2.5 Further Correlation & Regression (A Level only)

Adriana is a conservationist researching whether there is any correlation between the population sizes of king cobras, c, and their biggest enemy, the Indian grey mongoose, m.  She collects data on population sizes of both species from a sample of 15 wildlife reserves and calculates the product moment correlation coefficient to be  r = 0.3264.

The table below gives the critical values, for different significance levels, of the product moment correlation coefficient,  r, for a sample of size 15.

Conduct a hypothesis test at the 5% level of significance to test if there is linear correlation between c and  m. Clearly state your hypotheses, critical regions, and conclusion.

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, M kg, and its expected lifespan, L years.  The biologist suggests that there exists a relationship of the form L=AM^B,  where A and B are constants to be found.

Show that the relationship can be rewritten using logarithms as 

log L =log A +B log M

Using data from a wide range of animals, when y = log L   is plotted against x = log M on a scatter diagram there seems to be a strong positive correlation.  When the regression line of y on x is calculated, the equation is found to be y = 0.18x + 0.98.

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

Hence, predict 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 temperatures from the past 50 years and records the information in the graph below.

Explain why a model of  h = at + b  is unlikely to fit these data.

Hatter suggests that the equation for h in terms of t can be written in the form h = ab^t. He codes the data using  x = t  and  y = log h   and calculates the regression line of y on x to be y=1.903 –1.005x.

M.Hatter calculates the product moment correlation coefficient between x and y to be  r = 0.952 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 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 conduct a one-tailed hypothesis test at the 5% level of significance to test his theory.  

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

Mr Capazio calculates the regression line of t on d to be  t  = 2.75 + 0.619d.  

On 21^st January 2020, doctors in China started recording and reporting the number of new daily cases of an unknown virus. The doctors record the number of new cases, c, of the virus and the number of days, d, after 21^st January 2020.

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  r = 0.900.  

The table below gives the critical values, for different significance levels, of the product moment correlation coefficient,  r, for a sample size of 12. 

The equation for the regression line of c on d is found to be  c = 184.58d – 266.39.

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

Write down suitable null and alternative hypotheses to test Medhi’s claim.

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 are labelled as “tr”. Using your knowledge of the large data set, state the range of values Mehdi could assign to these values.

Medhi calculates the product moment correlation coefficient as r = 0.2659 using all the days in June.

Test, at the 5% level of significance, whether there is negative correlation between daily total rainfall and daily total sunshine.

Medhi uses this data to calculate the equation for the regression line of f on s. He plans to use the regression line to predict 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.

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

The product moment correlation coefficient for these data is r = 0.869.  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.

Test, at the 1% level of significance, whether Dave’s claim is justified, given that the relevant critical value is  0.7155.

Dave calculates the equation of the regression line of m on d to be  m = 50.7  0.642d.

State, giving a reason, whether or not the correlation coefficient is consistent with the use of a 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  r = 0.821.
The critical values for a 10% two-tailed test are ±0.5495.

State what is measured by the product moment correlation coefficient.

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 r = 0.107.

The table below gives the critical values, for different significance levels, of the product moment correlation coefficient,  r, 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.

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

During a zombie attack, Richard suspects that the number of flies in the area, f, is dependent on the number of zombies, z.

Richard is trying to decide whether the correlation is linear or non-linear, so he uses a graphical software package to plot two scatter graphs.  Figure 1 shows the graph of f plotted against z, and Figure 2 show the graph of log f plotted against log z .

Richard calculates the product moment correlation coefficient for each graph.  One value is found to be 0.847 while the other is 0.985.

State, with a reason, which PMCC value corresponds with Figure 2.

Test, using a 5% level of significance, whether there is positive linear correlation in the graph shown in Figure 2.  State your hypotheses clearly.
You are given that the critical value for this test is 0.1654.

State, with a reason, whether the relationship between number of zombies and number of flies is better represented as linear or non-linear.

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 r = 0.735.

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 In(B)   for the different values of t.

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

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

State, giving a reason, whether the conclusion to the test would be different if a 1% level of significance had been used.

Suggest one way in which Terry could improve his investigation.

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.

Another group of scientists code the data using  x = d  and  y = log c .  The regression line of y on x was found to be  y = 2.2476 + 0.1606x.  

Rory is studying the relationship between two variables x and y.  He believes they could be modelled by the equation  y = ax^m  where a and m are constants.  He codes his data and plots a scatter graph of  X = log x   against Y = log y .  Rory draws, by eye, a line of best fit between X and Y which passes through the points (2, 2.68) and (5, 3.10). 

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

Rory wants to conduct a test, using a 1% level of significance, to test whether there is a positive linear correlation between X and Y. The value of the product moment correlation coefficient for X and Y is calculated to be  r = 0.5047 for Rory’s data.

Rory later discovers the relationship between x and y is better suited to the equation model  y = kb^x.  From his raw data he calculates the regression line of log y on x  to be  log y  = 2.56 + 1.12x.  

Find the value of the constants k and b in Rory’s new model, giving your answers to 3 significant figures.

Alfie lives near a river and from observing the wildlife he believes that there is a relationship between the number of otters and the number of frogs.  He decides to investigate this further and gathers data on frog and otter populations from six wildlife centres around the UK. Alfie records the data in the table below. 

The product moment correlation coefficient for these data is r = 0.7401.

Alfie codes the data using logarithms and records the results in the table below.

Alife concludes that the evidence from the test suggests that there is not a non-linear relationship between the number of frogs and the number of otters.

Explain why Alfie could be incorrect by describing a type of relationship that might exist between the number of frogs and the number of otters

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

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

T = aG^m  or  T = bp^G

where a, b, m, and p are constants.

By calculating the product moment correlation coefficient for two different pairs of rows in the table, decide which of the two models better represents the relationship between T and G.

Charlie is interested to find out if there is positive correlation between the number of letters in someone’s name, l, and the time, t 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.

Charlie carries out the one-tailed test at the 1% level of significance and finds that the critical value for this test is 0.2324. 

Find the total number of names in Charlie’s sample and hence, find the value of  x.

Stating your hypotheses clearly, test, at the 1% level of significance, if there is evidence of positive linear correlation between the number of letters in someone’s name and the time it takes Charlie’s sister to guess the correct spelling.

Charlie calculates the equation of the regression line of t on l to be t = 1 1.2l 33.4.

When brewing beer the temperature that the beer is stored at during fermentation, T ℃, changes the alcohol content, A %, at the end of the fermentation process. A group of brewers collect data on T and A for their casks of beer. They suspect the data follows a model of the form A = bp^T where b and p are unknown constants. They plot the regression line of y = ln A   on  x = T  and find that the line has a gradient of 0.0392 and passes through the point (0, 0.811).

Using the line of regression, calculate the values of b and p, giving your answers correct to 3 significant figures.

In the data collected by the brewers, the range of values for  T was 15 and the range of values for A was 4. You are given that the minimum alcohol content occurred when the temperature was at its minimum and the maximum alcohol content occurred when the temperature was at its maximum.

Find estimates for the minimum values of T and A 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 50 ℃.