Sum of squared residuals and R² Problem set B

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.

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.

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.

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

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

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.
2. (b) Use the model to predict the orbital period of a satellite at a distance of 150 thousand km (r = 150).
3. (c) Comment on the reliability of your prediction.

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

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

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.
2. (b) Use the model to predict the average price in the year 2052.
3. (c) Comment on the reliability of your prediction.