does this help?
THE ANSWER: Margin of error = critical value * standard error = 2.626 * 0.7769 ≈ 2.038 Finally, the confidence interval can be calculated by subtracting and adding the margin of error to the best estimate. Confidence interval = (best estimate - margin of error, best estimate + margin of error) = (-3.1 - 2.038, -3.1 + 2.038) = (-5.138, -0.062) So, the 99% confidence interval for μ₁-μ₂ is approximately (-5.14, -0.06).
Step-by-step explanation:
The question is asking us to use the t-distribution to find a confidence interval for the difference in means μ₁-μ₂, given the sample results. The best estimate for μ₁-μ₂ is given as -3.1. To calculate the margin of error, we need to find the standard error. The standard error can be calculated using the formula: SE = sqrt((s₁²/n₁) + (s₂²/n₂)) where s₁ is the standard deviation of the first sample, n₁ is the size of the first sample, s₂ is the standard deviation of the second sample, and n₂ is the size of the second sample.
Using the given values, we have s₁ = 2.5, n₁ = 50, s₂ = 5.2, and n₂ = 50. Plugging these values into the formula, we get: SE = sqrt((2.5²/50) + (5.2²/50)) = sqrt(0.0625 + 0.5408) = sqrt(0.6033) ≈ 0.7769 The margin of error can be calculated by multiplying the standard error by the critical value from the t-distribution. The critical value depends on the desired confidence level and the degrees of freedom, which is equal to n₁ + n₂ - 2. Since the question asks for a 99% confidence interval, the critical value can be found using a t-table or calculator. For a 99% confidence level and 98 degrees of freedom (50 + 50 - 2), the critical value is approximately 2.626.