Model a linear relationship by using technology to fit a least-squares line to the data, in the form of 𝑦 = 𝑚𝑥 + 𝑐 where 𝑚 is slope (gradient) and 𝑐 is 𝑦-intercept.

A physiologist records the relationship between ambient temperature (in degrees Celsius) and the metabolic rate (in kJ/hour) of a small mammal. Using a calculator or computer, the least-squares regression line is determined to be $y = 4.2x + 187.5$, where $x$ is the ambient temperature and $y$ is the metabolic rate. (a) Identify the slope and $y$-intercept of this model. [1 mark] (b) Use the model to predict the metabolic rate when the ambient temperature is $18°C$. Give your answer to the nearest integer. [1 mark] (c) Explain what the slope value tells us about the relationship between temperature and metabolic rate in this context. [1 mark]

A graphics calculator is used to determine the equation of the least-squares line for data relating monthly rainfall (in mm) to the number of days of precipitation recorded during that month. The results are shown below. Which of the following is the correct equation of the least-squares line?

A local library collected data on the number of study rooms available and the total number of visitor bookings received per week. The data is shown in the table below. (a) Use technology to determine the equation of the least-squares line for this data, in the form \(y = mx + c\), where \(x\) is the number of study rooms and \(y\) is the total number of visitor bookings. Give the values of \(m\) and \(c\) correct to two decimal places. (2 marks) (b) Interpret the meaning of the gradient \(m\) in this context. (1 mark) (c) Use the model to predict the number of visitor bookings when 18 study rooms are available. Give your answer correct to the nearest whole number. (1 mark)

A local gym tracked the relationship between the number of weekly training sessions attended by members and their improvement in cardiovascular fitness (measured in beats per minute reduction in resting heart rate). (a) Use technology to find the equation of the least-squares line in the form $y = mx + c$, where $x$ is the number of weekly sessions and $y$ is the improvement in resting heart rate (in beats per minute). Give your answer correct to 1 decimal place. (2 marks) (b) Using your equation from part (a), predict the improvement in resting heart rate for a member who attends 6 weekly sessions. Give your answer to the nearest whole number. (1 mark)