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 = 185d – 266.

Jo, a transportation researcher, is investigating whether there is any correlation between the population size of an area and the proportion of its people who drive to work in a car or van. She collects data from 50 local authorities in the north of England and calculates the product moment correlation coefficient to be .
The table below gives critical values, for different significance levels, of the product moment correlation coefficient, , for a sample size of 50.

One tail	10%	5%	2.5%	1%	0.5%	One tail
Two tail	20%	10%	5%	2%	1%	Two tail
	0.1843	0.2353	0.2787	0.3281	0.3610

Test, using a 2% level of significance, whether there is evidence of any linear correlation between the population size of an area, , and the proportion of people who drive to work, , in the north of England. State your hypotheses clearly.

Jo calculates the equation for the regression line of  on  to be

One of the two values in the equation is incorrect. State which value is incorrect and give a reason for your answer.

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, , is dependent on the number of zombies, .

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  plotted against , and Figure 2 show the graph of log plotted against log .

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 for this data as r = 0.735. The table below gives the critical values, for different significance levels, of the product moment correlation coefficient, , for a sample of size 8.

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.

Given that the product moment correlation coefficient for these eight pairs of data is r = 0.999, comment on Mariam's claim that there is an exponential relationship between B and t.

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 r = 0.837. Using the scatter graph, explain how you know Terry's PMCC value is incorrect.

Terry corrects his mistake and calculates the correct PMCC as r = –0.837.
The table below gives the critical values, for different significance levels, of the product moment correlation coefficient, r, for a sample of size 7.

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.