Further Correlation & Regression (DP IB Maths: AI HL)

Exam Questions

5 hours31 questions
1a
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3 marks

Taro discovers a group of rocks and decides to measure their thickness, t, and weight, w. The results are shown below.

Thickness, bold italic t (cm)

2.2

4.5

3.2

2.8

3.7

Weight, bold italic w (kg)

2.6

3.6

2.7

3.0

2.9


Taro draws a line of best fit through the data points using the equation w equals 2 over 5 t plus 8 over 5.

Draw a scatter diagram of the data and sketch the given line of best fit on the same axes.
q1a_4-3_ib-maths-ai-hl

1b
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3 marks

Calculate the residual of each point.

1c
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1 mark

Hence find the sum of the squared residuals, S S subscript r e s end subscript.

1d
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2 marks

Comment on the fit of the model to the data.

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2a
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1 mark

Syafiqah wants to model the path of the water from a water fountain. She measures the horizontal and vertical distances of specific points on the path that the water takes with respect to the base of the water spout. These measurements are recorded in the table below.

Horizontal Distance, bold italic x (m)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Vertical Distance,
bold italic y
(m)

1.32

1.47

1.50

1.56

1.39

1.33

0.98

0.63

0.00

State an appropriate type of function to model the path way of the water.

2b
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2 marks

Use your graphic display calculator to find the best fit function for the data points.

2c
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2 marks

Find the coefficient of determination and comment on the closeness of fit to the original data.

2d
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3 marks

State the maximum height that the water reaches according to the model and the horizontal distance of this point from the base of the water spout.

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3a
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3 marks

The average price for cooking oil in USD per litre was recorded each year from 2007 until 2021 and the results put in the table below. Number of years from 2007.

Year

2007

2009

2011

2013

2015

2017

2019

2021

Price (USD/L)

0.78

1.32

1.31

1.22

0.99

0.85

0.84

1.24

Draw a scatter graph of the data points on the axes provided below, with the price in USD per litre against the number of years from 2007. ma3a_4-3_further-correlation-_-regression_medium_ib_ai_hl_maths

3b
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1 mark

State the type of function that would be appropriate to model the data.

3c
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2 marks

Using your graphic display calculator find the function of best fit for the data.

3d
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2 marks

Find the coefficient of determination and interpret the result in context.

3e
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3 marks

Sketch the model function on top of the scatter graph in part (a) and comment on the closeness of fit to the original data.

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4a
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3 marks

A specialist in infectious diseases is investigating the spread of a new disease. She is looking at the number of infections in a community based on the time, in days, since the first recorded case in the area.

She has focused on three sites and recorded the following information.

Site

1

2

3

Time since first recorded case, bold italic t

35

180

166

Number of infections, bold italic N

9

411

247

The scientist believes that the number of infections in a population can be modelled by an exponential function of the form

N equals A left parenthesis 1.2 right parenthesis to the power of b t end exponent, where A and b are constants.

Given that A equals 0.6 and b equals 0.2, find the predicted number of infections and hence the residuals for each site.

4b
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2 marks

A second model proposes that A equals 2.5 and b equals 0.15.

Find the residuals for each site for the second model.

4c
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3 marks

The scientist will use the model that has the lowest value for the sum of the squares of the residuals.

Determine which model the scientist should use.

4d
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1 mark

State one concern about the reliability of the model.

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5a
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2 marks

A unicorn leaves a magical sparkling trail behind him when he jumps off a rock that is 3 metres above ground level and flies away into the sky. The horizontal and vertical distances in metres from the base of the rock that the unicorn started from have been measured for several points in the sparkling trail and recorded in the table below.

bold italic x

0

2

4

6

7

10

bold italic y

3

3

4.2

9.5

15

53.3

Find an appropriate quadratic model for the data.

5b
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3 marks

Find the coefficient of determination for the quadratic model and comment on the closeness of fit to the data.

5c
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1 mark

State an alternative type of model that may be more suitable for the data.

