Statistics and Probability - Regression

A study finds the relationship between the number of hours spent studying ($x$) and the score on a math test ($y$). The mean study time is $\bar{x} = 6$ hours and the mean score is $\bar{y} = 72$. The gradient of the regression line of $y$ on $x$ is $a = 4.5$. Find the equation of the regression line and predict the score for a student who studies for $8$ hours.

1. Use the mean point $(\bar{x}, \bar{y})$ in the equation $y = ax + b$: $72 = 4.5(6) + b$ $72 = 27 + b$ $b = 72 - 27 = 45$
2. The regression equation is $y = 4.5x + 45$.
3. For $x = 8$: $y = 4.5(8) + 45$ $y = 36 + 45 = 81$

Since the regression line must pass through the mean point $(\bar{x}, \bar{y})$, we substitute these values along with the gradient $a$ to find the y-intercept $b$. Then, we substitute $x = 8$ into the resulting equation to find the predicted score.

Given a set of data where the correlation coefficient is $r = -0.85$, describe the relationship between the variables $x$ and $y$. If the regression line is $y = -2.5x + 100$, what is the predicted change in $y$ if $x$ increases by $4$ units?

1. The relationship is a strong negative linear correlation because $r$ is close to $-1$.
2. The gradient $a = -2.5$ represents the change in $y$ for an increase of $1$ unit in $x$.
3. Change in $y = a \times \Delta x$: $\Delta y = -2.5 \times 4 = -10$

The value $r = -0.85$ indicates that as $x$ increases, $y$ tends to decrease significantly. The gradient of the regression line tells us the rate of change; multiplying the gradient by the change in $x$ gives the total predicted change in $y$.