)

The vet used statistical software to draw this scatter diagram for her results.

From the scatter diagram the vet concluded that there is a relationship between the weight of a dog and its lifespan.

Using the given statistical information,

assess whether or not the vet’s conclusion is appropriate,

assess the reliability of the vet’s conclusion.

Gaby is researching the final position of a football team in the English Premier League and the mean value of all the players in the team.

Suggest a diagram that Gaby could draw to see if there is an association between the final position and the mean value.

The table gives information about the data for the 2018–2019 season that Gaby used.

(Source: www.transfermarket.com)

Calculate Spearman’s rank correlation coefficient for the information in the table. You may complete the d column and d² column of the table to help you.

Interpret your answer to part (b) in the context of Gaby’s research. You should refer to the effects of any anomalous data.

Amelia suggests that Pearson’s product moment correlation coefficient should be used instead of Spearman’s rank correlation coefficient to measure the correlation between the data Gaby is researching.

Discuss whether or not Amelia’s suggestion is appropriate.

Irina is investigating whether the percentage of the population of a country living in urban areas has an effect on the life expectancy in that country.

Suggest a hypothesis Irina could use for her investigation.

Irina collected the following information about 10 countries.

The Urban population (%) refers to the percentage of the population of the country who live in urban areas.

Irina used statistical software to draw a scatter diagram for the information in the table.

Give a reason why a scatter diagram is an appropriate diagram to use.

For this investigation, which variable is the explanatory variable? Give a reason for your answer.

The scatter diagram from the statistical software is shown below.

Explain, giving a statistical reason, whether or not this scatter diagram supports your hypothesis in part (a).

For these 10 countries, the double mean point of the data is (63.9, 77.8).

Using this information, draw a line of best fit on the scatter diagram.

Using statistical software, Irina finds that the gradient of the line of best fit should be 0.19

Interpret the gradient of the line of best fit.

Irina now finds that South Africa has Urban population 65% and Life expectancy 63 years.

Determine how this information for South Africa fits with the relationship shown in the scatter diagram for the other countries.

Irina is investigating whether the percentage of the population of a country living in urban areas has an effect on the life expectancy in that country.

Suggest a hypothesis Irina could use for her investigation.

Irina collected the following information about 10 countries.

The Urban population (%) refers to the percentage of the population of the country who live in urban areas.

Irina used statistical software to draw a scatter diagram for the information in the table.

Give a reason why a scatter diagram is an appropriate diagram to use.

For this investigation, which variable is the explanatory variable? Give a reason for your answer.

The scatter diagram from the statistical software is shown below.

Explain, giving a statistical reason, whether or not this scatter diagram supports your hypothesis in part (a).

For these 10 countries, the double mean point of the data is (63.9, 77.8).

Using this information, draw a line of best fit on the scatter diagram.

Using statistical software, Irina finds that the gradient of the line of best fit should be 0.19

Interpret the gradient of the line of best fit.

Irina now finds that South Africa has Urban population 65% and Life expectancy 63 years.

Determine how this information for South Africa fits with the relationship shown in the scatter diagram for the other countries.

Rachel collects data on the engine size (in litres) and the fuel economy during city driving (in miles per gallon) for each car in a sample of 61 cars of various types.

The scatter diagram was drawn for this information by statistical software.

Describe and interpret the type of correlation shown by the scatter diagram.

The statistical software worked out that the mean engine size for the 61 cars in the sample is 3.3 litres, correct to one decimal place. The statistical software worked out that the total of the fuel economies for the 61 cars in the sample is 1183

On the scatter diagram,

plot with a cross (X) the double mean point (, ) of the data,

draw a line of best fit through the double mean point.

Rachel wants to predict the fuel economy of a car with an engine size of 7 litres.

Explain why it is not appropriate to use the line of best fit on this scatter diagram to find this prediction.

Rachel collects data on the engine size (in litres) and the fuel economy during city driving (in miles per gallon) for each car in a sample of 61 cars of various types.

The scatter diagram was drawn for this information by statistical software.

Describe and interpret the type of correlation shown by the scatter diagram.

The statistical software worked out that the mean engine size for the 61 cars in the sample is 3.3 litres, correct to one decimal place. The statistical software worked out that the total of the fuel economies for the 61 cars in the sample is 1183

On the scatter diagram,

plot with a cross (X) the double mean point (, ) of the data,

draw a line of best fit through the double mean point.

Rachel wants to predict the fuel economy of a car with an engine size of 7 litres.

Explain why it is not appropriate to use the line of best fit on this scatter diagram to find this prediction.

A team of researchers investigated the time intervals between calls of tree frogs.

They investigated two species of tree frog: the Italian tree frog and the European tree frog.

They found that there was a correlation between air temperature, xC, and amount of time between calls, y milliseconds, for each of the two species of tree frog.

The table gives the equation of the regression line for the data for each of the two species of tree frog.

When x = 27, both equations give the same value of y, correct to the nearest whole number.

Explain what this means in context.

The equation of the regression line gives information about the relationship between temperature and time between calls for each of the two species of tree frog.

Using this information, compare the relationship for the Italian tree frog with the relationship for the European tree frog.

