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

The table shows the maximum daily temperature, x °C, and the number, y, of ice creams sold at a kiosk on six days. Fit a linear model and find \(R^2\) to 3 significant figures.

• From the calculator: linear regression: a = 3.29, b = 33.5
• r = 0.946
• \(R^2=r^2\) = 0.894

The height, y metres, of a football is measured x seconds after it is kicked. A quadratic model is suggested. Fit a quadratic model and find \(R^2\) to 3 significant figures.

• From the calculator: quadratic regression: a = −3.38, b = 17.0, c = 1.04
• \(R^2\) = 0.999

The depth of water, y metres, of a lake is measured at the start of each month for eight months (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 = 0.380, b = −4.02, c = 7.38, d = 32.4
• \(R^2\) = 0.993

A student finds that Pearson's product-moment correlation coefficient between the height and the arm span of 30 students is \(r=0.874\). A linear model is used.

1. (a) Find \(R^2\) to 3 significant figures.

   • \(R^2=r^2=(0.874)^2\) = 0.764

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

   • \(R^2\) as a percentage: 0.764 × 100 = 76.4%
   • About 76.4% of the variation is explained by the model

A courier models the cost, £D, of delivering a parcel k km by \(D=3.2+0.85k\). The table shows five deliveries.

1. (a) Find the residual for each delivery, the data value minus the model value.

   • Model values: 4.9, 6.6, 8.3, 10, 11.7
   • Residuals (data − model): 0.2, −0.2, 0.6, −0.3, 0.1

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

   • Squares: 0.04, 0.04, 0.36, 0.09, 0.01
   • \(SS_{res}=\) \(0.04 + 0.04 + 0.36 + 0.09 + 0.01 = 0.54\)

The table gives the distance r thousand km from the centre of the Earth of each of eight satellites, and the orbital period P minutes of the satellite.

1. (a) Find a cubic model \(P=ar^3+br^2+cr+d\), giving a, b, c and d to 3 significant figures.

   • From the calculator: cubic regression: a = −0.00447, b = 0.720, c = 11.8, d = −19.1
   • Model: \(P = -0.00447r^3 + 0.720r^2 + 11.8r - 19.1\)

2. (b) Use the model to predict the orbital period of a satellite at a distance of 150 thousand km (r = 150).

   • P(150) from the stored model = 2870

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

   • Data range: 6.8 to 36; 150 is outside it
   • r = 150 is outside the range of the data (6.8 to 36), so this is extrapolation and is not reliable. The cubic has a negative r³ coefficient: it reaches a maximum at r = 115 and then falls, but a satellite further from the Earth would not take less time to orbit it.

The monthly sales, y hundred items, of a seasonal product are recorded for nine consecutive months (x = 0 to 8). A quadratic model and a cubic model are both fitted.

1. (a) Find \(R^2\) for the quadratic model.

   • From the calculator: quadratic regression: \(R^2\) = 0.852

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

   • From the calculator: cubic regression: \(R^2\) = 0.956

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

   • 0.956 > 0.852
   • The model with the larger \(R^2\) fits the data better

The temperature, y °C, of a cooling drink is measured at the end of each of six minutes (x = 1 to 6). Two students suggest models: Model 1 is \(y=-6.8x+75.5\) and Model 2 is \(y=1.2x^2-14x+85\).

1. (a) Find \(SS_{res}\) for Model 1.

   • Model values: 68.7, 61.9, 55.1, 48.3, 41.5, 34.7
   • Residuals: 2.3, −3.4, −5.3, −4.4, −1.4, 1.3
   • \(SS_{res}\) = 67.95

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

   • Model values: 72.2, 61.8, 53.8, 48.2, 45, 44.2
   • Residuals: −1.2, −3.3, −4, −4.3, −4.9, −8.2
   • \(SS_{res}\) = 138.07

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

   • 67.95 and 138.07: the smaller \(SS_{res}\) belongs to the closer model
   • Model 1 fits better

The average price, y thousand pounds, of a house in a region is recorded each year for 11 years, from t = 0 (the year 2012) to t = 10. 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 = 5.97, b = 214, \(R^2\) = 0.950

2. (b) Use the model to predict the average price in the year 2052.

   • t = 40: y = 5.97 × 40 + 214 = 453

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

   • 40 is far outside the data range (0 to 10), so the prediction is extrapolation. House prices depend on wages, interest rates and supply, which do not follow one straight line for 40 years. The high \(R^2\) only describes the fit inside the data.