Further Correlation & Regression (A Level only) (AQA A Level Maths: Statistics)

Exam Questions

3 hours23 questions

Easy

Medium

Hard

2 marks

Explain what is measured by the Pearson product moment correlation coefficient.

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4 marks

The product moment correlation coefficient between two variables is denoted $r$ . Five different values of $r$ , rounded to four decimal places, are given below:

$r_{1} = 0.0000 r_{2} = 0.9812 r_{3} = - 1.0000 r_{4} = 0.7652 r_{5} = - 0.7098$

Match each of the following four scatter graphs, showing observations from different bivariate data sets, to one of the values of $r$ given above. You should use each given value of $r$ no more than once.

q1b-2-5-easy-aqa-a-level-maths-statistics

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2 marks

Sketch a scatter graph for the remaining value of $r$ from the list above.

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4 marks

Write suitable null and alternative hypotheses for each of the following situations.

(i)

A recording studio is interested in whether the increasing age of a band's lead singer decreases the number of records the band will sell.

(ii)

A researcher for an online gaming company believes that the higher the number of free revivals available in a game, the more time people will spend playing the game.

(iii)

A beach umbrella manufacturer is carrying out a test to see if there is correlation between temperature and the number of beach umbrellas sold.

(iv)

The developer of a new cryptocurrency tests, at the 5% level of significance, for any correlation between the new cryptocurrency's net value and that of a more popular cryptocurrency. She calculates the product moment correlation coefficient between the two cryptocurrency's net values to be

r = - 0.3452 .

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3 marks

For the following null and alternative hypotheses, state whether the test is a one-tailed or a two-tailed test and write down the type of correlation that is being tested for.

(i)

H_{0} : p = 0, H_{1} : p > 0 .

(ii)

H_{0} : p = 0, H_{1} : p \neq 0 .

(iii)

H_{0} : p = 0, H_{1} : p < 0 .

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3 marks

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

One tail	10%	5%	2.5%	1%	0.5%	One tail
Two tail	20%	10%	5%	2%	1%	Two tail
	0.2407	0.3061	0.3610	0.4226	0.4692

For each set of hypotheses in part (a), use the table above to determine the critical region for a hypothesis test at the 10% level of significance for a sample of size 30.

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3 marks

It is claimed that there is negative correlation between two variables $x$ and $y$ . A hypothesis test is carried out to test the claim and the null hypothesis is given as $H_{0} : ρ = 0 .$

(i)

Describe what a null hypothesis of

ρ = 0

means about the relationship between

x

and

y

(ii)

Describe what a negative correlation would suggest about the relationship between

x

and

y

(iii)

State a suitable alternative hypothesis to test for negative correlation between

x

and

y

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2 marks

The critical value for this hypothesis test is found to be –0.3674.

(i)

Explain what is meant by a critical value, within the context of hypothesis testing.

(ii)

Write down the critical region for this hypothesis test.

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1 mark

The product moment correlation coefficient, $r$ , between these two variables is calculated to be $r = - 0.3175$ .

Explain the difference between the statistic, $r$ , and the parameter, $ρ$ .

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1 mark

By comparing the test statistic with the critical value, conclude the hypothesis test.

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2 marks

Pim collects data on the amount of time she can hold plank each morning, $t$ minutes, and the amount of sleep, $s$ hours, she got the night before.

Amount of sleep, $s$ hours	6.21	8.15	7.52	7.19	6.18	5.28	9.03	6.01	7.55	8.39
Time holding plank, $t$ mins	0.92	1.13	1.07	$x$	0.99	0.96	1.12	0.98	1.20	1.09

The product moment correlation coefficient for these data is calculated as $r = 0.7536$ .

Pim plots this information on a scatter graph and draws, by eye, a line of best fit.

(i)

Describe the correlation between

s

and

t

(ii)

State, with a reason, whether Pim's line of best fit should have a positive or a negative gradient.

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4 marks

Pim calculates the equation of the regression line of $t$ on $s$ to be $t = 0.08 s + 0.45$ .

(i)

Using the regression line, estimate the value of

x

in the table above and explain why it is only an estimate.

(ii)

Give an interpretation of the value 0.45 in the equation of the regression line.

