To determine the p-value for the F-test, we follow these steps:

1. Understand the F-test setup: The F-statistic in this context is used to test the null hypothesis: $H_0: \beta_1 = 0$ against the alternative hypothesis: $H_a: \beta_1 \neq 0$

   Where $\beta_1$ is the slope of the regression line.

2. Determine the degrees of freedom (df):

   • df for the numerator (df$_1$): The number of predictor variables (in this case, 1).
   • df for the denominator (df$_2$): Total observations ($n$) minus the number of parameters estimated (2: the intercept and the slope), so $n - 2$.

   $df_1 = 1 \quad \text{and} \quad df_2 = n - 2 = 16 - 2 = 14$

3. Locate the F-statistic value: The F-statistic is given as $F = 42$.

4. Use an F-distribution table or software: Using the F-distribution with $df_1 = 1$ and $df_2 = 14$, the p-value for $F = 42$ can be computed.

   We can use statistical software or a calculator to find the exact p-value. For large F-values, the p-value will typically be very small.

   Let's calculate this.from scipy.stats import f

Parameters

F_stat = 42 # F statistic df1 = 1 # Numerator degrees of freedom df2 = 14 # Denominator degrees of freedom

Calculate the p-value for the F statistic

p_value = 1 - f.cdf(F_stat, df1, df2) p_value1.4471306468033518e-051.4471306468033518e-05successThe p-value for the F-test is approximately 0.0000145. This indicates a very small probability of observing such a large F-statistic under the null hypothesis $H_0: \beta_1 = 0$. Therefore, the regression slope is statistically significant.

Let me know if you'd like a further explanation or have questions about this result. Here are some related questions for practice:

1. How is the F-statistic computed in a regression analysis?
2. What does the p-value signify in the context of regression?
3. How do you interpret the degrees of freedom in an F-test?
4. Why is the null hypothesis $H_0: \beta_1 = 0$ important in regression analysis?
5. How would this test change if there were multiple predictor variables?

Tip: Always compare the p-value to the chosen significance level (e.g., $\alpha = 0.05$) to determine whether to reject the null hypothesis.