Product moment correlation coefficient

Kay is studying the variables Daily Total Sunshine (x) and Daily Total Rainfall (y) from the large data set for Leeming in 2015 Kay starts with 5th May and then selects every 10th day thereafter.

State the name of the sampling technique Kay uses. Kay wants to find the regression line of y on x for these data.

Using your knowledge of the large data set, explain how Kay might need to clean these data before finding the equation of the regression line. The equation of the regression line Kay finds is y = 0.741 + 0.199x

Using your knowledge of the large data set,

state the units of the gradient of the regression line,

give an interpretation of the y-intercept of the regression line. Kay’s teacher claimed that the greater the amount of sunshine in a day the lower the amount of rain there should be.

State, giving a reason, whether the teacher’s claim is true for Kay’s data. The teacher used all the data for these variables from the large data set for Leeming in 2015, as a sample. The teacher calculated the product moment correlation coefficient for x and y to be – 0.160 In a suitable test to determine whether there is evidence to support the teacher’s claim, the p-value was 0.015

Using a 5% level of significance, state the hypotheses and conclusion for this test.

For the test in part (e) describe

the sample,

a possible population.

Amar is studying the flight of a bird from its nest. He measures the bird’s height above the ground, h metres, at time t seconds for 10 values of t Amar finds the equation of the regression line for the data to be h = 38.6 – 1.28t

Interpret the gradient of this line. The product moment correlation coefficient between h and t is – 0.510

Test whether or not there is evidence of a negative correlation between the height above the ground and the time during the flight. You should

• state your hypotheses clearly

• use a 5% level of significance

• state the critical value used

Jane draws the following scatter diagram for Amar’s data.

With reference to the scatter diagram, state, giving a reason, whether or not the regression line h = 38.6 – 1.28 t is an appropriate model for these data. Jane suggests an improved model using the variable u = t k² where k is a constant. She obtains the equation h = 38.1 – 0.78u

Choose a suitable value for k to write Jane’s improved model for h in terms of t only.

Anna is investigating the relationship between exercise and resting heart rate. She takes a random sample of 19 people in her year at school and records for each person

• their resting heart rate, h beats per minute

• the number of minutes, m, spent exercising each week

Her results are shown on the scatter diagram.

Interpret the nature of the relationship between h and m Anna codes the data using the formulae

The product moment correlation coefficient between x and y is – 0.897

Test whether or not there is significant evidence of a negative correlation between x and y You should

• state your hypotheses clearly

• use a 5% level of significance

• state the critical value used

The equation of the line of best fit of y on x is y = – 0.05 x + 1.92

Use the equation of the line of best fit of y on x to find a model for h on m in the form

where a and k are constants to be found.

TOTAL FOR STATISTICS IS 50 MARKS

Marc took a random sample of 16 students from a school and for each student recorded

• the number of letters, x, in their last name

• the number of letters, y, in their first name

His results are shown in the scatter diagram on the next page.

Describe the correlation between x and y. Marc suggests that parents with long last names tend to give their children shorter first names.

Using the scatter diagram comment on Marc’s suggestion, giving a reason for your answer. The results from Marc’s random sample of 16 observations are given in the table below.

Use your calculator to find the product moment correlation coefficient between x and y for these data.

Test whether or not there is evidence of a negative correlation between the number of letters in the last name and the number of letters in the first name. You should

• state your hypotheses clearly

• use a 5% level of significance

A random sample of 15 days is taken from the large data set for Perth in June and July 1987. The scatter diagram in Figure 1 displays the values of two of the variables for these 15 days.

Describe the correlation. The variable on the x-axis is Daily Mean Temperature measured in °C.

Using your knowledge of the large data set,

suggest which variable is on the y-axis,

state the units that are used in the large data set for this variable. Stav believes that there is a correlation between Daily Total Sunshine and Daily Maximum Relative Humidity at Heathrow. He calculates the product moment correlation coefficient between these two variables for a random sample of 30 days and obtains r = −0.377

Carry out a suitable test to investigate Stav’s belief at a 5% level of significance. State clearly ● your hypotheses ● your critical value On a random day at Heathrow the Daily Maximum Relative Humidity was 97%

Comment on the number of hours of sunshine you would expect on that day, giving a reason for your answer.