(iii)

Give an interpretation of the value 0.08 in the equation of the regression line.

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2 marks

Pim says that if she sleeps for 13 hours then she will be able to hold plank for roughly 1.5 minutes. Give two reasons why Pim's claim could be incorrect.

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1 mark

One morning Pim can hold plank for one minute. Explain why the regression line should not be used to predict how long Pim slept the night before.

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3 marks

Andy, a preschool teacher, is exploring whether a new 'Mindfulness for Toddlers' course is helping the children to learn quicker. Andy devises a test where he times how long the nine toddlers in his class can sit in meditation and how long it takes them to solve a simple puzzle afterwards.

The table below shows the amount of time, $m$ to the nearest minute, each child spent meditating and how long, $p$ minutes, it took them to solve the puzzle afterwards.

$m$	5	4	2	10	3	5	1	2	4
$p$	2.8	3.6	4.5	1.8	5.1	2.8	7.0	8.0	2.5

Andy suspects that there is an exponential relationship between the times so he decides to code the data using the changes of variable $X = m$ and $Y = In p$ .

Complete the table below for $X$ and $Y$ , giving each value of $Y$ to two decimal places.

$X$	5	4	2	10	3	5	1	2	4
$Y$	1.03	1.28	1.50	0.59	1.63	1.03

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2 marks

Andy calculates the product moment correlation coefficient for the relationship between $m$ and $p$ to be $r_{1} = - 0.772$ and between $X$ and $Y$ to be $r_{2} = - 0.862$ .

State, giving a reason, whether there is stronger correlation between $m$ and $p$ or between $X$ and $Y$ .

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3 marks

Andy calculates the equation of the regression line of $Y$ on $X$ to be $Y = 1.98 - 0.162 X$ . A new student joins the class and spends 4 minutes meditating.

(i)

Write down the corresponding value of

X

(ii)

Use the regression line to estimate the value of

Y

(iii)

Hence estimate how long it takes the student to solve the puzzle.

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2 marks

It is believed that the relationship between two variables, $x$ and $y$ , can be modelled by $Y = b p^{x} .$

By first taking logarithms of both sides and then by using the laws of logarithms, show that $y = b p^{x}$ can be written as $\log y = \log b + x \log p$ .

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3 marks

The scatter diagram below shows the relationship between two sets of data, $x$ and log $y$ . The regression line of log $y$ on $x$ is shown and passes through the points (3, 0.7) and (7, 1.3).

q7b-2-5-easy-aqa-a-level-maths-statistics

(i)

Using the given coordinates, find the gradient of the regression line shown.

(ii)

Find the regression line of log

y

x

in the form log

y = a + m x

, where

a

and

m

are constants to be found.

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4 marks

(i)

By comparing the equations in (a) and (b) (ii) show that

b = 1.778

to three decimal places.

(ii)

Find the value of

p

to three decimal places.

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2 marks

The graph below shows the heights, $h$ metres, and the amount of time spent sleeping, $t$ hours, of a group of young giraffes. It is believed the data can be modelled using $t = k h^{n}$ .

q8a-2-5-easy-aqa-a-level-maths-statistics

By first taking logarithms of both sides, show that $t = k h^{n}$ can be written as $\log t = \log k + n \log h$ .

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5 marks

The data are coded using the changes of variables $x = \log h$ and $y = \log t$ . The regression line of $y$ on $x$ is found to be $y = 0.3 - 1.2 x .$

(i)

Find the values of

x

and

y

for a giraffe that is 2.1 metres tall and sleeps for 4.3 hours per day, giving your answers to four decimal places.

(ii)

Using the regression line, show that a giraffe of height 3.2 metres would be expected to sleep for approximately half an hour per day.

(iii)

State an assumption that was made in order to justify the use of the regression line in part (ii).

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3 marks

By first substituting log $h$ for $x$ and log $t$ for $y$ in the equation of the regression line and then by using part (a), show that the relationship between the height of a giraffe and the time it spends sleeping can be modelled by $t = 1.995 h^{- 1.2} .$

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2 marks

The table below shows the hydrocarbon emissions, $x$ , and the oxides of nitrogen, $y,$ for ten randomly selected cars from London with known values of $x$ and $y$ , taken from the large data set.

