Inferential Statistics · Paper 1 — Free Maths AI SL Practice Questions

Calculate the equation of the regression line of $y$ on $x$, giving the values of $a$ and $b$ in $y = ax + b$ correct to three significant figures. [2 marks]

A plant receives $x = 8$ hours of sunlight per day. A student uses the regression equation to predict its height after 30 days and states: "The model predicts a height of $52.5$ cm, so this is a reliable estimate." Evaluate the reliability of this prediction. [2 marks]

A car is 4.5 years old. Calculate its predicted selling price in dollars. [2 marks]

The actual selling price of the 4.5-year-old car is USD 6000. (i) Calculate the residual for this car in dollars. [1]
(ii) State what the sign of the residual indicates about the regression line's prediction for this car. [1 mark]

Calculate the coefficient of determination $r^2$. [1 mark]

Hence explain what the value of $r^2$ indicates about the relationship between temperature and chirping rate. [1 mark]

Determine the equation of the regression line of $D$ on $T$ in the form $D = a + bT$, with $a$ and $b$ correct to three significant figures. [2 marks]

Use your regression line to estimate the diving depth when $T = 9.5\,°\text{C}$, giving your answer correct to the nearest metre. [1 mark]

The biologist claims: "Temperature is a good predictor of diving depth because the correlation is strong." Evaluate this claim. In your response, refer to your value of $r$, the reliability of the regression model within the data range, and one limitation of using this model. The following formulae are provided: $$r = \frac{n\sum TD - \left(\sum T\right)\!\left(\sum D\right)}{\sqrt{n\sum T^2 - \left(\sum T\right)^2}\;\sqrt{n\sum D^2 - \left(\sum D\right)^2}}$$ $$b = \frac{n\sum TD - \left(\sum T\right)\!\left(\sum D\right)}{n\sum T^2 - \left(\sum T\right)^2}, \qquad a = \bar{D} - b\bar{T}$$ [3 marks]