5d
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2 marks

Find an equation for the alternative model.

5e
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2 marks

Comment on why this model is a better fit for the sparkly trail than the initial quadratic.

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6a
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2 marks

Grainne records the displacement,  in centimetres, of a particle from a fixed point O over the course of 9 seconds. She wishes to find an equation that will model the movement of the particle.

The coordinates of some of the data points of the particle are shown in the table below.

Time, bold italic t

0.9 1.2 2.7 3.9 4.2 5.1 5.7 7.7 8.3

Displacement, bold italic s

-5.9 -7.0 -8.2 -4.5 -3.0 -3.2 -5.8 -7.9 -6.0

State why a quadratic would not be a good fit for this model. Give a reason for your answer.

6b
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1 mark

State a type of function that would be appropriate to model the particle’s movement.

6c
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2 marks

Find an equation of the least squares regression for the type of function stated in part (b).

6d
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3 marks

Given that the displacement of the particle from point O is negative 4.7 cm, find the time(s) at which this occurs, for 0 less or equal than t less or equal than 9.

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7a
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3 marks

A professional skydiver is attempting to perform the highest skydive without the use of specialist breathing equipment. A sensor has been released from a height of 12,000 m to record the atmospheric pressure at different heights above sea level. The data from the sensor is recorded in the table below.

Height above sea level, bold italic h (m)

0

2885

5476

7732

9640

12000

Atmospheric pressure, bold italic p (Pa)

1013

766

483

350

301

215

An exponential model of the form p equals a left parenthesis b right parenthesis to the power of h is proposed to model the data, where a comma space b are constants to be found.

Find the values of a and b.

7b
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2 marks

Comment on any limits there might be for the domain of this model.

7c
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3 marks

It is agreed that an oxygen supply should be used when the atmospheric pressure is less than 365 Pa.

State whether a jump from a height of 8000 m could be attempted safely without an oxygen supply.

7d
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1 mark

Explain what action could be taken to increase the reliability of the model.

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8a
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4 marks

A digital artist has taken a photograph of a building and wishes to render it digitally. The photograph is imported to graphing software and the following points that follow the line of the roof are plotted.

bold italic x

0

0.7

1.8

2.3

3.3

4.0

4.5

bold italic y

1.2

1.5

1.7

2.9

6.5

8.0

13.8

One possible model for the roof is a cubic.

Another possible model for the roof is an exponential function, y equals a b to the power of x, where a comma b element of straight real numbers.

Find the equation of the least squares regression curve for

(i)
the cubic model

(ii)
the exponential model.
8b
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3 marks

Find the coefficient of determination for

(i)
the cubic model

(ii)
the exponential model.
8c
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2 marks

Hence state which function is more appropriate to model the roof line of the building. Give a reason for your answer.

8d
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2 marks

Using the model that was chosen in part (b), find the height of the roof when x equals 9.3.

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9a
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3 marks

Mo is looking at the growth rate of a particular type of bacteria. She records the number of cells of the bacteria and the time in minutes that has elapsed since the start of her observation. The results are recorded in the table below.

Time (minutes)

0

2

4

6

8

10

12

14

Number of cells

1

3

16

112

2667

6310

210863

914113

Express the number of cells as a logarithm. Give each value to 3 dp.

9b
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3 marks

Mo is not sure if a linear model or a quadratic model will fit the data best.

Find the coefficient of determination for

(i)
a linear model

(ii)
a quadratic model
9c
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1 mark

Comment on which model from part (c) best fits the graph.

9d
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2 marks

Find the equation of the least squares regression curve for the model specified in part (c).

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10a
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2 marks

Amelie wants to create a scale model of the cross section of a hill. She has measured the horizontal and vertical distances of several points on the hill from a fixed point O. These points can be seen in the diagram below. ma10a_4-3_further-correlation-_-regression_medium_ib_ai_hl_maths

Amelie decides to model the first section as a straight line and the second section as a quadratic with equation y equals negative 0.9 x squared plus 4.5 x minus 3.3.

