Use the equation of the least-squares line to make predictions.

A study tracked the relationship between the number of training sessions attended and the time (in minutes) taken to complete a fitness task. The least-squares regression line is given by \(t = -1.8n + 45.2\), where \(t\) is the time in minutes and \(n\) is the number of training sessions attended. Which of the following gives the predicted time for someone who has attended 10 training sessions?

A mobile phone company analysed the relationship between years of customer loyalty and monthly spending. The least-squares line equation for the data is \(y = 8.5x + 24.3\), where \(x\) is the number of years a customer has been with the company and \(y\) is the predicted monthly spending in dollars. (a) Use this equation to predict the monthly spending of a customer with 6 years of loyalty. (b) Hence, calculate the total spending predicted over 12 months for this customer, to the nearest dollar.

The number of visitors to a wildlife sanctuary over several years has been modelled using a least-squares line with equation $y = 1{,}250x + 8{,}500$, where $x$ represents the year (with year 1 being 2015) and $y$ represents the number of annual visitors. (a) Use the equation to predict the number of annual visitors in 2024. [1 mark] (b) Calculate the increase in predicted visitors from 2019 to 2023. [2 marks]

A market researcher models the relationship between years since launch (x) and annual sales (y, in thousands of units) for a new product. Data from the first seven years gives the least-squares line equation y = 23.8 + 6.4x, with a coefficient of determination, R² = 0.88. (a) Use the equation of the least-squares line to predict the annual sales in year 10. (2 marks) (b) The product will be discontinued if annual sales fall below 50,000 units. Evaluate the reasonableness of continuing the product to year 10, based on your prediction in part (a) and the strength of the linear model. (2 marks)

A local cinema analysed the relationship between the number of cinema tickets sold per week and the average daily temperature during that week. The least-squares line equation for this data is $y = 8.2x + 156$, where $x$ is the average daily temperature in degrees Celsius and $y$ is the number of tickets sold per week. (a) Use the equation to predict the number of tickets sold in a week when the average daily temperature is $18°C$. (1 mark) (b) In another week, the average daily temperature is $24°C$. Use the equation to predict the number of tickets sold. (1 mark) (c) State one limitation of using this equation to make predictions. (1 mark)