Sum of squared residuals and R² Problem set A One question at a time

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.

• From the calculator: linear regression: a = 4.98, b = 45.1
• r = 0.940
• \(R^2=r^2\) = 0.883

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.

• From the calculator: quadratic regression: a = 0.212, b = −6.83, c = 124
• \(R^2\) = 0.969

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.

• From the calculator: cubic regression: a = 0.186, b = −2.83, c = 11.0, d = 6.57
• \(R^2\) = 0.888

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.

   • \(R^2=r^2=(−0.823)^2\) = 0.677

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

   • \(R^2\) as a percentage: 0.677 × 100 = 67.7%
   • About 67.7% of the variation is explained by the model

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.

   • Model values: 4.3, 6.1, 7.9, 9.7, 11.5
   • Residuals (data − model): 0.2, 0.3, 0.2, 0.3, −0.2

2. (b) Find \(SS_{res}\).

   • Squares: 0.04, 0.09, 0.04, 0.09, 0.04
   • \(SS_{res}=\) \(0.04 + 0.09 + 0.04 + 0.09 + 0.04 = 0.3\)

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.

   • From the calculator: cubic regression: a = −0.0121, b = 0.489, c = 1.75, d = −0.500
   • Model: \(T = -0.0121x^3 + 0.489x^2 + 1.75x - 0.500\)

2. (b) Use the model to predict the orbital period of a moon at a distance of 3 500 000 km (x = 35).

   • T(35) from the stored model = 140

3. (c) Comment on the reliability of your prediction.

   • Data range: 0.6 to 10; 35 is outside it
   • x = 35 is outside the range of the data (0.6 to 10), so this is extrapolation and is not reliable. The cubic has a negative x³ coefficient: it reaches a maximum at x = 28.6 and then falls, but a moon further from the planet would not take less time to orbit it.

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.

   • From the calculator: quadratic regression: \(R^2\) = 0.962

2. (b) Find \(R^2\) for the cubic model.

   • From the calculator: cubic regression: \(R^2\) = 0.992

3. (c) Which model fits the data better? Give a reason.

   • 0.992 > 0.962
   • The model with the larger \(R^2\) fits the data better

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.

   • Model values: 2.7, 5.8, 8.9, 12, 15.1, 18.2
   • Residuals: 0.4, −0.6, −0.5, −0.7, −1.2, −0.7
   • \(SS_{res}\) = 3.19

2. (b) Find \(SS_{res}\) for Model 2.

   • Model values: 3.1, 5.6, 8.3, 11.2, 14.3, 17.6
   • Residuals: 0, −0.4, 0.1, 0.1, −0.4, −0.1
   • \(SS_{res}\) = 0.35

3. (c) Which model fits the data better? Give a reason.

   • 3.19 and 0.35: the smaller \(SS_{res}\) belongs to the closer model
   • Model 2 fits better

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.

   • From the calculator: linear regression: a = 0.533, b = 14.7, \(R^2\) = 0.991

2. (b) Use the model to predict the population of the town in the year 2068.

   • t = 60: y = 0.533 × 60 + 14.7 = 46.7

3. (c) Comment on the reliability of your prediction.

   • 60 is far outside the data range (0 to 14), so the prediction is extrapolation. A town's population does not usually keep growing by the same amount for 60 years: housing, jobs and land all limit it. The high \(R^2\) only describes the fit inside the data.