To answer the questions that follow, download an SPSS® spreadsheet containing the demographic data for a sample of 30 adults by clicking on the following words in bold: Download SPSS® File (ZIP file). (Note: If you have already downloaded the data set adult_demographics1.sav, you do not need to download it again, as long as you have not changed the data. Using Internet Explorer is not recommended when downloading SPSS® files.)
The questions you will answer using SPSS®
Use SPSS® to obtain an estimated regression equation predicting the value of income from age.
The regression equation is Ŷ =    +    X, where Ŷ is the predicted value of    and X is the value of    .
The Pearson correlation between age and income is    .
You believe there is a linear relationship between age and income. You conduct a hypothesis test with the null hypothesis H₀: β₁ = 0 versus the alternative hypothesis H₁: β₁ ≠ 0. Based on these results, with a significance level of α = .05, you    reject the null hypothesis. You    conclude that age and income are linearly related.

To help with your SPSS® analysis, follow these step-by-step instructions for obtaining the estimated regression equation, Pearson correlation, and conducting a hypothesis test on the relationship between **age** and **income**.

---

### **Step-by-Step Instructions in SPSS**  

#### 1. **Load the Data File**  
- Open **SPSS®**.
- Go to **File → Open → Data**, and select your `adult_demographics1.sav` file.  
- Once the data is loaded, ensure you can see the variables **age** and **income** in the Data View and Variable View tabs.

#### 2. **Generate the Regression Equation**  
- Click on **Analyze → Regression → Linear**.
- In the Linear Regression dialog box:
  - Move **income** to the **Dependent** box (since this is the variable we want to predict).
  - Move **age** to the **Independent(s)** box (the predictor variable).
- Click on **OK**.

After running the analysis, SPSS will produce an **output window** with the following tables:
  - **Coefficients table**: This contains the intercept and slope for the regression equation.
  
#### **Interpret the Output:**
The regression equation is in the form:  
\[
\hat{Y} = a + bX
\]
- **a** = Intercept (Constant term in the coefficients table)
- **b** = Coefficient of the predictor (age)  
Thus, your regression equation will be:  
\[
\hat{Y} = a + b \cdot \text{age}
\]

---

#### 3. **Obtain Pearson Correlation**  
- Click on **Analyze → Correlate → Bivariate**.
- Move **age** and **income** into the Variables box.
- Ensure the box for **Pearson** correlation is checked.
- Click **OK**.

The Pearson correlation coefficient will appear in the output window under **Correlation Table**.

---

#### 4. **Hypothesis Test for Linear Relationship**  
The null and alternative hypotheses are:

- **H₀**: β₁ = 0 (No linear relationship between age and income)
- **H₁**: β₁ ≠ 0 (There is a linear relationship between age and income)

In the **Coefficients Table** (from step 2), look at:
- **p-value** associated with the predictor variable (age).
- If **p-value ≤ 0.05**, reject the null hypothesis (H₀), indicating that age and income are linearly related.
- If **p-value > 0.05**, do not reject the null hypothesis.

---

#### 5. **Summary and Conclusion**  
- Fill in the missing parts of your question based on the output:
  - **Regression equation**: Ŷ = Intercept + (Slope * Age)
  - **Pearson correlation**: r = [Your Value from Correlation Table]
  - **Hypothesis Test Conclusion**:
    - Reject or do not reject H₀ based on the p-value from the coefficients table.

---

### **Questions to Explore Further**  
1. What does the Pearson correlation value indicate about the strength of the relationship between age and income?
2. How does the slope coefficient interpret the change in income per additional year of age?
3. If the p-value was very close to 0.05, how should we interpret the results?
4. How could multicollinearity affect the regression output if more variables were added?
5. What assumptions must be checked to validate a linear regression model?

---

### **Tip:**  
Always check for **outliers** or unusual data points in the scatterplot of age vs. income before relying entirely on the regression results!

To answer the questions that follow, download an SPSS® spreadsheet containing the demographic data for a sample of 30 adults by clicking on the following words in bold: Download SPSS® File (ZIP file). (Note: If you have already downloaded the data set adult_demographics1.sav, you do not need to download it again, as long as you have not changed the data. Using Internet Explorer is not recommended when downloading SPSS® files.) The questions you will answer using SPSS® Use SPSS® to obtain an estimated regression equation predicting the value of income from age. The regression equation is Ŷ =    +    X, where Ŷ is the predicted value of    and X is the value of    . The Pearson correlation between age and income is    . You believe there is a linear relationship between age and income. You conduct a hypothesis test with the null hypothesis H₀: β₁ = 0 versus the alternative hypothesis H₁: β₁ ≠ 0. Based on these results, with a significance level of α = .05, you    reject the null hypothesis. You    conclude that age and income are linearly related.

Learn how to perform linear regression and hypothesis testing in SPSS to predict income from age. This guide covers calculating the regression equation, finding the Pearson correlation, and conducting a hypothesis test to determine if age and income are linearly related.

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