$x$	0.044	0.031	0.031	0.013	0.067	0.060	0.040	0.056	0.044	0.013
$y$	0.008	0.010	0.010	0.006	0.041	0.047	0.025	0.051	0.008	0.006

Use your knowledge of the large data set to:

(i)

state the units of both

x

and

y

(ii)

write down the most likely propulsion type for the two cars with hydrocarbon emissions of 0.013.

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3 marks

The product moment correlation coefficient between the two variables is calculated to be $r = 0.7544$ . The critical value for a sample size of 10 at the 0.5% level of significance is 0.7646.

(i)

State suitable null and alternative hypotheses to test whether there is evidence of positive correlation between

x

and

y

(ii)

Test, at the 0.5% level of significance, whether there is evidence of a positive correlation between hydrocarbon emissions and oxides of nitrogen.

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1 mark

The linear regression line of $y$ on $x$ is given as $y = 0.7433 x - 0.006357$ .

Give an interpretation of the value 0.7433 in the context of the question.

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1 mark

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.

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2 marks

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, $s$ %, and the amount of revision they did, $h$ hours.

Ms Pearson calculates the product moment correlation coefficient for these data as $r = 0.874$ .

Given that the p-value for the test statistic $r = 0.874$ is $0.0217$ , test at the 5% level of significance whether Ms Pearman's claim is justified.

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2 marks

Ms Pearman decides to use a linear regression model for these data. She calculates the equation for the regression line of $s$ on $h$ to be $s = 21.3 + 5.29 h$ .

(i)

Give an interpretation of the value 21.3 in context.

(ii)

Give an interpretation of the value 5.29 in context.

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3 marks

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.

$d$	48	51	51	57	61	68	70	72	73	75
$m$	19	21	17	12	8	16	7	4	0	1

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.

(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 –0.7155.

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1 mark

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

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

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2 marks

(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.

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1 mark

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:

Year	2005	2006	2007	2008	2009	2010	2011	2012	2013	2014
Chicken consumption (pounds)	86.4	86.9	85.5	83.8	80.0	82.8	83.3	80.8	82.3	83.8
Unemployment rate (%)	5.08	4.62	4.62	5.78	9.25	9.63	8.95	8.07	7.38	6.17

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.

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3 marks

(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.

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1 mark

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.

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1 mark

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.

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3 marks

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.

One tail	10%	5%	2.5%	1%	0.5%	One tail
Two tail	20%	10%	5%	2%	1%	Two tail
	0.2653	0.3365	0.3961	0.4622	0.5052

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.

How did you do?

1 mark

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

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2 marks

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 $z$ .

q5a-2-5-medium-aqa-a-level-maths-statistics

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.

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3 marks

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.

How did you do?

1 mark

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

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3 marks

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.

Hours (t)	1	2	3	4	5	6	7	8
Number of bacteria (B)	10	50	170	520	1730	5200	17020	58140

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, $r$ , for a sample of size 8.

One tail	10%	5%	2.5%	1%	05%	One tail
Two tail	20%	10%	5%	2%	1%	Two tail
	0.5067	0.6215	0.7067	0.7887	0.8343

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.

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2 marks

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.

$t$	1	2	3	4	5	6	7	8
$In (B)$	2.303	3.912	5.136	6.254	7.456	8.556

How did you do?

1 mark

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 $t$ .

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1 mark

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:

q6a-2-5-medium-aqa-a-level-maths-statistics

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.

How did you do?

3 marks

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.

One tail						One tail
Two tail						Two tail
	0.5509	0.6694	0.7545	0.8329	0.8745

(i)

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

(ii)

Test Terrys claim using a 5% level of significance.

How did you do?

1 mark

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

How did you do?

1 mark

Suggest one way in which Terry could improve his investigation.

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2 marks

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, $d$ , increases as the number of days until the school holidays, $h$ , decreases.

(i)

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

(ii)

State which of the two variables is the explanatory variable.

How did you do?

4 marks

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
$r = - 0.4187 .$

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

Level	10%	5%	2.5%	1%	0.5%
$n = 30$	0.2407	0.3061	0.3610	0.4226	0.4629

(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.

