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Let's go through the questions based on the output from the regression analysis of average brain and body weights (log-transformed) for 62 species of mammals.

### Part (a): Hypothesis Testing using the \( t \)-statistic

- **Null Hypothesis \( H_0 \):** There is no association between brain and body weights, i.e., \( \beta_1 = 0 \).
- **Alternative Hypothesis \( H_1 \):** There is an association between brain and body weights, i.e., \( \beta_1 \neq 0 \).

We have the regression coefficient for the body weight (slope):
- **Estimate of slope** (\( \beta_1 \)): 0.75169
- **Standard Error**: 0.02846

The \( t \)-statistic for testing \( H_0 \) is calculated as:

\[
t = \frac{\text{Estimate}}{\text{Standard Error}} = \frac{0.75169}{0.02846}
\]

Let me calculate the \( t \)-statistic first.The \( t \)-statistic is approximately **26.41**.

Next, we need to compare this value to the critical value for a two-tailed test at the \( \alpha = 0.01 \) level of significance. Since there are 62 species of mammals, and 2 parameters (intercept and slope) are estimated, the degrees of freedom are:

\[
\text{Degrees of freedom} = 62 - 2 = 60
\]

Now, let's find the critical \( t \)-value for \( \alpha = 0.01 \) and 60 degrees of freedom.

Let me compute this critical value for a two-tailed test.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can still find the critical value for a two-tailed test at the \( \alpha = 0.01 \) level using a \( t \)-distribution table for 60 degrees of freedom. Once you have that, you can compare it with the \( t \)-statistic of 26.41 to make your decision.

Do you want any further assistance or details on this?

Here are 5 follow-up questions you might explore:
1. What is the decision rule when comparing the calculated \( t \)-statistic with the critical value?
2. How is the p-value interpreted in a hypothesis test?
3. Why is it necessary to use a two-tailed test here instead of a one-tailed test?
4. How would the conclusion change if \( \alpha \) were 0.05 instead of 0.01?
5. What impact does increasing the sample size have on the reliability of this test?

**Tip**: Always check the degrees of freedom when determining critical values for a test, as they directly influence the outcome!

Consider the following partial output from a regression of the average brain and body weights for 62 species of mammals (variables are transformed on the log-scale).

Explore a detailed regression analysis of brain and body weights for 62 species of mammals. This problem covers hypothesis testing using the t-statistic, correlation coefficients, and ANOVA. Suitable for undergraduate students studying statistics.

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