Regression lines

Two students are experimenting with some water in a plastic bottle. The bottle is filled with water and a hole is put in the bottom of the bottle. The students record the time, t seconds, it takes for the water level to fall to each of 10 given values of the height, h cm, above the hole.

The time it takes the water level to fall to a height of 9 cm above the hole is 47 seconds.

Calculate the residual for this data point. Give your answer to 2 decimal places.

calculate the RSS for these 10 data points.

Student B models the data with an equation of the form t = c + dh

The regression line of t on h is calculated and the residual sum of squares (RSS) is found to be 980 to 3 significant figures.

With reference to part (c) state, giving a reason, whether Student B’s model or Student A ’s model is the more suitable for these data.

Baako is investigating the times taken by children to run a 100 m race, x seconds, and a 500 m race, y seconds. For a sample of 20 children, Baako obtains the time taken by each child to run each race.

Here are Baako’s summary statistics.

Calculate the product moment correlation coefficient between the times taken to run the 100 m race and the times taken to run the 500 m race.

Show that the equation of the regression line of y on x can be written as

y = 5.12 x + 7.42

where the gradient and y intercept are given to 3 significant figures.

The child who completed the 100 m race in 20 seconds took 104 seconds to complete the 500 m race.

Find the residual for this child.

The table below shows the signs of the residuals for the 20 children in order of finishing time for the 100 m race.

Explain what the signs of the residuals show about the model’s predictions of the 500 m race times for the children who are fastest and slowest over the 100 m race.

Kwame is investigating a possible relationship between average March temperature, t °C, and tea yield, y kg/hectare, for tea grown in a particular location. He uses 30 years of past data to produce the following summary statistics for a linear regression model, with tea yield as the dependent variable.

Residual Sum of Squares (RSS) = 1 666 567 S_tt = 52.0 S_yy = 1 774 155 least squares regression line: gradient = 45.5 y-intercept = 2080

Use the regression model to predict the tea yield for an average March temperature of 20 °C

He also produces the following residual plot for the data.

Explain what you understand by the term residual.

Calculate the product moment correlation coefficient between t and y

Explain why the linear model may not be a good fit for the data

with reference to your answer to part (c)

with reference to the residual plot.

Question 1 continues on page 4

Kwame also collects data on total March rainfall, w mm, for each of these 30 years.

For a linear regression model of w on t the following summary statistic is found.

Residual Sum of Squares (RSS) = 86 754

Kwame concludes that since this model has a smaller RSS, there must be a stronger linear relationship between w and t than between y and t (where RSS = 1 666 567)

State, giving a reason, whether or not you agree with the reasoning that led to Kwame’s conclusion.

A researcher is investigating the relationship between elevation, x metres, and annual mean temperature, t °C.

From a random sample of 20 weather stations in Switzerland, the following results were obtained

The product moment correlation coefficient for these data is found to be −0.959

Interpret the value of this correlation coefficient.

Show that the equation of the regression line of t on x can be written as

t = 14.3 − 0.00681 x

The random variable W represents the elevations of the weather stations in kilometres.

Write down the equation of the regression line of t on w for these 20 weather stations in the form t = a + bw

Show that the residual sum of squares (RSS) for the model for t and x is 35.7 correct to one decimal place.

One of the weather stations in the sample had a recorded elevation of 1100 metres and an annual mean temperature of 1.4 °C

Calculate this weather station’s contribution to the residual sum of squares. Give your answer as a percentage.

Comment on the data for this weather station in light of your answer to part (e)(i).

A large field of wheat is split into 8 plots of equal area. Each plot is treated with a different amount of fertiliser, grams /m². The yield of wheat, w tonnes, from each plot is recorded. The results are summarised below.

Calculate the product moment correlation coefficient between f and w

Interpret the value of your product moment correlation coefficient.

Find the equation of the regression line of w on f in the form w = a + bf

Using your equation, estimate the decrease in yield when the amount of fertiliser decreases by 0.5 grams /m²

The residuals of the data recorded are calculated and plotted on the graph below.

With reference to this graph, comment on the suitability of the model you found in part (c).

Suggest how you might be able to refine your model.