A group of climbers collected information about the height above sea level, h metres, and the air temperature, t C, at the same time at 8 different points on the same mountain. The data are summarised by

Show that S_th = – 17501.25 and S_tt = 227.875 The product moment correlation coefficient for these data is –0.985

State, giving a reason, whether or not this value supports the use of a regression equation to predict the air temperature at different heights on this mountain.

Find the equation of the regression line of t on h, giving your answer in the form t = a + bh. Give the value of your coefficients to 3 significant figures.

Give an interpretation of your value of a. One of the climbers has just stopped for a short break before climbing the next 150 metres.

Estimate the drop in temperature over this 150 metre climb.

A clothes shop manager records the weekly sales figures, £ s, and the average weekly temperature, t °C, for 6 weeks during the summer. The sales figures were coded so that

The data are summarised as follows

Write down the value of S_ss and the value of S_st

Find the product moment correlation coefficient between s and t.

The manager of the clothes shop believes that a linear regression model may be appropriate to describe these data.

State, giving a reason, whether or not your value of the correlation coefficient supports the manager’s belief.

Find the equation of the regression line of w on t, giving your answer in the form w = a + bt

Hence find the equation of the regression line of s on t, giving your answer in the form s = c + dt, where c and d are correct to 3 significant figures.

Using your equation in part (f), interpret the effect of a 1°C increase in average weekly temperature on weekly sales during the summer.

A biologist is studying the behaviour of bees in a hive. Once a bee has located a source of food, it returns to the hive and performs a dance to indicate to the other bees how far away the source of the food is. The dance consists of a series of wiggles. The biologist records the distance, d metres, of the food source from the hive and the average number of wiggles, w, in the dance.

[You may use wdw = 13833 S_dd = 394600 S_ww = 80.481 (to 3 decimal places)]

State, giving a reason, which is the response variable.

Calculate the product moment correlation coefficient for these data.

Calculate the equation of the regression line of w on d, giving your answer in the form w = a + bd

A new source of food is located 350 m from the hive.

Use your regression equation to estimate the average number of wiggles in the corresponding dance.

Comment, giving a reason, on the reliability of your estimate.

Before going on holiday to Seapron, Tania records the weekly rainfall (x mm) at Seapron for 8 weeks during the summer. Her results are summarised as

Find the standard deviation, _x, for these data.

Tania also records the number of hours of sunshine (y hours) per week at Seapron for these 8 weeks and obtains the following y – = 58 _yy FRUUHFW WR VLJQLILFDQW ILJXUHV ™ xy = 4900.5

Show that S_yy = 716 (correct to 3 significant figures)

Calculate the product moment correlation coefficient, r, for these data.

During Tania’s week-long holiday at Seapron there are 14 mm of rain and 70 hours of sunshine.

State, giving a reason, what the effect of adding this information to the above data would be on the value of the product moment correlation coefficient.

An estate agent recorded the price per square metre, p £/m², for 7 two-bedroom houses. He then coded the data using the coding q = , where a and b are positive constants. His results are shown in the table below.

Find the value of a and the value of b

The estate agent also recorded the distance, d km, of each house from the nearest train station. The results are summarised below.

Calculate the product moment correlation coefficient between d and q

Write down the value of the product moment correlation coefficient between d and p

The estate agent records the price and size of 2 additional two-bedroom houses, H and J.

Suggest which house is most likely to be closer to a train station. Justify your answer.

A meteorologist believes that there is a relationship between the height above sea level, h m, and the air temperature, t °C. Data is collected at the same time from 9 different places on the same mountain. The data is summarised in the table below.

[You may assume that h = 7150, t =110, h² = 7171500, t² = 1716, th = 64980 and S_tt = 371.56]

Calculate S_th and S_hh. Give your answers to 3 significant figures.

Calculate the product moment correlation coefficient for this data.

State whether or not your value supports the use of a regression equation to predict the air temperature at different heights on this mountain. Give a reason for your answer.

Find the equation of the regression line of t on h giving your answer in the form t = a + bh.

Interpret the value of b.

Estimate the difference in air temperature between a height of 500 m and a height of 1000 m.

A researcher believes that parents with a short family name tended to give their children a long first name. A random sample of 10 children was selected and the number of letters in their family name, x, and the number of letters in their first name, y, were recorded. The data are summarised as:

Calculate the product moment correlation coefficient, r, between x and y.

State, giving a reason, whether or not these data support the researcher’s belief.

The researcher decides to add a child with family name “Turner” to the sample.

your answer.

