Sum of squared residuals and R² Problem set A

The table shows the number of hours, x, that eight students revised for a test, and their score, y, out of 100. Fit a linear model and find \(R^2\) to 3 significant figures.

The table shows the average outdoor temperature, x °C, in a month, and the household's electricity bill, £y, for that month. A quadratic model is suggested. Fit a quadratic model and find \(R^2\) to 3 significant figures.

A shopping centre counts the shoppers, y hundred, present at the start of each hour after opening (x = 0 to 7). A cubic model is suggested. Fit a cubic model and find \(R^2\) to 3 significant figures.

A student finds that Pearson's product-moment correlation coefficient between the outdoor temperature and a household's heating bill is \(r=-0.823\). A linear model is used.

1. (a) Find \(R^2\) to 3 significant figures.
2. (b) Write down the percentage of the variation in the heating bill that is explained by the model, to 3 significant figures.

A print shop models the cost, £C, of printing n copies by \(C=2.5+0.18n\). The table shows five orders.

1. (a) Find the residual for each order, the data value minus the model value.
2. (b) Find \(SS_{res}\).

The table gives the distance x, in hundreds of thousands of km, of each of eight moons from the centre of the planet Zorin, and the orbital period T days of the moon.

1. (a) Find a cubic model \(T=ax^3+bx^2+cx+d\), giving a, b, c and d to 3 significant figures.
2. (b) Use the model to predict the orbital period of a moon at a distance of 3 500 000 km (x = 35).
3. (c) Comment on the reliability of your prediction.

A shop's annual profit, £y thousand, is recorded for each of the first eight years after it opens (x = 0 to 7). A quadratic model and a cubic model are both fitted.

1. (a) Find \(R^2\) for the quadratic model.
2. (b) Find \(R^2\) for the cubic model.
3. (c) Which model fits the data better? Give a reason.

The height, y cm, of a plant is measured at the end of each of six weeks (x = 1 to 6). Two students suggest models: Model 1 is \(y=3.1x-0.4\) and Model 2 is \(y=0.1x^2+2.2x+0.8\).

1. (a) Find \(SS_{res}\) for Model 1.
2. (b) Find \(SS_{res}\) for Model 2.
3. (c) Which model fits the data better? Give a reason.

The population, y thousand, of a town is recorded every two years from t = 0 (the year 2008) to t = 14. A linear model is used.

1. (a) Find the linear model \(y=at+b\) and \(R^2\), to 3 significant figures.
2. (b) Use the model to predict the population of the town in the year 2068.
3. (c) Comment on the reliability of your prediction.