use a 95% confidence interval to estimate the difference between the population means. Interpret the confidence interval. random sample 390 x1 5258 s1 152 x2 5220 s2 192
To use a 95% confidence interval to estimate the difference between population means, we need to calculate the standard error and the margin of error. Here's how you can do it:
1. Calculate the standard error (SE) for the difference between the means using the formula:
SE = √[(s1^2 / n1) + (s2^2 / n2)]
Where s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.
For example, let's calculate the standard error using the given values:
s1 = 152
s2 = 192
n1 = 390
n2 = 525
SE = √[(152^2 / 390) + (192^2 / 525)]
2. Calculate the margin of error (ME) using the formula:
ME = Z * SE
Where Z is the Z-score corresponding to the desired confidence level. For a 95% confidence interval, Z is approximately 1.96.
ME = 1.96 * SE
Now we can interpret the confidence interval:
The confidence interval can be calculated by subtracting and adding the margin of error to the difference between the sample means.
CI = (x1 - x2) ± ME
Where x1 and x2 are the sample means.
Let's calculate the confidence interval using the given data:
x1 = 5258
x2 = 5220
CI = (5258 - 5220) ± ME
Substituting the value of ME (calculated earlier) into the equation:
CI = 38 ± (1.96 * SE)
Finally, we can calculate the confidence interval by substituting the value of SE:
CI = 38 ± (1.96 * SE)
Note that the final confidence interval will be in terms of the difference between the population means.