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

Stav believes that there is a correlation between Daily Total Sunshine and Daily Maximum Relative Humidity at Heathrow.

He calculates the product moment correlation coefficient between these two variables for a random sample of 30 days and obtains .

Carry out a suitable test to investigate Stav’s belief at a 5% level of significance.

State clearly

• your hypotheses

• your critical value

On a random day at Heathrow the Daily Maximum Relative Humidity was 97%.

Comment on the number of hours of sunshine you would expect on that day, giving a reason for your answer.

Marc took a random sample of 16 students from a school and for each student recorded

• the number of letters, , in their last name

• the number of letters, , in their first name

His results are shown in the scatter diagram.

Describe the correlation between and .

Marc suggests that parents with long last names tend to give their children shorter first names.

Using the scatter diagram comment on Marc’s suggestion, giving a reason for your answer.

The results from Marc’s random sample of 16 observations are given in the table below.

Use your calculator to find the product moment correlation coefficient between and for these data.

Test whether or not there is evidence of a negative correlation between the number of letters in the last name and the number of letters in the first name.

You should

• state your hypotheses clearly

• use a 5% level of significance

Tessa owns a small clothes shop in a seaside town. She records the weekly sales figures, £, and the average weekly temperature, °C, for 8 weeks during the summer.

The product moment correlation coefficient for these data is −0.915

Stating your hypotheses clearly and using a 5% level of significance, test whether or not the correlation between sales figures and average weekly temperature is negative.

Suggest a possible reason for this correlation.

Tessa suggests that a linear regression model could be used to model these data.

State, giving a reason, whether or not the correlation coefficient is consistent with Tessa’s suggestion.

State, giving a reason, which variable would be the explanatory variable.

Tessa calculated the linear regression equation as

Give an interpretation of the gradient of this regression equation.

A teacher, Ms Pearman, claims that there is a positive correlation between the number of hours spent studying for a test and the percentage scored on it.

Write down suitable null and alternative hypotheses to test Ms Pearman's claim.

Ms Pearman takes a random sample of 25 students and gives them a week to prepare for a test. She records the percentage they score in the test, %, and the amount of revision they did, hours.

Ms Pearman calculates the product moment correlation coefficient for these data as .

Given that the p-value for the test statistic is , test at the 5% level of significance whether Ms Pearman's claim is justified.

Ms Pearman decides to use a linear regression model for these data. She calculates the equation of the regression line of on to be .

(i) Give an interpretation of the value 21.3 in context.

(ii) Give an interpretation of the value 5.29 in context.

The following table shows the number of hours spent learning to drive, , and the number of mistakes made in the driving test, , of ten college students.

The product moment correlation coefficient for these data is . A driving instructor, Dave, believes there is a negative correlation between the number of hours spent learning to drive and the number of mistakes made in the driving test.

(i) Write down suitable null and alternative hypotheses to test Dave's claim.

(ii) Test, at the 1% level of significance, whether Dave's claim is justified, given that the relevant critical value is .

Dave calculates the equation of the regression line of on to be .

State, giving a reason, whether or not the correlation coefficient is consistent with the use of a linear regression model.

(i) Explain why the linear regression model could be unreliable for predicting the number of mistakes a student would make on their driving test after learning for 30 hours.

(ii) By considering a student who has spent 80 hours learning to drive, give a limitation to the linear regression model.

The table below shows data from the United States regarding annual per capita chicken consumption (in pounds) and the unemployment rate (% of population) between the years 2005 and 2014.

The product moment correlation coefficient for these data is . The critical values for a 10% two-tailed test are .

State what is measured by the product moment correlation coefficient.

(i) Write down suitable null and alternative hypotheses for a two-tailed test of the correlation coefficient.

(ii) Show that, at the 10% level of significance, there is evidence that the correlation coefficient is different from zero.

A newspaper's headline states:

"Eating chicken is the secret to reducing the unemployment rate in the US!"

Explain whether this headline is fully justified.

Jessica is researching whether there is a correlation between the productivity of university students and the number of hours sleep they get per night.

Write suitable null and alternative hypotheses to test for linear correlation.