Given that the addition of the child with family name “Turner” to the sample leads to an increase in S_yy

will increase, decrease or stay the same. Give a reason for your answer.

A teacher asked a random sample of 10 students to record the number of hours of television, t, they watched in the week before their mock exam. She then calculated their grade, g, in their mock exam. The results are summarised as follows.

Calculate, to 3 significant figures, the product moment correlation coefficient between t and g.

The teacher also recorded the number of hours of revision, v, these 10 students completed during the week before their mock exam. The correlation coefficient between t and v was –0.753

Describe, giving a reason, the nature of the correlation you would expect to find between v and g.

A bank reviews its customer records at the end of each month to find out how many customers have become unemployed, u, and how many have had their house repossessed, h, during that month. The bank codes the data using variables x = and y = The results for the 12 months of 2009 are summarised below.

Calculate the value of the product moment correlation coefficient for x and y.

Write down the product moment correlation coefficient for u and h.

The bank claims that an increase in unemployment among its customers is associated with an increase in house repossessions.

State, with a reason, whether or not the bank’s claim is supported by these data.

On a particular day the height above sea level, x metres, and the mid-day temperature, y °C, were recorded in 8 north European towns. These data are summarised below

Find S yy

Calculate, to 3 significant figures, the product moment correlation coefficient for these data.

Give an interpretation of your coefficient.

A student thought that the calculations would be simpler if the height above sea level, h, was measured in kilometres and used the variable h instead of x.

Write down the value of S hh

Write down the value of the correlation coefficient between h and y.

A random sample of 50 salmon was caught by a scientist. He recorded the length l cm and weight w kg of each salmon. The following summary statistics were calculated from these data.

Calculate, to 3 significant figures, the product moment correlation coefficient between l and w.

Give an interpretation of your coefficient.

Gary compared the total attendance, x, at home matches and the total number of goals, y, scored at home during a season for each of 12 football teams playing in a league. He correctly calculated:

Calculate the product moment correlation coefficient for these data.

Interpret the value of the correlation coefficient.

Helen was given the same data to analyse. In view of the large numbers involved she decided to divide the attendance figures by 100. She then calculated the product moment

Write down the value Helen should have obtained.

The volume of a sample of gas is kept constant. The gas is heated and the pressure, p, is measured at 10 different temperatures, t. The results are summarised below.

calculate the product moment correlation coefficient.

Give an interpretation of your answer to part (b).

In a study of how students use their mobile telephones, the phone usage of a random sample of 11 students was examined for a particular week. The total length of calls, y minutes, for the 11 students were 17, 23, 35, 36, 51, 53, 54, 55, 60, 77, 110

Find the median and quartiles for these data.

A value that is greater than Q₃ + 1.5 × (Q₃ – Q₁) or smaller than Q₁ – 1.5 × (Q₃ – Q₁) is defined as an outlier.

Show that 110 is the only outlier.

Using the graph paper on page 15 draw a box plot for these data indicating clearly the position of the outlier.

The value of 110 is omitted.

Show that S_yy for the remaining 10 students is 2966.9

These 10 students were each asked how many text messages, x, they sent in the same week. The values of S_xx and S_xy for these 10 students are S_xx = 3463.6 and S_xy = –18.3.

Calculate the product moment correlation coefficient between the number of text messages sent and the total length of calls for these 10 students.

A parent believes that a student who sends a large number of text messages will spend fewer minutes on calls.

Comment on this belief in the light of your calculation in part (e).

Crickets make a noise. The pitch, v kHz, of the noise made by a cricket was recorded at 15 different temperatures, t °C. These data are summarised below.

Find S_tt, S_vv and S_tv for these data.

Find the product moment correlation coefficient between t and v.

State, with a reason, which variable is the explanatory variable.

Give a reason to support fitting a regression model of the form v = a + bt to these data.

Find the value of a and the value of b. Give your answers to 3 significant figures.

Using this model, predict the pitch of the noise at 19 °C.

A personnel manager wants to find out if a test carried out during an employee’s interview and a skills assessment at the end of basic training is a guide to performance after working for the company for one year. The table below shows the results of the interview test of 10 employees and their performance after one year.

Showing your working clearly, calculate the product moment correlation coefficient between the interview test and the performance after one year.

The product moment correlation coefficient between the skills assessment and the performance after one year is –0.156 to 3 significant figures.

Use your answer to part (a) to comment on whether or not the interview test and skills assessment are a guide to the performance after one year. Give clear reasons for your answers.