Find the equation of the straight line passing through points A and B.

10b
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4 marks

Find the residuals for each point B, C, D and E for the quadratic model.

10c
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3 marks

Find the equation of the least squares quadratic curve for points B, C, D and E.

10d
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4 marks

By examining the sum of the squares of the residuals, show that the model found in part (c) is a better fit for the data.

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11a
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2 marks

The number of active profiles, N, a new social media platform has t months after launching are shown in the table below.

t 3 5 8 12 15 24
N 542 1122 2345 4254 8621 22 324
log N            


Complete the third row of the table above.

11b
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3 marks

Draw a scatter diagram of log N versus t.

11c
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3 marks

The regression line of log N on t can be written in the form log N = a + bt.

Find the values of a and b.

11d
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3 marks

Calculate the Pearson’s product-moment correlation coefficient, r, between log N and t and comment on the validity of the regression line found in part (c).

11e
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1 mark

Based on part (d) suggest a suitable type of regression model for N on t.

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1a
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3 marks

Jun is investigating the growth patterns of a certain species of slug and measures their thickness, t, and length, l. The results are shown below.

 

Thickness, t (mm)

9

8

4

3

6

10

Length, l (cm)

5.9

4.1

2.1

1.8

3.0

7.8

 

Draw a scatter diagram of the data and explain why a line of best fit should not be used for this data.

mi_q1a_4-3_further-correlation-_-regression_hard_dig
1b
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4 marks

Jun thinks that an exponential model of the form l equals 2 open parentheses 1.15 close parentheses to the power of t may fit the data, his sister Lily thinks that a quadratic model of the form l equals 0.06 t squared plus 1 is a better fit for the data.

Calculate the sum of the squared residuals, S S subscript r e s end subscript, for

(i)
Jun’s model, 
(ii)
Lily’s model.

 

1c
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2 marks

Hence, choose whether Jun’s model or Lily’s model best fits the data, giving a reason for your answer.

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2a
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2 marks

Wallace wants to model the path of some cylindrical cheeses as they roll down a hill. He measures the vertical distance from the bottom of the hill and the horizontal distance from the point at which the cheese was released at specific points on the path that the cheese takes. These measurements are recorded in the table below.

 

Horizontal Distance, x (m)

0

1

2

3

4

5

6

7

8

Vertical Distance,y (m)

8.00

7.47

6.08

4.19

2.23

0.23

0.00

0.29

1.46

By first plotting a scatter diagram on your GDC, choose whether a sinusoidal model or a power model would be a better fit for the data.

2b
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2 marks

Use your graphic display calculator to find the best fit function for the data points.

2c
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2 marks

Find the coefficient of determination and comment on the closeness of fit to the original data.

2d
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1 mark

Explain why this model could not be used outside of the range of the data collected.

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3a
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2 marks

A café in the UK begins selling ice creams on the 1st April each year and keeps track of their ice cream sales each month until they stop selling them at the end of October. The results from the year 2021 are shown in the table below. 

Month

April

May

June

July

August

Sept

Oct

Sales

82

142

391

516

728

312

64

 

Suggest a reason why the café owner may choose to use a quadratic function to model the monthly number of ice cream sales.

3b
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2 marks

Use technology to find the quadratic function of best fit for the data.

3c
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2 marks

Find the coefficient of determination and interpret the result in context.

3d
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2 marks

The café owner believes he may have miscalculated the ice cream sales in August. Use the model to find an estimate for the true number of sales of ice creams in August. Comment on the reliability of using the model in this context.

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4a
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3 marks

An ecologist is researching the connection between the mass of different species of rabbits and the spread of their population. She is looking at time taken for a population of 100 to increase to 1000 in four different species of rabbit. The table below shows the average mass of an adult male and the time, in months, for the population to reach 1000.

