A regression equation is given below.
yequals=149149plus+55x
a. What is the change in y when x changes by plus+22​?
b. What is the change in y when x changes by minus−22​?
c. What is the predicted value of y when xequals=1414​?
d. What is the predicted value of y when xequals=2222​?
e. Does this equation prove that a change in x causes a change in​ y?

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.

---

### (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.

---

### (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.

---

### (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.

---

### (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.

---

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.

A regression equation is given below.
y = 149 + 55x
a. What is the change in y when x changes by +22?
b. What is the change in y when x changes by −22?
c. What is the predicted value of y when x = 14?
d. What is the predicted value of y when x = 22?
e. Does this equation prove that a change in x causes a change in y?

Explore how to solve a linear regression equation y = 149 + 55x with a step-by-step approach. Learn how changes in x affect y, predict values of y, and understand why this equation does not prove causality. Suitable for students in Grades 9-12.

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