The regression line and prediction

When there is a linear relation, $\displaystyle y$ is predicted from $\displaystyle x$ with one straight line that runs “through the middle” of the points – the regression line. The dashed line is the regression line of the son’s height on the father’s height:

• The prediction of $\displaystyle y$ for a value of $\displaystyle x$ is the point on the line above $\displaystyle x$.
• The line is one straight line for every value of $\displaystyle x$, not broken lines as in the means graph.
• There is only one line for which the sum of the squares of the points’ vertical distances from it is the smallest – that is the regression line.

The regression line predicting $\displaystyle y$ from $\displaystyle x$ passes through the point of the means $\displaystyle (\bar{x},\bar{y})$, and its slope $\displaystyle m$ is set by the correlation coefficient and the standard deviations:

• The sign: $\displaystyle S_x$ and $\displaystyle S_y$ are positive, so the slope has the sign of $\displaystyle r$ – the line rises in a positive relation and falls in a negative one.
• The size: when $\displaystyle {r}={1}$ the points lie on the line, and an $\displaystyle x$ one standard deviation larger predicts a $\displaystyle y$ one standard deviation larger: $\displaystyle {m}={\frac{S_y}{S_x}}$.
• When the relation is weaker, the prediction is “pulled” towards the mean $\displaystyle \bar{y}$: for the son of a very tall father the predicted height is less extreme. When $\displaystyle {r}={0}$, $\displaystyle {m}={0}$ – the prediction is always $\displaystyle \bar{y}$.

Pupils sat two maths tests.
In the first test: $\displaystyle {\bar{x}}={70}$, $\displaystyle {S_x}={5}$.
In the second test: $\displaystyle {\bar{y}}={75}$, $\displaystyle {S_y}={8}$.
The correlation coefficient between the two tests is $\displaystyle {r}={0.7}$.
A pupil scored 80 in the first test. What is the prediction of the pupil’s mark in the second test?

• $\displaystyle {m}={0.7\cdot\frac{8}{5}}={1.12}$.
• $\displaystyle {y-75}={1.12(x-70)}$, that is $\displaystyle {y}={1.12x-3.4}$.
• $\displaystyle {1.12\cdot80-3.4}={86.2}$: the prediction is 86.2.
• For a pupil who scored 70, the mean, the prediction is 75 – the mean of the second test.

A large company recorded every day the number of employees who caught flu ($\displaystyle x$) and the number of employees who worked from home ($\displaystyle y$).
$\displaystyle {\bar{x}}={10}$, $\displaystyle {S_x}={1.2}$.
$\displaystyle {\bar{y}}={25}$, $\displaystyle {S_y}={2.3}$.
$\displaystyle {r}={0.65}$.

Work out the slope of the regression line, to two decimal places.

Write the equation of the regression line in the form $\displaystyle {y}={mx+b}$. Start from $\displaystyle \bar{y}+m(x-\bar{x})$, with the rounded slope.

By the regression line, how many employees will work from home on a day when 15 employees caught flu?

Why, in your opinion, is $\displaystyle r\neq1$?

Among 300 teenagers, the relation between age ($\displaystyle x$, in years) and the number of minutes a day they spend on social networks ($\displaystyle y$) was examined.
The mean age is 15.4 years, and the standard deviation 1.2 years.
The mean time is 75 minutes, and the standard deviation 24 minutes.
$\displaystyle {r}={0.72}$.

Find the equation of the regression line predicting the time on social networks from the age, and write it on the board.

By the regression line, how many minutes a day does a girl aged 16 years and 3 months spend on social networks?

A lecturer looked at whether there is a linear relation between students’ number of absences from lectures ($\displaystyle x$) and their mark in the final exam ($\displaystyle y$).
The mean number of absences is $\displaystyle {\bar{x}}={10}$, and the mean mark $\displaystyle {\bar{y}}={70}$.
The correlation coefficient is negative.
The equation of the regression line is one of these: $\displaystyle {y}={10x-30}$, $\displaystyle {y}={x+60}$, $\displaystyle {y}={-2x+70}$, $\displaystyle {y}={-2x+90}$.

Which one? Explain.

The standard deviations are $\displaystyle {S_y}={10}$, $\displaystyle {S_x}={4}$. Work out the correlation coefficient.

For which number of absences does the regression line predict the mark 80?