 

Rabbit Species

Brush Rabbit

Lionhead

Swamp Rabbit

English Lop

Average mass (k g)

0.7

1.2

2.1

5.2

Time (months)

7

8

11

24

 

The ecologist believes that the amount of time for a population to reach 1000 can be modelled by an exponential function of the form T equals 5.89 space e to the power of 0.273 m end exponent where T is the time in months, and m is the mass in kg.

Using the exponential model, find the predicted time taken for each species to reach a population of 1000 and hence the sum of the squared residuals, S S subscript r e s end subscript, for the model.

4b
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3 marks

Using technology, find the coefficient of determination for

(i)
the exponential model,
(ii)
the least squares regression quadratic model.
4c
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2 marks

Hence state which function is more appropriate to model the amount of time for a population to reach 1000. Give a reason for your answer.

4d
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1 mark

State one concern about the reliability of the model.

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5a
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3 marks

As a unicorn moves through the sky its magical sparkling trail becomes more prominent and draws a crowd of elves out to see it. The unicorn flies up high into the sky for a short while, drops down to a height of just above the elves’ heads and then comes to its landing place on a cliff top 10 metres above ground level. The horizontal and vertical distances, in metres, of the sparkling trail from the base of the rock that the unicorn started from have been measured for several points and recorded in the table below. 

 x

10

20

30

40

50

60

 y

53

82

116

95

3

10

Find an appropriate cubic model and quartic model for the data, giving all coefficients correct to four significant figures.

5b
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3 marks

Find the coefficient of determination for both models found in part (a) and comment on the reliability of each model.

5c
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3 marks

State an alternative type of model that may be more suitable for the data and find

(i)
an equation for this model,
(ii)
the coefficient of determination for this model.
5d
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2 marks

Comment on why this model is a better fit for the sparkly trail than the initial cubic and quartic models.

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6a
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5 marks

Scientists are collecting information about oxygen levels and temperature in the ocean. They collect data from various different sites. The information is recorded in the table below. 

Temperature at surface, T (°C)

13.2

11.1

23.6

15.1

4.2

29.1

19.1

Dissolved Oxygen Content, D( mgL-1)

9.1

10.9

8.7

9.0

13.6

8.1

8.8

 

Use your graphic display calculator to

(i)
find an appropriate quadratic model and logarithmic model for the data,
(ii)
investigate whether a quadratic or a logarithmic model best fits the data, giving reasons for your answer.
6b
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1 mark

Explain what action the scientists could take to further investigate which model is best.

6c
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3 marks

For both models found in part (a), find an approximation for the dissolved oxygen content in an ocean of 35 degree C and comment on the validity of your answers.

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7a
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4 marks

A new company believes that their net profit, t months after opening, can be modelled by the equation

 P open parentheses t close parentheses equals A t e to the power of b t end exponent comma space space space space space space space A comma b element of straight real numbers 

Their net profit at different times over the first year is given in the table below.

 

 t (months)

2

5

10

12

P ($1000)

0.2

2.5

51.6

201.9

 

Given A equals 0.04 and b equals 0.5, calculate the sum of the squared residuals,S S subscript r e s end subscript for this model and comment on its suitability.

7b
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4 marks

Show that a second model for the net profit over time can be given as

 ln space P equals 0.6764 t minus 2.764

7c
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2 marks

Choosing the model from part (a) or (b) that you believe to fit the data best, find an estimate for the net profit gained by the company after 20 months.

7d
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1 mark

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

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8a
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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 equals A M to the power of B,  where A and B are constants to be found.

Show that the relationship can be rewritten using logarithms as

log space L equals log space A space plus B space log space M

8b
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2 marks

Using data from a wide range of animals, when y equals log space L is plotted against x equals log space 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 equals 0.18 x plus 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

8c
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1 mark

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

8d
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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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9a
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4 marks

A virologist is studying the growth rate of a particular type of virus when attached to a particular host cell. They record the number of cells of the virus and the time in minutes that has elapsed since the virus attached to the host cell. The results are recorded in the table below. 

