4.3 Further Correlation & Regression

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.

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. 

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.

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. 

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. 

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.

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.

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

The biologist concludes the research by suggesting that one way to increase your lifespan is to increase your mass.

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 .

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.