Jessica takes a random sample of 25 students, measures their productivity during the day, and records how many hours sleep they had during the previous night. She calculates the product moment correlation coefficient and finds that .

The table below gives the critical values, for different significance levels, of the product moment correlation coefficient, , for a sample of size 25.

Jessica wishes to test, at the 10% level of significance, whether there is evidence that the correlation coefficient for the population is different from zero.

(i) Find the critical regions for Jessica's test.

(ii) Show that, at the 10% level of significance, there is no evidence of a linear correlation.

State, with a reason, whether there could be a relationship between students’ hours of sleep and their productivity.

Nicole is a biologist studying the growth of bacteria. She records the number of bacteria on an organism every hour. The table below shows her results for the first eight hours.

Nicole calculates the product moment correlation coefficient as .

Nicole claims that there is a positive linear correlation between the number of hours and the number of bacteria.

Test, at the 1% level of significance, whether Nicole's claim is justified. State your hypotheses clearly.

Mariam, Nicole's lab assistant, claims that there is an exponential relationship between the two variables. To test this, Mariam calculates the values of for the different values of .

Complete the table, giving your answers to three decimal places.

(i) Calculate the product moment correlation coefficient between and .

(ii) Comment on Mariam's claim that there is an exponential relationship between and .

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 results are recorded below.

Calculate the product moment correlation coefficient for this sample.

(i) Write down suitable null and alternative hypotheses for a two-tailed test to investigate Terry's claim.

(ii) Test Terry's claim using a 5% level of significance.

Suggest one way in which Terry could improve his investigation.

A student studying plants measures the hours of sunshine per day, , and the growth of a specific plant species, mm, over a two-week period.

A random sample of 12 days is selected. The product moment correlation coefficient (PMCC) for these data is .

State, with a reason, whether a linear regression model is appropriate for this data.

The student believes there is a positive correlation between hours of sunshine and plant growth.

Stating your hypotheses clearly, test at the 5% significance level whether there is evidence to support the student's belief.

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

Give an interpretation of:

(i) the value 3.8 in this model,

(ii) the value 1.2 in this model.

The student converts the growth data from millimetres to centimetres. State the value of the new product moment correlation coefficient.

State one limitation of using the product moment correlation coefficient to test for a relationship between two variables.

Barbara is investigating the relationship between average income (GDP per capita), US dollars, and average annual carbon dioxide (CO₂) emissions, tonnes, for different countries.

She takes a random sample of 24 countries and finds the product moment correlation coefficient between average annual CO₂ emissions and average income to be 0.446

Stating your hypotheses clearly, test, at the 5% level of significance, whether or not the product moment correlation coefficient for all countries is greater than zero.

Barbara believes that a non-linear model would be a better fit to the data.
She codes the data using the coding and and obtains the model

The product moment correlation coefficient between and is found to be 0.882

Explain how this value supports Barbara’s belief.

Show that the relationship between and can be written in the form where and are constants to be found.

Anna is investigating the relationship between exercise and resting heart rate.

She takes a random sample of 19 people in her year at school and records for each person

• their resting heart rate, beats per minute

• the number of minutes, , spent exercising each week

Her results are shown on the scatter diagram.

Interpret the nature of the relationship between and .

Anna codes the data using the formulae

The product moment correlation coefficient between and is

Test whether or not there is significant evidence of a negative correlation between and .

You should

• state your hypotheses clearly

• use a 5% level of significance

• state the critical value used

The equation of the line of best fit of on is

Use the equation of the line of best fit of on to find a model for on in the form

where and are constants to be found.

A snack shop owner has noticed that the sale of energy drinks seems to increase later in the school term. He conducts a hypothesis test at the 1% level of significance to see if the sale of the drinks, , increases as the number of days until the school holidays, , decreases.

(i) What type of correlation is the snack shop owner testing for?

(ii) State which of the two variables is the explanatory variable.

Over the final thirty days of term the owner keeps a record of the number of sales of energy drinks and, using this data, calculates the product moment correlation coefficient to be .

The table below gives the critical values, for different significance levels, of the product moment correlation coefficient, , for a sample size of 30.

(i) Write down the critical region for the hypothesis test.