Time ( t hours)

0

1

2

3

4

5

Number of cells (n)

1

22

604

11270

125242

1007518

The virologist wants to linearise the data. They take logarithms of the number of cells for 1 less or equal than t less or equal than 5 and draw a semi-log graph of their calculated data. 

Draw a semi-log graph of the calculated data y equals ln space n, against time, t, for 1 less or equal than t less or equal than 5.

9b
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2 marks

Find the equation of the regression line of y on t, giving your answer in the form y equals a t plus b, where a and b are constants to be found.

9c
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3 marks

By substituting for into the answer for part (b), find a model for t and n in the form n equals A e to the power of c t end exponentwhere A and c are constants to be found.

9d
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2 marks

Find an estimate for the number of virus cells present after 3.5 hours, comment on the reliability of this estimate.

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10a
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5 marks

Olivia is modelling a new bowl for her pottery shop. She models the outline of one side of the bowl on a 2D Cartesian coordinate grid and plans to rotate the design 360° about the y- axis.
The coordinates Olivia uses to plot the cross-section are given in the table below.  

Point

A

B

C

D

E

F

G

H

I

x

5

6

8

10

12

14

16

18

18

y

0

2

4

5

6

6

8

10

12

 

Point A is connected to the origin and point is connected to the point open parentheses 0 comma 12 close parentheses with a straight, horizontal line.

Olivia initially models all of the points using a cubic curve.

(i)
Find the equation of the least squares regression cubic curve for all nine points.
(ii)
Find the coefficient of determination for the cubic model.
(iii)
Explain why this model is not a good model for all of the points and state which points Olivia should use a different model for. Use mathematical reasoning to validate your argument.
10b
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3 marks

Olivia decides instead to use two different linear models between points A and C and C and E and then use a quadratic model to connect points E, F, G, H and I. Find the equation of this quadratic model and write down a problem with using this model instead.

10c
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1 mark

State which model Olivia should use and any limitations to this model.

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1a
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1 mark

Effie is carrying out some research into dragon egg incubation patterns. She studies five dragon eggs and records the number of days it takes for the egg to hatch and the length of the wingspan on the newly hatched dragon. Some of Effie’s data is displayed in the following table.

 

Incubation time, t days

22

58

41

13

96

11

Wingspan length, lcm

52.368

x

59.151

y

34.016

16.761

 

Effie uses a least squares regression curve in the form l equals a t cubed plus b t squared plus c t minus 52 to model the data and calculates the sum of the squared residuals,  S S subscript r e s end subscript, to be zero.

Explain what a value of S S subscript r e s end subscript equals 0 means about using this least squares regression curve to model the data and hence write down the coefficient of determination.

1b
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4 marks

Use the data given in the table to find the values of a comma space band c and hence, find the values of xand y.

1c
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3 marks

Effie manages to collect data on one more dragon egg which has an incubation period of 33 days. Using the same model Effie now calculates the value of the sum of the squared residuals to be 0.0441.

Find the two possible wingspan lengths of this newly hatched dragon.

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2a
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2 marks

Gromit is researching the perfect basketball shot to get the basketball into a net of height 3.5 metres. He records himself taking ten shots from the same position and uses tracking software to measure the vertical and horizontal distances from the net at time t seconds after he shoots the ball. The mean distances are recorded in the table below. A positive value means the ball is above or in front of the net and a negative value means the ball is below or behind the net. 

Time, t seconds

0.0

0.3

0.6

0.9

1.2

1.5

1.8

Horizontal Distance,x (m)

2.52

2.44

2.07

1.12

0.28

0.01

-0.02

Vertical Distance, y (m)

- 2.05

0.56

2.98

4.09

2.23

0.23

-0.08

 

Find the height from which Gromit throws the ball.

