The given regression equation is:

$y = 149 + 5x$

Let's solve each part step-by-step:

Part (a): Change in $y$ when $x$ changes by $+2$

The coefficient of $x$ in the regression equation is 5, which represents the rate of change of $y$ with respect to $x$. Therefore, when $x$ increases by 2:

$\text{Change in } y = 5 \times 2 = 10$

So, $y$ increases by 10 when $x$ changes by $+2$.

Part (b): Change in $y$ when $x$ changes by $-2$

Similarly, when $x$ decreases by 2:

$\text{Change in } y = 5 \times (-2) = -10$

So, $y$ decreases by 10 when $x$ changes by $-2$.

Part (c): Predicted value of $y$ when $x = 14$

To find the predicted value of $y$, substitute $x = 14$ into the equation:

$y = 149 + 5 \cdot 14 = 149 + 70 = 219$

So, the predicted value of $y$ when $x = 14$ is 219.

Part (d): Predicted value of $y$ when $x = 22$

Similarly, substitute $x = 22$ into the equation:

$y = 149 + 5 \cdot 22 = 149 + 110 = 259$

So, the predicted value of $y$ when $x = 22$ is 259.

Part (e): Does this equation prove that a change in $x$ causes a change in $y$?

No, this equation does not prove that a change in $x$ causes a change in $y$; it only describes a linear relationship between $x$ and $y$. The regression equation shows that there is an association between $x$ and $y$, meaning that if $x$ changes, $y$ is expected to change in a predictable way. However, correlation does not imply causation; there could be other underlying factors influencing both $x$ and $y$.

Do you want more details or have any questions?

Here are 5 additional questions to expand on this topic:

1. What is the interpretation of the intercept (149) in the regression equation?
2. How would the equation change if the slope were negative?
3. What would happen to the predicted value of $y$ if $x = 0$?
4. Can you find the correlation coefficient given this regression line?
5. How does the standard error of the regression line affect the reliability of the predictions?

Tip:

Always remember that a regression equation shows a relationship but not necessarily causation. Further statistical analysis, like controlled experiments, are needed to establish causation.