To calculate a 95% confidence interval for the difference in means, $\mu_1 - \mu_2$, we'll follow these steps:

Given Information

• Sample sizes: $n_1 = 7$, $n_2 = 7$
• Sample means: $\bar{x}_1 = 240$, $\bar{x}_2 = 210$
• Sample standard deviations: $s_1 = 5$, $s_2 = 6$
• Significance level: $\alpha = 0.05$, so the confidence level is 95%.

Step 1: Calculate the Pooled Standard Deviation

Since we assume equal variances, we use the pooled standard deviation formula:

$s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}$

Plugging in the values:

$s_p = \sqrt{\frac{(7 - 1) \cdot 5^2 + (7 - 1) \cdot 6^2}{7 + 7 - 2}}$

Calculating this:

$s_p = \sqrt{\frac{6 \cdot 25 + 6 \cdot 36}{12}} = \sqrt{\frac{150 + 216}{12}} = \sqrt{\frac{366}{12}} = \sqrt{30.5} \approx 5.524$

Step 2: Calculate the Standard Error for the Difference in Means

The standard error (SE) for the difference in means is:

$SE = s_p \cdot \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}$

Substitute the values:

$SE = 5.524 \cdot \sqrt{\frac{1}{7} + \frac{1}{7}} = 5.524 \cdot \sqrt{\frac{2}{7}} = 5.524 \cdot 0.5345 \approx 2.954$

Step 3: Determine the t-Score for a 95% Confidence Level

With $n_1 + n_2 - 2 = 12$ degrees of freedom, the two-tailed critical t-value at 95% confidence is approximately $t_{0.025, 12} = 2.179$.

Step 4: Calculate the Confidence Interval

The confidence interval for $\mu_1 - \mu_2$ is given by:

$(\bar{x}_1 - \bar{x}_2) \pm t \cdot SE$

Substitute the values:

$(240 - 210) \pm 2.179 \cdot 2.954$

Calculating this:

$30 \pm 2.179 \cdot 2.954 = 30 \pm 6.433$

So, the 95% confidence interval for $\mu_1 - \mu_2$ is:

$(23.567, 36.433)$

Interpretation

Since the entire confidence interval $(23.567, 36.433)$ is greater than 20, we can be 95% confident that $\mu_1 - \mu_2$ is greater than 20.

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Do you want more details on any part of the process, or have any questions?

Here are related questions to expand on this topic:

1. How would the confidence interval change if the sample sizes were larger?
2. What would the confidence interval be if we did not assume equal variances?
3. How does the choice of confidence level affect the width of the confidence interval?
4. Can a confidence interval include zero, and what would that imply about the means?
5. How can a hypothesis test complement this confidence interval analysis?

Tip: When calculating confidence intervals, remember that larger sample sizes generally yield narrower intervals, improving precision.