Final answer:
To estimate the difference in population means with a 95% confidence interval, we use the known sample means, population standard deviations, and sample sizes to calculate the interval using the formula for a confidence interval with known population standard deviations.
Step-by-step explanation:
The objective here is to construct a 95% confidence interval to estimate the difference in population means based on two sets of sample data. We can apply the formula for a confidence interval when the population standard deviations are known and the population is normally distributed or the sample sizes are large (Central Limit Theorem).
Given:
- Sample mean 1 (¯x1) = 29
- Population standard deviation 1 (σ1) = 5
- Sample size 1 (n1) = 37
- Sample mean 2 (¯x2) = 24
- Population standard deviation 2 (σ2) = 4
- Sample size 2 (n2) = 32
The confidence interval is estimated by the formula:
(¯x1 - ¯x2) ± z*√((σ12/n1) + (σ22/n2))
Where z is the z-value from the standard normal distribution corresponding to the desired confidence level (for 95% confidence, z is approximately 1.96).
With the provided data, the calculation of the confidence interval would involve substituting the given values into the formula and then rounding to two decimal places as needed. It's important to remember that this confidence interval estimates the difference between two population means, not a single mean.