Understand and use 𝑚 = 𝑟 𝑠𝑦 𝑠𝑥 and 𝑐 = 𝑦 − 𝑚𝑥 to determine the equation of a least-squares line, where 𝑚 is slope (gradient), 𝑟 is correlation coefficient, 𝑠𝑦 is (sample) standard deviation of 𝑦 values, 𝑠𝑥 is (sample) standard deviation of 𝑥 values, 𝑐 is 𝑦-intercept, 𝑦 is mean of 𝑦 values and 𝑥 is mean of 𝑥 values.

A fitness centre recorded the number of minutes spent exercising per week and the corresponding resting heart rate (in beats per minute) for a sample of members. Analysis of the data produced the following statistics: • Correlation coefficient: $r = -0.88$ • Mean exercise time: $\bar{x} = 180$ minutes • Mean resting heart rate: $\bar{y} = 72$ beats per minute • Standard deviation of exercise time: $s_x = 45$ minutes • Standard deviation of resting heart rate: $s_y = 8$ beats per minute Determine the equation of the least-squares line to predict resting heart rate from exercise time. Give your answer in the form $y = mx + c$, where $m$ and $c$ are correct to two decimal places.

A fitness researcher collected data on the number of hours spent exercising per week and the resting heart rate (in beats per minute) for a sample of participants. For this data: • Mean hours exercising: $\bar{x} = 4.2$ hours • Mean resting heart rate: $\bar{y} = 72$ beats per minute • Standard deviation of hours: $s_x = 1.8$ hours • Standard deviation of resting heart rate: $s_y = 8.5$ beats per minute • Correlation coefficient: $r = -0.92$ Determine the equation of the least-squares line in the form $y = mx + c$, giving both $m$ and $c$ to 1 decimal place.

A fitness researcher collected data on weekly gym attendance (\(x\) hours) and weight loss (\(y\) kilograms) for eight participants over a study period. The data is summarised in the table below. For this data set, the correlation coefficient (\(r\)) is \(0.92\), the mean weekly gym attendance (\(\bar{x}\)) is \(4.5\) hours, and the mean weight loss (\(\bar{y}\)) is \(2.8\) kg. Determine the equation of the least-squares line for this data in the form \(y = mx + c\), where \(m\) and \(c\) are given correct to two decimal places.

A fitness centre tracked the relationship between the number of weekly training sessions and average heart rate recovery time in seconds for a group of clients over an eight-week period. The data is shown in the table below. The correlation coefficient for this data is r = -0.92. Determine the equation of the least-squares line in the form y = mx + c, where m and c are expressed correct to two decimal places.

A sports scientist collects data on athletes' training hours per week and their performance scores. For a sample of 15 athletes, the mean training hours is \(\bar{x} = 8.5\) hours, the mean performance score is \(\bar{y} = 73.2\), the standard deviation of training hours is \(s_x = 2.1\) hours, the standard deviation of performance scores is \(s_y = 5.8\), and the correlation coefficient is \(r = 0.88\). Determine the equation of the least-squares line in the form \(\hat{y} = mx + c\), where values are correct to one decimal place.

A dataset contains information about hours spent training per week and the corresponding performance score for 8 athletes. The correlation coefficient is $r = 0.92$, the mean training hours is $\bar{x} = 5.5$, the mean performance score is $\bar{y} = 73.4$, the standard deviation of training hours is $s_x = 2.1$, and the standard deviation of performance scores is $s_y = 8.6$. Which of the following is the equation of the least-squares regression line (where $x$ is training hours and $y$ is performance score)?