4.3 Further Correlation & Regression

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.

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

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.

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.

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.

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.

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

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

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

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

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.

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

, where  and  are constants.

Given that and , find the predicted number of infections and hence the residuals for each site.

A second model proposes that  and .

Find the residuals for each site for the second model.

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.

State one concern about the reliability of the model.

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.

	0	2	4	6	7	10
	3	3	4.2	9.5	15	53.3

Find an appropriate quadratic model for the data.

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

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

Find an equation for the alternative model.

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

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.

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

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

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

Given that the displacement of the particle from point O is cm, find the time(s) at which this occurs, for .

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.

An exponential model of the form  is proposed to model the data, where  are constants to be found.

Find the values of and .

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

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

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

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

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.

	0	0.7	1.8	2.3	3.3	4.0	4.5
	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, , where .

Find the equation of the least squares regression curve for

Find the coefficient of determination for

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

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

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.

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

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

Find the coefficient of determination for

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

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

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.

Amelie decides to model the first section as a straight line and the second section as a quadratic with equation 

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

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

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

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.

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

Complete the third row of the table above.

Draw a scatter diagram of log N versus t.

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.

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).

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

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.