Regression lines

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

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.

Statistical models can provide a cheap and quick way to describe a real world situation.

Give two other reasons why statistical models are used.

A scientist wants to develop a model to describe the relationship between the average daily temperature, x °C, and her household’s daily energy consumption, y kWh, in winter. A random sample of the average daily temperature and her household’s daily energy consumption are taken from 10 winter days and shown in the table.

Find S_xy for these data.

Find the equation of the regression line of y on x in the form y = a + bx Give the value of a and the value of b to 3 significant figures.

Give an interpretation of the value of a

Estimate her household’s daily energy consumption when the average daily temperature is 2°C

The scientist wants to use the linear regression model to predict her household’s energy consumption in the summer.

Discuss the reliability of using this model to predict her household’s energy consumption in the summer.

Sammy is studying the number of units of gas, g, and the number of units of electricity, e, used in her house each week. A random sample of 10 weeks use was recorded and the data for each week were coded so that x = and y = The results for the coded data are summarised below

Find the equation of the regression line of y on x in the form y = a + bx. Give the values of a and b correct to 3 significant figures.

Hence find the equation of the regression line of e on g in the form e = c + dg. Give the values of c and d correct to 2 significant figures.

Use your regression equation to estimate the number of units of electricity used in a week when 100 units of gas were used.

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 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 biologist is comparing the intervals (m seconds) between the mating calls of a certain species of tree frog and the surrounding temperature (t °C). The following results were obtained.

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

Use your regression line to estimate the time interval between mating calls when the surrounding temperature is 10 °C.

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

A scientist is researching whether or not birds of prey exposed to pollutants lay eggs with thinner shells. He collects a random sample of egg shells from each of 6 different nests and tests for pollutant level, p, and measures the thinning of the shell, t. The results are shown in the table below.

[You may use p² = 1967 and pt = 694]

Draw a scatter diagram on the axes on page 7 to represent these data.

Explain why a linear regression model may be appropriate to describe the relationship between p and t.

Calculate the value of S_pt and the value of S_pp.

Find the equation of the regression line of t on p, giving your answer in the form t = a + bp.

Plot the point ( ) and draw the regression line on your scatter diagram.

The scientist reviews similar studies and finds that pollutant levels above 16 are likely to result in the death of a chick soon after hatching.

Estimate the minimum thinning of the shell that is likely to result in the death of a chick.

A teacher took a random sample of 8 children from a class. For each child the teacher recorded the length of their left foot, f cm, and their height, h cm. The results are given in the table below.

(You may use f =186 h =1085 S_ff = 39.5 S_hh =139.875 fh = 25291)

Calculate S fh

Find the equation of the regression line of h on f in the form h = a + bf. Give the value of a and the value of b correct to 3 significant figures.

Use your equation to estimate the height of a child with a left foot length of 25 cm.

Comment on the reliability of your estimate in (c), giving a reason for your answer.

The left foot length of the teacher is 25 cm.

Give a reason why the equation in (b) should not be used to estimate the teacher’s height.

A farmer collected data on the annual rainfall, x cm, and the annual yield of peas, p tonnes per acre. The data for annual rainfall was coded using v = and the following statistics were found.

Find the equation of the regression line of p on v in the form p = a + bv.

Using your regression line estimate the annual yield of peas per acre when the annual rainfall is 85 cm.

A travel agent sells flights to different destinations from Beerow airport. The distance d, measured in 100 km, of the destination from the airport and the fare £ f are recorded for a random sample of 6 destinations.

Using the axes below, complete a scatter diagram to illustrate this information.

Explain why a linear regression model may be appropriate to describe the relationship between f and d.

Calculate the equation of the regression line of f on d giving your answer in the form f a + bd.

Give an interpretation of the value of b.

Jane is planning her holiday and wishes to fly from Beerow airport to a destination km t away. A rival travel agent charges 5p per km.

Find the range of values of t for which the first travel agent is cheaper than the rival.

The weight, w grams, and the length, l mm, of 10 randomly selected newborn turtles are given in the table below.

(You may use S_ll = 33.381 S_wl = 59.99 S_ww = 120.1)

Find the equation of the regression line of w on l in the form w = a + bl.

Use your regression line to estimate the weight of a newborn turtle of length 60 mm.

Comment on the reliability of your estimate giving a reason for your answer.

A teacher is monitoring the progress of students using a computer based revision course. The improvement in performance, y marks, is recorded for each student along with the time, x hours, that the student spent using the revision course. The results for a random sample of 10 students are recorded below.

Find the equation of the least squares regression line of y on x in the form y = a + bx.

Give an interpretation of the gradient of your regression line.

Rosemary spends 3.3 hours using the revision course.

Predict her improvement in marks.

Lee spends 8 hours using the revision course claiming that this should give him an improvement in performance of over 60 marks.

Comment on Lee’s claim.

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 second hand car dealer has 10 cars for sale. She decides to investigate the link between the age of the cars, x years, and the mileage, y thousand miles. The data collected from the cars are shown in the table below.

Find the equation of the least squares regression line in the form y = a + bx. Give the values of a and b to 2 decimal places.

Give a practical interpretation of the slope b.

Using your answer to part (b), find the mileage predicted by the regression line for a 5 year old car.