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Q2c) t-Test of the Coefficients

SOLUTION

Given:

\beta_{\text{INCOME}} = 0.0000166514

SE_{\text{INCOME}} = 7.30326 \times 10^{-6}

Degrees of Freedom (Residual): df = 105

Hypotheses

H_{0}: \beta_{\text{INCOME}} = 0

(Income does not significantly affect voter intent.)

H_{1}: \beta_{\text{INCOME}} \neq 0

(Income significantly affects voter intent.)

Rejection Region

Critical Value

For (\alpha=0.01) and (df=105), the critical values of t from the table are t =\pm2.626.

Decision Rule

Reject H_{0} if the calculated t-value is greater than 2.626 or less than -2.626.

Test Statistic

The t-statistic is calculated using the coefficient and standard error:

    \[ t = \frac{\beta_{\text{INCOME}}}{SE_{\text{INCOME}}} \]


    \[ t = \frac{0.0000166514}{7.30326 \times 10^{-6}} \]


    \[ t = 2.279999319 \]

Conclusion

Since the calculated t-value (t = 2.2799) is less than 2.626, we fail to reject the null hypothesis.

This suggests that income does not significantly affect voter intent at the 1% significance level.

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