Question

To construct a confidence interval for the difference between two population means ​, use the formula shown below when both population standard deviations are​ known, and either both populations are normally distributed or both and . ​also, the samples must be randomly selected and independent. the descriptive statistics for the annual salaries from a random sample of microbiologists from two regions are shown below. construct a​ 95% confidence interval for the difference between the mean annual salaries. ​$​, ​, and ​$​; ​$​, ​, and ​$

Options:
A) $X1 - $X2 ± Z*(√((S1^2 / n1) + (S2^2 / n2))) (where Z* is the critical value from the standard normal distribution for a 95% confidence level)
B) $X1 - $X2 ± t*(√((S1^2 / n1) + (S2^2 / n2))) (where t* is the critical value from the t-distribution for a 95% confidence level with degrees of freedom calculated using a pooled standard deviation)
C) $X1 - $X2 ± Z*(√((S1^2 / n1) - (S2^2 / n2))) (where Z* is the critical value from the standard normal distribution for a 95% confidence level)
D) $X1 - $X2 ± t*(√((S1^2 / n1) - (S2^2 / n2))) (where t* is the critical value from the t-distribution for a 95% confidence level with degrees of freedom calculated using individual standard deviations)

Answer 1

To construct a 95% confidence interval for the difference between two population means when standard deviations are unknown, use the formula $X1 - $X2 ± Z*(√((S1^2 / n1) + (S2^2 / n2))) with Z* as the critical value for a 95% level. The necessary values for sample means, standard deviations, and sizes are required to complete the calculation.

Step-by-step explanation:

The student has provided descriptive statistics for two samples and seeks to construct a 95% confidence interval for the difference between two population means. As the standard deviations of the populations are estimated using the sample standard deviations, and given not all information necessary to complete the calculation was provided in the question, the general approach for constructing the interval uses the sample means and sample standard deviations.

The correct formula for constructing a confidence interval when population standard deviations are unknown and estimated from the sample is: $X1 - $X2 ± Z*(√((S1^2 / n1) + (S2^2 / n2))), where Z* is the critical value from the standard normal distribution for a 95% confidence level.

To perform the calculations for the confidence interval, you would need the specific values of the sample means ($X1, $X2), sample standard deviations (S1, S2), and sample sizes (n1, n2) for each region. The Z* value for a 95% confidence level is approximately 1.96. You would then plug these values into the formula to calculate the confidence interval for the difference in means.