Changes in the data and the limits of prediction

In the last lesson the regression line between the two tests was $\displaystyle {y}={1.12x-3.4}$. The teacher added 5 points to every mark in the second test:

• $\displaystyle \bar{y}$ grows by 5: $\displaystyle {75+5}={80}$.
• $\displaystyle S_y$ does not change, and neither does $\displaystyle r$ – all the points move 5 up, and their shape stays.
• The slope $\displaystyle {m}={r\cdot\frac{S_y}{S_x}}$ stays 1.12, and the line moves 5 up: $\displaystyle {y}={1.12x+1.6}$.
• Every prediction grows by 5.

Had the teacher multiplied every mark in the second test by 0.9 (taking off 10%):

• $\displaystyle \bar{y}$ and $\displaystyle S_y$ would be multiplied by 0.9: $\displaystyle {\bar{y}}={67.5}$, $\displaystyle {S_y}={7.2}$.
• $\displaystyle r$ does not change: multiplying by a positive number is like changing units.
• The slope is multiplied by 0.9: $\displaystyle {m}={0.7\cdot\frac{7.2}{5}}={1.008}$, and every prediction is multiplied by 0.9.

Prediction with the regression line is useful only when there is a linear relation. A researcher counted how many seeds sprouted at different temperatures:

• There is a clear relation: up to 30 degrees more plants sprout, and above 30 – fewer.
• The relation is not linear, $\displaystyle r$ is close to 0, and the regression line is almost horizontal: it predicts about the same number for every temperature, and is wrong at almost every point.
• When there is no linear relation, the regression line has no practical meaning. So look at the scatter diagram first.

• Prediction far from the data is dangerous: the relation was checked only in the range where the data was measured. A regression line of height on age for children aged 2 to 10 predicts a height of several metres for a 40-year-old.
• One outlier, far from all the other points, can change $\displaystyle r$ and the line a lot.
• Even when the relation is linear, the prediction is almost never exact: the relation is perfect only when $\displaystyle {r}={\pm1}$.
• A regression line does not prove a causal relation, as you learned about a statistical relation.

Researchers studied the relation between the weight of a mouse’s daily portion of food ($\displaystyle x$, in grams) and the mouse’s weight ($\displaystyle y$, in grams), for 10 mice.
$\displaystyle {\bar{x}}={3.4}$, $\displaystyle {S_x}={1.2}$.
$\displaystyle {\bar{y}}={19.4}$, $\displaystyle {S_y}={6.086}$.

One of these four numbers is the correlation coefficient $\displaystyle r$: $\displaystyle 0$, $\displaystyle -0.123$, $\displaystyle 0.923$, $\displaystyle 1$.
In the scatter diagram the mice that eat more weigh more, but the points do not lie on one straight line. Choose the correlation coefficient, and explain without working it out.

Find the equation of the regression line predicting a mouse’s weight from its portion of food. Round the slope to two decimal places, and start from $\displaystyle \bar{y}+m(x-\bar{x})$.

It then turned out that the scales on which the mice were weighed added 2 grams to every mouse’s weight. What is the equation of the regression line once the mice’s weights are corrected?

By the corrected line, what is the predicted weight of a mouse whose portion of food is 3.5 grams?

At the tablet shop, from the lesson on working out the correlation coefficient, a new salesperson was taken on, and thanks to him every customer’s decision time was cut by 20%.
For each of these measures – does it grow, shrink or stay the same? Explain.

1. The correlation coefficient $\displaystyle r$.
2. The standard deviation of the decision time $\displaystyle y$.
3. The slope of the regression line predicting the decision time from the screen size.

For the pupils of a grade-12 class: $\displaystyle {\bar{x}}={71}$, $\displaystyle {S_x}={23}$ – the mark for a project; $\displaystyle {\bar{y}}={75}$, $\displaystyle {S_y}={14}$ – the final mark.
It was decided to raise every pupil’s final mark by 5 points.
Did the standard deviation $\displaystyle S_y$ change? What happened to the regression line predicting the final mark from the project mark? Explain.

The regression line of time on social networks on age was worked out from data on teenagers aged 13 to 18.
Is it a good idea to use it to predict the time a 45-year-old spends on social networks? Explain.