A second group of researchers investigated the relationship between temperature and the time between calls for a third species of tree frog. They also obtained the equation of the regression line for this species of tree frog.

The results of the different teams of researchers are to be compared.

Give one potential limitation of doing this.

Mike is investigating whether or not there is a relationship between the distance from the nearest car park to a restaurant and the annual profit made by the restaurant.

He finds out the distance, in metres, from the nearest car park to a restaurant and the annual profit, in £, made by the restaurant. He does this for twenty restaurants in a city.

Mike then draws a scatter diagram for this information.

Explain why annual profit is the response variable for this scatter diagram.

Mike’s hypothesis is that, for these restaurants, the further the restaurant is from its nearest car park the less is its annual profit.

Explain, giving a statistical reason, whether or not the scatter diagram supports Mike’s hypothesis.

Mike wants to draw a line of best fit on the scatter diagram. Using statistical software he obtains the following information about these restaurants.

Using this information, draw a line of best fit on the scatter diagram.

Interpret the value of the intercept of the line of best fit on the Annual profit axis.

Restaurant A and restaurant B are two other restaurants in the city. Restaurant A is 250 m from its nearest car park. Restaurant B is 700 m from its nearest car park. Mike uses the scatter diagram to find an estimate for the annual profit of each of these restaurants.

Explain which of these two estimates will be the more reliable estimate.

Mike finds a positive correlation between the number of tables at a restaurant and its annual profit. He concludes that as the number of tables increases this causes the annual profit to increase.

Explain whether or not this conclusion is valid.

Mike reads an article in a newspaper that says that restaurant profits for the top 100 restaurants had fallen from £345 million to £125 million in the past year. (Source: telegraph.co.uk)

Using the data in the newspaper article, calculate the percentage decrease in restaurant profits for the top 100 restaurants in the past year.

Mike is investigating whether or not there is a relationship between the distance from the nearest car park to a restaurant and the annual profit made by the restaurant.

He finds out the distance, in metres, from the nearest car park to a restaurant and the annual profit, in £, made by the restaurant. He does this for twenty restaurants in a city.

Mike then draws a scatter diagram for this information.

Explain why annual profit is the response variable for this scatter diagram.

Mike’s hypothesis is that, for these restaurants, the further the restaurant is from its nearest car park the less is its annual profit.

Explain, giving a statistical reason, whether or not the scatter diagram supports Mike’s hypothesis.

Mike wants to draw a line of best fit on the scatter diagram. Using statistical software he obtains the following information about these restaurants.

Using this information, draw a line of best fit on the scatter diagram.

Interpret the value of the intercept of the line of best fit on the Annual profit axis.

Restaurant A and restaurant B are two other restaurants in the city. Restaurant A is 250 m from its nearest car park. Restaurant B is 700 m from its nearest car park. Mike uses the scatter diagram to find an estimate for the annual profit of each of these restaurants.

Explain which of these two estimates will be the more reliable estimate.

Mike finds a positive correlation between the number of tables at a restaurant and its annual profit. He concludes that as the number of tables increases this causes the annual profit to increase.

Explain whether or not this conclusion is valid.

Mike reads an article in a newspaper that says that restaurant profits for the top 100 restaurants had fallen from £345 million to £125 million in the past year. (Source: telegraph.co.uk)

Using the data in the newspaper article, calculate the percentage decrease in restaurant profits for the top 100 restaurants in the past year.

Here is a scatter diagram.

The Spearman’s rank correlation coefficient and the Pearson’s product moment correlation coefficient are each going to be calculated for the data shown in the scatter diagram.

How would you expect the values of these two correlation coefficients to compare?

The Australian Government collected data on crop yields from fields. The data collected were the yield of wheat grown in a field, in tonnes per hectare (t/ha), and the yield of other crops grown in the same field (t/ha) in a different year.

The table gives the Pearson’s product moment correlation coefficient for the wheat yield and the barley yield and for the wheat yield and the oats yield. The table also gives the equation of the regression line in each case.

Compare the relationship between wheat yield and barley yield with the relationship between wheat yield and oats yield.

You should refer to both correlation coefficients and to the equations of both regression lines in your comparison.

This year Louise has a field planted with wheat. She wants to use the information in the table to decide whether to plant barley or to plant oats in the field next year. She plans to plant the crop which is predicted to give the greatest yield.

Louise solves the equation 1.24x – 0.30 = 1.52x – 1.05 She gets the answer

Show that the answer Louise gets is correct.

Explain how Louise could use her answer to decide which of barley or oats to plant next year.

Give a limitation of the data that could affect the result of Louise’s decision.

Naomi recorded the weight, in thousands of tons, and the maximum number of passengers, in hundreds, for 11 cruise ships.

Naomi drew the scatter diagram below for her results.

One of the 11 cruise ships has a weight of 116 000 tons.

For this ship, write down its maximum number of passengers.

Draw a line of best fit on the scatter diagram.

Describe and interpret the type of correlation shown by the scatter diagram.

A new cruise ship is being built. The ship will have a weight of 170 000 tons.

Naomi plans to use the line of best fit on her scatter diagram to predict the maximum number of passengers for the new cruise ship.

Explain whether or not it is appropriate to use the line of best fit to make her prediction.