1. Interpreting the Estimated Slope Coefficient

The regression equation is given as: $f = 38.03 + 0.07 \cdot t$ The slope coefficient $a_1 = 0.07$ means that for each unit increase in $t$, the expected value of $f$ increases by 0.07 units on average.

2. Null and Alternative Hypothesis

To test for correlation (i.e., to determine whether the slope $a_1$ is significantly different from zero), the hypotheses are:

• Null Hypothesis $H_0$: $a_1 = 0$ (No correlation, meaning $t$ does not affect $f$)
• Alternative Hypothesis $H_1$: $a_1 \neq 0$ (There is a correlation, meaning $t$ affects $f$)

3. Test to Conduct and Why

We need to perform a t-test on the slope coefficient $a_1$. This is because we want to determine if the slope is significantly different from zero, which directly tests for the presence of a linear relationship.

4. Test Statistic for the Slope $a_1$

The test statistic for a slope $a_1$ in linear regression is given by: $t = \frac{a_1}{S_{a_1}}$ Where:

• $a_1 = 0.07$ (estimated slope coefficient)
• $S_{a_1} = 0.05$ (standard error of the slope)

Let's calculate the $t$-statistic:

$t = \frac{0.07}{0.05} = 1.4$

5. Critical Value for $t$-distribution

With 55 samples, the degrees of freedom $df = 55 - 2 = 53$. For a two-tailed test at a 1% significance level ($\alpha = 0.01$), we need to compare the calculated $t$-statistic against the critical value of $t$ for 53 degrees of freedom.

Using a $t$-table, the critical $t$-value for a two-tailed test with $df = 53$ at $\alpha = 0.01$ is approximately $t_{crit} \approx 2.67$.

6. Decision: Reject or Fail to Reject $H_0$?

Since $|t| = 1.4$ is less than $t_{crit} = 2.67$, we fail to reject the null hypothesis at the 1% significance level.

This means that there is insufficient evidence to conclude that there is a statistically significant correlation between $t$ and $f$ at the 1% level.

Conclusion:

• Interpretation of the slope: The slope $0.07$ indicates a small positive relationship between $t$ and $f$, but this relationship is not statistically significant at the 1% level.
• Correlation conclusion: There is no statistically significant correlation between $t$ and $f$ at the 1% significance level.

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Would you like more details on any part of this process?

Here are 5 related questions you might find interesting:

1. How does increasing the sample size affect the hypothesis test results?
2. What is the interpretation of the $r^2$ value of 0.41 in this context?
3. How would the decision change if the significance level were 5% instead of 1%?
4. What does the value $S_{a_0} = 14.2$ tell us about the intercept?
5. What are the assumptions of linear regression that need to be satisfied for this test to be valid?

Tip: Always check the $r^2$ value to understand how much of the variation in the dependent variable is explained by the independent variable in regression models.