(ii) Stating your hypotheses clearly, test the snack shop owner's suspicion that more energy drinks are sold closer to the school holidays.

The snack shop owner calculates 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.

Adriana is a conservationist researching whether there is any correlation between the population sizes of king cobras, , and their biggest enemy, the Indian grey mongoose, . She collects data on population sizes of both species from a random sample of 15 wildlife reserves and calculates the product moment correlation coefficient to be .

Carry out a suitable test to investigate if there is a linear correlation between and . You should:

• state your hypotheses clearly

• use a 5% level of significance

• state the critical value used

Adriana concludes that the test indicates that there is no correlation between population sizes of king cobras, , and the Indian grey mongoose, .

Explain why Adriana's conclusion is not fully correct.

A biologist is researching a connection between the mass of an animal, kg, and its expected lifespan, years. The biologist suggests that there exists a relationship of the form , where and are constants to be found.

Show that the relationship can be rewritten using logarithms as

Using data from a wide range of animals, when is plotted against on a scatter diagram there seems to be a strong positive correlation. When the regression line of on is calculated, the equation is found to be .

By relating the equation of the regression line to the equation found in (a), or otherwise, find the constants and correct to 2 decimal places where appropriate.

Hence, estimate the lifespan of a horse with a mass of 600 kg to the nearest year.

The biologist concludes the research by suggesting that one way to increase your lifespan is to increase your mass.

Explain, based on these data, why the biologist may be incorrect.

M. 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, , and the global mean temperature, °C, from the past 50 years and represents the information in the graph below.

Explain why a model of is unlikely to fit these data.

Hatter suggests that the equation for in terms of can be written in the form . He codes the data using and and calculates the regression line of on to be .

(i) Show that correct to 3 significant figures.

(ii) Find the value of to 3 significant figures.

(iii) Give an interpretation, in context, of the value of in your answer to (b)(i).

M. Hatter calculates the product moment correlation coefficient between and to be and 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.

A restaurant owner, Mr Capazio, suspects that there is positive correlation between the number of drinks consumed, , and the amount of time taken to pay the bill, minutes. He decides to conduct a one-tailed hypothesis test at the 5% level of significance to test his theory.

In the context of this question, describe what positive correlation would mean.

The table below shows the number of drinks consumed, , and the amount of time taken to pay the bill, minutes, for a random sample of 10 visitors to the restaurant.

(i) Find the product moment correlation coefficient for these data.

(ii) Carry out a suitable test at the 5% level of significance to investigate whether there is evidence to support Mr Capazio's theory. You should:

• state your hypotheses clearly

• state the critical value used

Mr Capazio calculates the regression line of on to be .

(i) Give an interpretation of the values 2.75 and 0.619 in the context of the question.

(ii) A person took 4.5 minutes to pay their bill. Explain why the regression line should not be used to estimate the number of drinks they had had.

Medhi is a meteorologist investigating the weather in Heathrow and claims that there is negative correlation between the daily total rainfall, mm, and daily total sunshine, hours.

Medhi decides to use the large data set to investigate this claim and forms a sample using all the days in June 2015 relating to Heathrow.

Some values for the daily total rainfall, mm, are labelled as 'tr'. Using your knowledge of the large data set, state the range of values Medhi could assign to these values.

Medhi uses all the days in June 2015 as a sample and calculates the product moment correlation coefficient to be .

Carry out a suitable test at the 5% level of significance to investigate Medhi's claim. You should:

• state your hypotheses clearly

• state the critical value used

Medhi uses this data to calculate the equation for the regression line of on . He plans to use the regression line to estimate the amount of rainfall there will be in Heathrow during a day in December.

Give two reasons why this is unlikely to produce a reliable estimate.

A doctor is collecting data on how a certain illness affects the weight of a person. Let w be the number of weeks that a patient had the illness and d kg be the amount of weight that the patient lost whilst they were ill. The doctor A doctor is collecting data on how a certain illness affects the weight of a person. Let be the number of weeks that a patient had the illness and kg be the amount of weight that the patient lost whilst they were ill. The doctor suspects that and have a relationship of the form , where and are constants to be found.

After plotting against , the doctor found there to be a strong correlation and the equation of the regression line of on was

Explain why the relationship between and must have shown positive correlation.

