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To solve the problem of finding the 99% confidence interval for the difference in means \(\mu_1 - \mu_2\), we can follow these steps:

### Given Information:
- Sample 1: \(n_1 = 55\), \(\bar{x}_1 = 2.73\), \(s_1 = 0.64\)
- Sample 2: \(n_2 = 40\), \(\bar{x}_2 = 2.59\), \(s_2 = 0.53\)

### Step 1: Determine the Difference in Sample Means
\[
\bar{x}_1 - \bar{x}_2 = 2.73 - 2.59 = 0.14
\]

### Step 2: Calculate the Standard Error of the Difference Between Means
The formula for the standard error (SE) is:
\[
SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = \sqrt{\frac{0.64^2}{55} + \frac{0.53^2}{40}}
\]

### Step 3: Determine the Critical Value
For a 99% confidence interval and a large sample size, the critical value (z*) can be taken from the standard normal distribution:
\[
z^* = 2.576
\]

### Step 4: Calculate the Confidence Interval
The confidence interval is calculated using the formula:
\[
\text{CI} = (\bar{x}_1 - \bar{x}_2) \pm z^* \times SE
\]

Let me compute the values for the standard error and the confidence interval.The 99% confidence interval for the difference in means \(\mu_1 - \mu_2\) is approximately \((-0.17, 0.45)\), rounded to two decimal places.

This means that we are 99% confident that the true difference between the population means falls within this interval.

Would you like more details or have any questions? Here are some additional questions you might consider:

1. What assumptions are made when calculating the confidence interval for the difference between two means?
2. How does changing the confidence level affect the width of the confidence interval?
3. What is the impact of sample sizes on the standard error in this context?
4. How would the confidence interval change if the sample variances were equal?
5. What is the interpretation of a confidence interval that includes zero?
6. How would you calculate the confidence interval if the population standard deviations were known?
7. How does the choice of a one-tailed vs. two-tailed test affect the critical value?
8. What is the relationship between the critical value and the significance level?

**Tip:** Always check if the conditions for applying the formula (such as normality or large sample sizes) are met before interpreting the confidence interval.

Learn how to calculate the 99% confidence interval for the difference in means with a detailed step-by-step solution. This problem involves statistical concepts such as confidence intervals and hypothesis testing. Suitable for college-level students and researchers.

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