2b
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6 marks

By first finding the equation of a quadratic, cubic and quartic least squares regression curve, investigate which is best for Gromit to use to model the trajectory of the ball. Give a reason why each equation is either suitable or unsuitable.

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3a
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4 marks

Paul moors his sailboat in a harbour and wants to come up with a model for the daily tidal pattern at a certain time of year. He collects data on the depth of the water at every hour over a 24-hour period and uses it to calculate a least squares regression curve in the form D equals a space sin open parentheses straight pi over 12 open parentheses t minus 2 close parentheses close parentheses plus b, where D is the water depth in metres and is the number of hours after midnight. Some of his data is given in the table below.

 

Time, t hours

3

8

16

20

Depth,D metres

x

12.87

7.12

4.93

Residual

0.065

-0.13

0.12

y

 

Use the information in the table to find the values of a and b.

3b
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3 marks

Hence, find the values of x and y.

3c
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2 marks

Find an estimate for the maximum and minimum tidal depths and the times at which they should occur.

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4a
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3 marks

Jusef is modelling the height of some water as it drains out of a conical tank. He believes that the vertical distance of the water in the tank from the top of the tank, H cm over time, t seconds, can be modelled using the equation H equals A b to the power of t, where A comma space b element of straight real numbers

Time, t seconds

1

5

7

10

12

14

15

Distance from top, H cm

0.1

1.3

2.2

5.1

9.4

16.2

26.0

 

Find the equation of the least squares regression curve and hence write down the values of A and b.

4b
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2 marks

Use mathematical reasoning to comment on the suitability of the model found in part (a).

4c
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4 marks

Jusef linearises the data and models it using a semi-log graph.

Calculate

(i)
the equation of the least squares regression line for the semi-log graph, 
(ii)
the Pearson’s product-moment correlation coefficient, r, and comment on the validity of the regression line found in part (c)(i).

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5a
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1 mark

Oakley records the vertical height, h in metres, of a red kite in flight over the course of 20 seconds. He wishes to find an equation that will model the movement of the red kite.

The height measurements of some points during the flight of the red kite are shown in the table below. 

Time, t

0.5 2.2 4.7 8.2 12.1 14.6 16.3 17.7 19.1

Height, h

19.3 19.2 19.1 18.7 18.2 1.9 1.8 1.4 1.5

 

Suggest a reason why Oakley may choose to use a logistic function to model the flight of the red kite.

5b
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3 marks

Use the data given to find an appropriate logistic model for the flight of the red kite, giving limits for the domain of this model.

5c
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1 mark

Use the model to estimate the height of the red kite at t equals 9.8 seconds.

5d
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1 mark

Oakley noted that the red kite was level with his window at a certain point on its flight. If Oakley’s window is 11.1 metres above ground level, find an approximation for the time at which the red kite was level with his window.

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6a
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3 marks

The velocity, v space ms to the power of negative 1 end exponent of a vehicle was recorded over 6 seconds and the results given in the table and plotted in the velocity time graph below. 

 mi_q6a_4-3_further-correlation-_-regression_dig 

Use the trapezoidal rule to find an estimate for the distance travelled by the vehicle.

6b
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2 marks

Explain whether the answer found in part (a) will be an over or underestimate of the actual distance travelled.

6c
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3 marks

Use the points on the graph to find the equation of the least squares quadratic regression curve.

6d
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2 marks

Write down the R squaredvalue and interpret what this tells you about the model.

6e
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2 marks

Use the answer found in part (c) to find a better estimate for the distance travelled by the vehicle.

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7a
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3 marks

Each year during the dry season in a particular country a man-made trapezoidal reservoir empties completely and then begins to refill when the rains reappear. In 2021 Bea collected information about the height of the water level, a certain number of days after the beginning of the rainy season. Bea believes thatthe vertical height of the water from the bottom of the reservoir, H metres, can be modelled using the equation H equals A t to the power of b, where A comma b element of straight real numbers and t is the number of days since the beginning of the rainy season.

