The regression equation is given as:
$y = 149 + 55x$

Let's address each part of the question.

(a) What is the change in $y$ when $x$ changes by $+22$?

The change in $y$ can be determined by multiplying the change in $x$ by the coefficient of $x$ in the regression equation.

$\Delta y = 55 \times 22 = 1210$

Thus, when $x$ increases by 22, $y$ increases by 1210.

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(b) What is the change in $y$ when $x$ changes by $-22$?

Similarly, for $x$ decreasing by $-22$:

$\Delta y = 55 \times (-22) = -1210$

So, when $x$ decreases by 22, $y$ decreases by 1210.

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(c) What is the predicted value of $y$ when $x = 14$?

Substitute $x = 14$ into the equation:

$y = 149 + 55 \times 14 = 149 + 770 = 919$

Thus, the predicted value of $y$ when $x = 14$ is 919.

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(d) What is the predicted value of $y$ when $x = 22$?

Substitute $x = 22$ into the equation:

$y = 149 + 55 \times 22 = 149 + 1210 = 1359$

Thus, the predicted value of $y$ when $x = 22$ is 1359.

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

The regression equation shows a statistical relationship between $x$ and $y$. It suggests that changes in $x$ are associated with changes in $y$, but it does not prove causality. Regression models can only show correlations, not cause-and-effect relationships. To establish causality, additional evidence and experimental data are required.

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Do you need further clarification on any part of the solution?

Here are 5 related questions to extend your understanding:

1. What assumptions are needed for a linear regression to be valid?
2. How do we test if the coefficient of $x$ is statistically significant in this model?
3. What is the interpretation of the intercept (149) in the regression equation?
4. How would the regression equation change if $x$ had a quadratic relationship with $y$?
5. Can this model predict values of $y$ for very large or small $x$? Why or why not?

Tip: In regression, it's essential to check if the relationship between variables is linear by plotting the data points and residuals.