To calculate the pooled variance and the estimated standard error for the mean difference, we can follow these steps:

Step 1: Calculate the pooled variance

The formula for pooled variance when we have independent samples is:

$S_p^2 = \frac{SS_1 + SS_2}{n_1 + n_2 - 2}$

Where:

• $SS_1$ and $SS_2$ are the sum of squares for sample 1 and sample 2, respectively.
• $n_1$ and $n_2$ are the sample sizes for sample 1 and sample 2, respectively.

Given:

• $SS_1 = 72$, $n_1 = 4$
• $SS_2 = 168$, $n_2 = 8$

Plugging in the values:

$S_p^2 = \frac{72 + 168}{4 + 8 - 2} = \frac{240}{10} = 24$

Thus, the pooled variance $S_p^2 = 24$.

Step 2: Calculate the estimated standard error for the mean difference

The formula for the estimated standard error for the mean difference between two independent samples is:

$SE_{\text{diff}} = \sqrt{\frac{S_p^2}{n_1} + \frac{S_p^2}{n_2}}$

Substitute the known values:

$SE_{\text{diff}} = \sqrt{\frac{24}{4} + \frac{24}{8}} = \sqrt{6 + 3} = \sqrt{9} = 3$

Thus, the estimated standard error for the mean difference is $SE_{\text{diff}} = 3$.

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Final Results:

• Pooled variance $S_p^2 = 24$
• Estimated standard error for the mean difference $SE_{\text{diff}} = 3$

Would you like further details on any of the steps, or do you have any other questions about this?

Related Questions:

1. How would you calculate the t-statistic for a two-sample independent t-test using the pooled variance?
2. What are the assumptions required for using pooled variance in hypothesis testing?
3. How do you interpret the results of a t-test when the pooled variance is used?
4. How do degrees of freedom affect the interpretation of t-tests and confidence intervals?
5. How can you calculate the confidence interval for the mean difference between two independent samples?

Tip:

When calculating pooled variance, ensure that both samples are reasonably similar in variance and size. If the variances differ widely, using a pooled variance might not be appropriate, and you might need to use a Welch's t-test instead.