Explanation

The correct test statistic to use in this scenario is the t-statistic.

Why t-statistic?

When testing hypotheses about individual regression coefficients in a linear regression model, the appropriate test statistic is the t-statistic. This is because:

1. Single coefficient testing: The hypothesis test is about a single regression coefficient (β = 1)
2. Unknown population variance: In regression analysis, the population variance is typically unknown
3. Normal distribution assumption: Under the classical linear regression assumptions, the estimated coefficients follow a t-distribution

Calculation

The t-statistic is calculated as:

t = (β_estimated - β_hypothesized) / SE(β_estimated)

t = (β_estimated - β_hypothesized) / SE(β_estimated)

Where:

• β_estimated = 0.86 (the estimated coefficient from the regression)
• β_hypothesized = 1 (the null hypothesis value)
• SE(β_estimated) = 0.80 (standard error of the coefficient)

t = (0.86 - 1) / 0.80 = -0.175

t = (0.86 - 1) / 0.80 = -0.175

Why not the other options?

• B. Chi-squared test statistic: Used for testing multiple restrictions or variance parameters, not individual coefficients
• C. Jarque-Bera test statistic: Used for testing normality of residuals, not coefficient significance
• D. Sum of squared residuals: A measure of model fit, not a test statistic for hypothesis testing

Decision Rule

With t = -0.175 and |t| < 1.96 (the critical value for a 95% confidence level), we cannot reject the null hypothesis that β = 1.