Questions: Section 2 of Chapter 6, Problem 7: We always assume throughout the test, that the is true. It depends on the wording of the problem. Sometimes it is the null hypothesis; sometimes it is the alternative hypothesis. Neither hypothesis The Null Hypothesis The Alternative Hypothesis

Section 2 of Chapter 6, Problem 7:
We always assume throughout the test, that the is true.
It depends on the wording of the problem. Sometimes it is the null hypothesis; sometimes it is the alternative hypothesis.
Neither hypothesis
The Null Hypothesis
The Alternative Hypothesis

Transcript text: Section 2 of Chapter 6, Problem 7: We always assume throughout the test, that the $\qquad$ is true. It depends on the wording of the problem. Sometimes it is the null hypothesis; sometimes it is the alternative hypothesis. Neither hypothesis The Null Hypothesis The Alternative Hypothesis

Solution

Solution Steps

Step 1: Standard Error Calculation

The Standard Error $ (SE) $ is calculated using the formula:

\[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = \sqrt{\frac{0.133}{5} + \frac{0.157}{5}} = 0.2408 \]

Step 2: Test Statistic Calculation

The test statistic $ (t) $ is computed as follows:

\[ t = \frac{\bar{x}_1 - \bar{x}_2}{SE} = \frac{2.76 - 2.62}{0.2408} = 0.5813 \]

Step 3: Degrees of Freedom Calculation

The degrees of freedom $ (df) $ are calculated using the formula:

\[ df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}} = \frac{0.0034}{0.0004} = 7.9456 \]

Step 4: P-value Calculation

The p-value $ (P) $ is determined using the formula:

\[ P = 2(1 - T(|t|)) = 2(1 - T(0.5813)) = 0.5771 \]

Step 5: Summary of Results

The results of the Welch's t-test are summarized as follows: