4.2 Correlation & Regression

Jennifer sells cups of tea at her shop and has noticed that she sells more tea on cooler days.  On five different days, she records the maximum daily temperature, T, measured in degrees Celsius, and the number of cups of teas sold, C. The results are shown in the following table.

The relationship between and can be modelled by the regression line of on with equation

Find the value of  and the value of .

Use your regression equation from part (a)(i) to estimate the number of teas that Jennifer will sell on a day when the maximum temperature is 11°C.

Being sure to consider the result from part (a)(ii) in your answer, state how confident you would be in your estimate from part (b).

The following table shows the mean height, y cm, of primary school children who are age x years old.

The relationship between and can be modelled by the regression line of on with equation 

Using the regression equation with your values of  and  from part (a)(i), estimate the height of a child aged 9 years old.

Explain why it is not appropriate to use the regression equation to estimate the age of a child who is 133 cm tall.

Rebecca, a regular jogger, ran the “Thao Dien Loop” on 7 consecutive days. The following table shows the distance, x km, that she ran and the corresponding number of calories, y, that she was able to burn during the run. 

The number of calories burnt during a run is dependent upon on the length of the run.  The relationship between and can be modelled by the regression line of on  with equation   

Interpret, in the context of the question, the value of a found in part (a)(i).

On the 8th day, Rebecca is only able to run for 8 kilometres.

Use the result from part (a)(i) to estimate the number of calories Rebecca will lose.

Comment on the validity of using the result from part (a)(i) to answer part (c).

The percentage of people who are willing to get a particular vaccine is dependent on their age. The following table shows the age, A years old, and the corresponding percentage of people, V. that are willing to receive a vaccine for 6 different ages. 

 
The relationship between and  can be modelled by the regression line of on with equation

Interpret, in the context of the question, the value of  found in part (a)(i).

Use the result from part (a)(i) to estimate the percentage of people aged 95 years old who are in willing to receive a vaccine.

Comment on the validity of using the result from part (a)(i) to answer part (c).

The price, $ , of an airline ticket is dependent on the distance, km, between two cities.  The table below shows the airfares in US dollars from Prague in the Czech Republic, to eight different destinations in Europe.

Distance	885	340	835	330	1270	295	650	1930
Price	99	50	90	45	119	42.5	59	139

Let be the regression line of on .  The equation of the line  can be written in the form  

Let be the regression line of on .  The equation of the line can be written in the form 

Find the value of and the value of 

Find the value of and the value of 

Write down the value of Pearson’s product-moment correlation coefficient,

Use the result from part (a)(i) to estimate the price of an airline ticket for a flight from Prague to a destination that is 2635 km away.

The lines and both pass through the same point with coordinates 

Find the value of  and the value of 

The table below shows the lengths, in km, of 5 taxi rides in Melbourne, Australia and the corresponding prices, in AUD.

Length, in km	12.1	4.2	9.1	3.7	6.2
Price, in AUD	26.75	5.75	8.50	5.50	6.95

Draw a scatter diagram for the above data on the axes below.

Calculate

, the mean taxi ride length

, the mean price

Plot the point on your scatter diagram.

Show that the point lies on the regression line on .

A health study was conducted on 5 male and 5 female participants, measuring their average daily caffeine intake, in milligrams (mg), and their resting heart rate, in beats per minute (BPM).

The following table shows the results of the study.

Average daily caffeine intake, in mg – male	222	312	211	190	120
Resting heart rate, in BPM – male	57	72	60	48	50
Average daily caffeine intake, in mg – female	202	411	254	81	52
Resting heart rate, in BPM – female	57	81	71	45	49

Calculate the Pearson’s product-moment correlation coefficient for,

the male participants, ,

the female participants, .

Write down the equation of the regression line

on 

on .

Find the intersection of the two regression lines found in part (b) and interpret its meaning.

The following table shows the distance, in km, to 5 different ferry destinations from Rostock, Germany and the corresponding price of the cruise, in €.

Destination	Copenhagen	Oslo	Stockholm	Helsinki	Riga
Distance, in km ()	174	620	730	933	810
Price, in € ()	30.50	65.00	45.75	85.50	125.00

The regression line on can be written in the form .

Calculate the values of and and interpret their meanings

The distance to Aberdeen from Rostock is 1093 km.

Estimate the cost of the ferry to Aberdeen.

Comment on the reliability of your estimate found in part (b).

The following table shows the total revenue, , in £, obtained weekly during the first 7 weeks of 2021 by Larry, an independent financial consultant, and the number of clients, , served.

Week	1	2	3	4	5	6	7
Revenue, in £ ()	2452	5751	6429	1203	4587	9786	6911
Clients,	7	11	14	4	5	8	9

Write down the equation of the regression line  on .

Larry’s weekly operating costs are £2250 and the cost of serving each client is £35.

 Find an expression for the profit Larry makes when serving  clients in a week.

Estimate the least number of clients required to generate a minimum of £1000 profit.

Sandy Café is located on a beach and is open all year. The manager wants to see whether the daily average temperature, in , is correlated with the average tip they receive, as a percentage of the customer’s total bill. He records this data over 9 days and details it in the table below.

Daily average temperature, in	22.4	27.8	15.4	12.2	8.8	2.1	33.4	14.7	19.4
Average tip as a percentage of the total bill	20.1	16.3	12.4	12.8	10.1	9.4	18.8	13.1	15.9

Find the Pearson’s product-moment correlation coefficient, , for this data.

Write down the equation of the regression line  on .

On the 10^th day, the average temperature is 25  and a customer tips their waiter $20.

Use the regression line to estimate the customer’s total bill. Give your answer to 2 decimal places.

The customer’s total bill was $98.50.

Calculate the tip as a percentage of the actual total bill. Give your answer to the nearest integer.

The table below shows the petrol prices, in New Zealand dollars (NZD) per litre, for 6 different petrol stations (labelled A to F) along with their distance south of Auckland’s city centre.

Petrol station	A	B	C	D	E	F
Distance south of Auckland, in km	122	314	456	231	178	392
Petrol price, in NZD per litre	1.94	1.88	1.78	1.84	1.99	1.81

Calculate the mean petrol price, .

The equation of the regression line  on  can be written in the form .

Calculate the value of

Calculate the value of , giving your answer in the form , where .

The distance between Auckland’s city centre and a new petrol station, G, is 200 km and the bearing of G from Auckland’s city centre is .

Estimate the petrol price at G.