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.

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.

Use your regression equation from part (a)(i) to 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.

Write down the equation of the regression line of y  on x, giving your answer in the form where a and b are constants to be found.

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. 

Write down the equation of the regression line of V on A, giving your answer in the form where a and b are constants to be found.

Interpret, in the context of the question, the value of a 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, $P , of an airline ticket is dependent on the distance, d 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.

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

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

Madlenka buys a ticket for a flight from Prague to Cairo, a distance of 2635 km. The airfare in US dollars is $245.

Compare this price to your result from part (b), suggesting possible reasons for any discrepancies.

A snack company is trialling a new series of crisp flavours. The eight flavours have been assigned the letters A through H, and the company has conducted taste tests in which volunteers assign each flavour a score on a scale from 1 to 10 (where ‘1’ indicates ‘I hate it!’, and ‘10’ indicates ‘This is my new favourite!’).  The following table collates the scores assigned to the flavours by each of three volunteers – Idris, Jameel and Kevin. 

The company would like to find the Spearman’s rank correlation coefficients for these taste testers’ rankings.

Complete the information in the following table. 

Find the value of the Spearman’s rank correlation coefficient, r_s, for:

 (i)      Idris’ and Jameel’s ranks 

(ii)      Idris’ and Kevin’s ranks 

(iii)     Jameel’s and Kevin’s ranks.

Comment, in the context of the question, on the results obtained for  in part (b)(i), (ii) and (iii).

It is noticed that a greasy fingerprint has caused Kevin’s score of 9 for flavour A to be misread.  The score actually assigned by Kevin to flavour A was 8.

Explain, with a reason, whether this will change any of the values for the Spearman’s rank correlation coefficient r_s that were calculated in part (b).

The following table collates the scores achieved on a recent maths test and a recent physics test by a group of 10 students.

The scatter diagram for these scores is shown in the diagram below:

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

 (i)      with student J included

 (ii)     without student J included.

Complete the information in the following table, ranking the scores from highest to lowest: 

Find the value of the Spearman’s rank correlation coefficient, r_s,

(i)      with student J included 

(ii)     without student J included.

Comment on the results of parts (a) and (c).

A school decides to introduce a test for students aged 14 as they start their Mathematics course to predict how well they will perform in their final Mathematics examination at age 16.10 students are randomly selected to sit the test and their results are compared to their final exam score.

The results of the 10 students are shown in the table below.

State the name for this test of validity.

Calculate Pearson’s product moment correlation coefficient for this data.

Hence determine if the test is a valid indicator of future performance in the formal Mathematics examinations. Give a reason for your answer.

A company is considering 8 candidates for a role and asks them to complete a set of questions designed to identify potential leadership qualities at the first stage of interview. A few days later, they ask them to complete a second set of similarly designed questions in order to test the reliability of the questionnaire.

The results have been recorded in the table below.

State the name of this test for reliability.

State a possible disadvantage of this test for reliability.

Find Pearson’s product moment correlation coefficient for the data in the table above.

Hence determine if the data from the personality test is reliable. Give a reason for your answer.

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

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

Calculate

Show that the point  lies on the regression line y on x.

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.

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

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 €.

The regression line P on D can be written in the form

Calculate the values of a and b 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 ferry company decides to charge €135 to travel to Aberdeen.

Calculate the percentage error between your estimate found in part (b) and the actual price.

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

Write down the equation of the regression line R on x.

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 x 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 °C, 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.

On the 10^th day, the average temperature is 25° C 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.

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.

Calculate the mean petrol price, .

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

Calculate the value of 

Calculate the value of b, 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 166°.

Estimate the petrol price at G.

Petrol station G decides to set their price at 0.44 NZD per litre below the mean petrol price calculated in part (a).

Calculate the percentage error between the estimated petrol price for G calculated in part (c) and the actual petrol price at G.

Jezebel is testing the accuracy and consistency of her new thermometer when it measures the temperature of an object. To test her thermometer’s readings, Jezebel uses specialist scientific equipment to heat an object to different precise temperatures and uses the thermometer to read the temperature. She repeats this process twice using the same conditions to measure the reliability of her thermometer’s reading. The data is shown in the table below.

State the name of this test for reliability.

Determine whether the thermometer’s reading is reliable by calculating the Pearson’s product-moment correlation coefficient between relevant sets of data.

Explain why using only Pearson’s product-moment correlation coefficient is not necessarily a good measure of the criterion-related validity of the thermometer’s readings.

Explain whether or not Jezebel’s experiment is likely to have content validity for assessing the thermometer’s ability of measuring the temperature accurately.