Explanation

To test whether the slope coefficient is different from 0, we use a t-test with the following hypothesis:

• Null hypothesis (H₀): β = 0
• Alternative hypothesis (H₁): β ≠ 0

The test statistic is calculated as:

$T = \frac{\hat{\beta} - \beta_{H_0}}{\text{SEE}_{\hat{\beta}}}$

Where:

• $\hat{\beta}$ = 1.65 (the estimated slope coefficient)
• $\beta_{H_0}$ = 0 (the hypothesized value under the null hypothesis)
• $\text{SEE}_{\hat{\beta}}$ = standard error of $\hat{\beta}$

Step 1: Calculate the standard error of $\hat{\beta}$

The standard error is given by:

$\text{SEE}_{\hat{\beta}} = \sqrt{\frac{s^2}{n \hat{\sigma}^2_X}}$

Where:

• $s^2$ = 20.45 (estimated variance of the error term)
• $\hat{\sigma}^2_X$ = 18.65 (sample variance of the independent variable)
• n = number of observations = 10 years × 12 months = 120

Plugging in the values:

$\text{SEE}_{\hat{\beta}} = \sqrt{\frac{20.45}{120 \times 18.65}} = \sqrt{\frac{20.45}{2238}} = \sqrt{0.009139} = 0.0956$

Step 2: Calculate the t-statistic

$T = \frac{1.65 - 0}{0.0956} = 17.2594$

Step 3: Interpretation

The test statistic of 17.2594 is quite large, which provides strong evidence against the null hypothesis that the slope coefficient is zero. This suggests that the slope coefficient is statistically significantly different from zero at conventional significance levels.

Note: The value $\bar{\mu}_X = 0.61$ (the sample mean of the independent variable) is not needed for this calculation, as it doesn't appear in the formula for the standard error of the slope coefficient in simple linear regression.