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

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.

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

Jun thinks that an exponential model of the form may fit the data, his sister Lily thinks that a quadratic model of the form  is a better fit for the data.

Calculate the sum of the squared residuals, , for

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

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.

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.

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

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

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

A café in the UK begins selling ice creams on the 1^st 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. 

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

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

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

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.

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.

The ecologist believes that the amount of time for a population to reach 1000 can be modelled by an exponential function of the form  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, , for the model.

Using technology, find the coefficient of determination for

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.

State one concern about the reliability of the model.

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. 

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

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

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

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

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. 

Use your graphic display calculator to

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

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

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

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

Given  and , calculate the sum of the squared residuals, for this model and comment on its suitability.

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

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.

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

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 ,  where  and  are constants to be found.

Show that the relationship can be rewritten using logarithms as

Using data from a wide range of animals, when  is plotted against on a scatter diagram there seems to be a strong positive correlation. When the regression line of  on  is calculated, the equation is found to be .

By relating the equation of the regression line to the equation found in (a), or otherwise, find the constants and  correct to 2 decimal places where appropriate

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

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.

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. 

The virologist wants to linearise the data. They take logarithms of the number of cells for  and draw a semi-log graph of their calculated data. 

Draw a semi-log graph of the calculated data , against time, t, for .

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

By substituting for into the answer for part (b), find a model for  and  in the form where  and  are constants to be found.

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

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
	5	6	8	10	12	14	16	18	18
	0	2	4	5	6	6	8	10	12

Point A is connected to the origin and point I  is connected to the point  with a straight, horizontal line.

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

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.

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

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, cm	52.368		59.151		34.016	16.761

Effie uses a least squares regression curve in the form  to model the data and calculates the sum of the squared residuals,  , to be zero.

Explain what a value of means about using this least squares regression curve to model the data and hence write down the coefficient of determination.

Use the data given in the table to find the values of and  and hence, find the values of and .

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.

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. 

Find the height from which Gromit throws the ball.

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.

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 , where 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		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  and .

Hence, find the values of and .

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

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, cm over time, t seconds, can be modelled using the equation , where . 

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

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

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

Calculate

(ii)

the Pearson’s product-moment correlation coefficient, , and comment on the validity of the regression line found in part (c)(i).

Oakley records the vertical height,  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. 

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

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.

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

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.

The velocity,  of a vehicle was recorded over 6 seconds and the results given in the table and plotted in the velocity time graph below. 

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

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

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

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

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

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 , where and t is the number of days since the beginning of the rainy season.

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

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

Bea wants to linearise the data using logarithms. 

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

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

	119	54	92	25	442	340	9	261
	50	15	25	12	129	22	8	21

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

where and  are constants.

By finding the values of and the coefficient of determination for the two different models, decide which better represents the relationship between  and .

Without carrying out any further calculations,

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

When brewing beer the temperature that the beer is stored at during fermentation, , changes the alcohol content, A %, at the end of the fermentation process. A group of brewers collect data on and A for their casks of beer. They suspect the data follows a model of the form  where  and  are unknown constants. They plot the regression line of  on  and find that the line has a gradient of 0.0392 and passes through the point .

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

In the data collected by the brewers, the range of values for  was 15 and the range of values for   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  and  to 2 significant figures.

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 .

Nora is building a skate ramp for a project at school. She models the first part of the ramp using the coordinates and . These four points fit a cubic model in the form .

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

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

Given that the skate ramp goes through the point  with a residual of  to six significant figures, find the values of  and .

The final section of Nora’s skate ramp goes through the coordinates  and . Nora models this using a quadratic model of .

Find the equation of the least squares trigonometric curve for points  and .

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.