Explanation

Step 1: Formulate the hypotheses

• Null hypothesis (H₀): β₁ = 0 (slope coefficient is zero)
• Alternative hypothesis (H₁): β₁ ≠ 0 (slope coefficient is statistically different from zero)

Step 2: Calculate the test statistic The t-statistic formula for testing a slope coefficient is:

$t = \frac{\hat{\beta}_1 - \beta_0}{SE(\hat{\beta}_1)}$

Where:

• $\hat{\beta}_1 = 0.8823$ (estimated slope coefficient)
• $\beta_0 = 0$ (hypothesized value under H₀)
• $SE(\hat{\beta}_1) = 0.0931$ (standard error)

$t = \frac{0.8823 - 0}{0.0931} = 9.477$

Step 3: Determine the critical value

• Sample size (n) = 10
• Degrees of freedom = n - 2 = 10 - 2 = 8
• Significance level (α) = 5% = 0.05
• Since this is a two-tailed test, α/2 = 0.025
• Critical t-value for t₀.₀₂₅,₈ = ±2.306 (from t-distribution table)

Step 4: Decision rule

• Reject H₀ if |t-statistic| > critical t-value
• 9.477 > 2.306
• Therefore, we reject the null hypothesis

Step 5: Conclusion The test statistic of 9.477 exceeds the critical value of 2.306, so we reject the null hypothesis. This provides sufficient evidence to conclude that the slope coefficient is statistically different from zero at the 5% significance level.