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.

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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:
  $\hat{Y} = a + b \cdot \text{age}$

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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.

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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.

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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.

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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?

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Tip:

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