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2 marks

The snack shop owner calculates the regression line of $d$ on $h$ 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.

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4 marks

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.

One tail	10%	5%	2.5%	1%	0.5%	One tail
Two tail	20%	10%	5%	2%	1%	Two tail
$n = 15$	0.3507	0.4409	0.5140	0.5923	0.6411	$n = 15$

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.

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1 mark

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.

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3 marks

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 $L = A M^{B}$ , where $A$ and $B$ are constants to be found.

Show that the relationship can be rewritten using logarithms as

$\log L = \log A + B \log M$

How did you do?

2 marks

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 $y = 0.18 x + 0.98$ .

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.

How did you do?

1 mark

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

How did you do?

1 mark

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.

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1 mark

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.

q4a-2-5-medium-aqa-a-level-maths-statistics

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

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5 marks

Hatter suggests that the equation for $h$ in terms of $t$ can be written in the form $h = a b^{t}$ . He codes the data using $x = t$ and $y = \log h$ and calculates the regression line of $y$ on $x$ to be $y = 1.903 - 1.005 x$ .

(i)

Show that

a = 80.0

correct to 3 significant figures.

(ii)

Find the value of

b

to 3 significant figures.

(iii)

Give an interpretation, in context, of the value of

a

in your answer to (b) (i).

How did you do?

1 mark

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.

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2 marks

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.

(i)

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

(ii)

Write down suitable null and alternative hypotheses to test Mr Capazio's theory.

How did you do?

4 marks

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 sample of 10 visitors to the restaurant on a particular night.

Number of drinks, $d$	0	1	3	2	8	4	2	0	3	2
Time taken, $t$ minutes	2.6	3.1	5.3	2.0	6.1	9.3	1.5	3.2	5.7	4.2

The product moment correlation coefficient for these data is $r = 0.6099$ .
The critical values for a 5% one-tailed test are ±0.5494.

(i)

Test, at the 5% level of significance, whether there is evidence to suggest Mr Capazio's theory is correct, stating your conclusion clearly.

(ii)

Suggest a reason why Mr Capazio's conclusion may be unreliable.

How did you do?

3 marks

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

(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.

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4 marks

On 21^st January 2020, doctors in China started recording and reporting the number of new daily cases of an unknown virus. The table below shows the number of new cases, $c$ , of the virus in China for each day, $d$ , after 21^st January 2020.

$d$	1	2	3	4	5	6
$c$	278	48	221	92	277	1084

$d$	7	8	9	10	11	12
$c$	700	1700	1600	1700	2000	1500

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.

One tail	10%	5%	2.5%	1%	0.5%
$n = 12$	0.3981	0.4973	0.5760	0.6581	0.7079

(i)

Clearly stating suitable null and alternative hypotheses, show that there is evidence of linear correlation between the number of days and the number of new cases at a 1% level of significance.

(ii)

Give a reason why a linear regression line is suitable to model the relationship between the number of days and the number of new cases reported each day.

How did you do?

3 marks

The equation for the regression line of $c$ on $d$ is found to be $c = 185 d - 266 .$

(i)

Interpret the value of 185 in context.

(ii)

Explain why the equation for the regression line should not be used to estimate how many new cases there were on 19^th January 2020.

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1 mark

Elena is an environmental agent, and she is investigating whether there is any correlation between the mass of a car, $m$ kg, and the particulate emissions, $p$ g/km.

Write down suitable null and alternative hypotheses for Elena's investigation.

How did you do?

2 marks

Elena uses the large data set and the random numbers to randomly select a sample of 15 cars.

Use your knowledge of the large data set to,

(i)

explain why Elena needs to subtract 75 kg from the mass of each car,

(ii)

explain why the data for the mass of the cars needs to be cleaned before forming the sample.

How did you do?

2 marks

Elena calculates the product moment correlation coefficient for her sample as $r = 0.1169$ . The p-value for this test statistic is 0.6782.

Test, using a 10% level of significance, whether there is any correlation between the mass of a car and the particulate emissions.

How did you do?

1 mark

Suggest one way in which Elena could make better use of the large data set.

How did you do?

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