By using the equation of the regression line of on , or otherwise, find the values of the constants and correct to 3 significant figures where appropriate.

Stating any assumptions you make, estimate the weight loss expected of a patient who has been sick for 20 days, to the nearest whole kilogram.

Scientists in Wuhan, China, started tracking the total number of cases of the CoViD-19 virus in January 2020. The graph below shows the number of days, d, after the first reported case, and the total number of cases, c, of the virus for a period of 12 days.Scientists in Wuhan, China, started tracking the total number of cases of the CoViD-19 virus in January 2020. The graph below shows the number of days, , after the first reported case, and the total number of cases, , of the virus for a period of 12 days.

(i) Give a reason why the scientists should not use a regression line to model the relationship between the number of days and the total number of cases.

(ii) After two days the scientists tried to model the relationship using an exponential model of the form . Given that after 1 day there were 278 cases and after 2 days there were 326 cases, calculate the values of and .

(iii) After 11 days there were 9700 reported cases in China. Comment on the suitability of this model.

Another group of scientists code the data using and . The regression line of on was found to be

(i) Using the regression line of on , find an equation for in terms of in the form . State the values of and to 4 significant figures.

(ii) Explain what the value of represents in your answer to (b)(i).

(iii) One of the scientists used this model and estimated that after three months there would have been over cases. This is more than the world's population. Give a reason to explain why this estimate was unreliable.

Rory is studying the relationship between two variables and . He believes they could be modelled by the equation where and are constants. He codes his data and plots a scatter graph of against . Rory draws, by eye, a line of best fit between and which passes through the points and .

Using Rory's line of best fit, find the values of the constants and in his model. Give your answers correct to 3 significant figures where appropriate.

The product moment correlation coefficient between and is for Rory's data.

(i) Given that the critical value at 1% significance (one-tailed) is , write down the size of Rory's sample.

(ii) Carry out a suitable test at the 1% level of significance to investigate whether there is positive linear correlation between and .

You should:

• state your hypotheses clearly

• state the critical value used

(iii) Comment on the suitability of Rory's equation .

Rory later discovers the relationship between and is better suited to the equation model . From his raw data he calculates the regression line of on to be

Find the values of the constants and in Rory's new model, giving your answers to 3 significant figures.

A researcher has been collecting data within a particular city on the number of sleeveless T-shirts sold per week, , and the number of new gym memberships per week, . The data is shown in the table below along with the values of and .

The researcher suspects that and are related in one of two ways:

where and are constants.

By calculating the product moment correlation coefficient for two appropriate coded variable pairs from the table, decide which of the two models better represents the relationship between and .

Charlie is interested to find out if there is positive correlation between the number of letters in someone's name, , and the time seconds (rounded to the nearest 5 seconds) it takes her six-year-old sister to correctly guess the spelling of the name. She looks at a random sample of names and records the results below.

Charlie carries out a one-tailed test at the 1% level of significance and finds that the critical value for this test is .

Find the total number of names in Charlie's sample and hence find the value of .

Carry out a suitable test at the 1% level of significance to investigate whether there is positive linear correlation between the number of letters in a name and the time it takes Charlie's sister to guess the correct spelling.

You should:

• state your hypotheses clearly

• state the PMCC

Charlie calculates the equation of the regression line of on to be

(i) Give an interpretation of the value of the gradient of this regression line.

(ii) Explain why Charlie should not use this equation to estimate the number of letters in someone's name if it took her sister 70 seconds to guess the spelling.

When brewing beer, the temperature at which beer is stored during fermentation, °C, affects the alcohol content, %, at the end of the process. A group of brewers model the relationship as , where and are unknown constants.

They plot the regression line of on and find it has a gradient of and passes through the point .

Calculate the values of and , giving your answers to 3 significant figures.

In the data collected by the brewers, the range of values for was and the range of values for was . The minimum alcohol content occurred at the minimum temperature and the maximum alcohol content occurred at the maximum temperature.

Find estimates for the minimum values of and , giving your answers to 2 significant figures.

Hence explain why it would not be appropriate to use the model to predict the alcohol content of beer when the temperature during fermentation is °C.