 

Time, t days

1

2

3

5

10

20

40

Height, H m

1.5

3.8

5.7

8.3

14.5

24.9

30.0

 

Find the equation of the least squares regression curve and hence write down the values of A and b.

7b
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2 marks

Use mathematical reasoning to comment on the suitability of the model found in part (a).

7c
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3 marks

Bea wants to linearise the data using logarithms. 

(i)
Suggest, with a reason, whether Bea should use a semi-log graph or a log-log graph to linearise her data.

 

          (ii)      For this graph, find the equation of the regression line.

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8a
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4 marks

A researcher in a particular fitness centre has been collecting data on the number of sleeveless T-shirts sold per week, T,  and the number of new gym memberships per week, G. The data is shown in the table below. 

T

119

54

92

25

442

340

9

261

G

50

15

25

12

129

22

8

21

 

The researcher suspects that T  and G are related in one of two ways:

 T equals a G to the power of m space or space T equals b p to the power of G 

where a comma space b comma space mand p  are constants.

By finding the values of a comma space b comma space m comma space p and the coefficient of determination for the two different models, decide which better represents the relationship between T  and G.

8b
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3 marks

Without carrying out any further calculations,

(i)
Write down which model has a higher value of the product moment correlation coefficient and give a reason for your answer,
(ii)
explain whether it would be best to represent the data on a semi-log graph or a log-log graph.
8c
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1 mark

Hence, calculate the value of the product moment correlation coefficient for the chosen model in part (a).

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9a
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5 marks

When brewing beer the temperature that the beer is stored at during fermentation, T degree C , changes the alcohol content, A %, at the end of the fermentation process. A group of brewers collect data on T and A for their casks of beer. They suspect the data follows a model of the form A equals b p to the power of T where b and p are unknown constants. They plot the regression line of y equals ln space A on x equals T and find that the line has a gradient of 0.0392 and passes through the point open parentheses 0 comma space 0.811 close parentheses.

Using the line of regression, calculate the values of b and p.

9b
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6 marks

In the data collected by the brewers, the range of values for T was 15 and the range of values for A  was 4. The minimum alcohol content occurred when the temperature was at its minimum and the maximum alcohol content occurred when the temperature was at its maximum.

Find estimates for the minimum values of T and A to 2 significant figures.

9c
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1 mark

Hence explain why it would not be appropriate to use the model to predict the alcohol content of beer when the temperature during fermentation is 50 degree C.

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10a
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3 marks

Nora is building a skate ramp for a project at school. She models the first part of the ramp using the coordinates A open parentheses 0 comma space 5 close parentheses comma space B open parentheses 1 comma space 8 close parentheses comma space C open parentheses 5 comma space 15 close parentheses and D open parentheses 10 comma space 18 close parentheses. These four points fit a cubic model in the form y equals a x cubed plus b x squared plus c x plus d.

Find the model for this section of Nora’s skate ramp, giving a comma space b comma space c space and d as exact values. Hence, write down the value of the sum of the squares residual.

10b
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4 marks

Nora models the next part of her skate ramp from the point D using a least squares regression curve with equation

 y equals a space cos space 0.1 open parentheses x minus 10 close parentheses plus b 

Given that the skate ramp goes through the point E open parentheses 23 comma 7 close parentheses with a residual of negative 0.0124824 to six significant figures, find the values of a and b.

10c
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2 marks

The final section of Nora’s skate ramp goes through the coordinates F open parentheses 30 comma space 2 close parentheses comma space G open parentheses 35 comma space 0 close parentheses comma space H left parenthesis 40 comma space 2 right parenthesis and I open parentheses 60 comma space 15 close parentheses. Nora models this using a quadratic model of y equals 0.028 x squared minus 2.1 x plus 40.

Find the equation of the least squares trigonometric curve for points F comma space G comma space H and I.

10d
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4 marks

By examining the sum of the squares of the residuals, show that the model found in part (c) is a better fit for the data than Nora’s quadratic model.

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