Question

In Exercises 3.51 to 3.56 , information about a sample is given. Assuming that the sampling distribution is symmetric and bell-shaped, use the information to give a \(95 \%\) confidence interval, and indicate the parameter being estimated. \(\bar{x}_{1}-\bar{x}_{2}=3.0\) and the margin of error for \(95 \%\) confidence is 1.2

Short answer

The 95% confidence interval for the difference between the population means is \((1.8, 4.2)\). This interval is estimating the true difference of the population means.

Step-by-step solution

Identify given values

In this problem, we are given the difference between two sample means, \(\bar{x}_{1}-\bar{x}_{2}=3.0\) and the margin of error for 95% confidence is 1.2.

Understand the concept of confidence interval

A confidence interval is a range of values, derived from a given set of data or from a probability distribution, that is likely to contain an estimated parameter. The width of the interval gives us some idea about the uncertainty of the estimated parameter. In this problem, we are to find a 95% confidence interval, which means that we can be 95% confident that this interval contains the true difference of the population means.

Calculate the lower and upper limits of the confidence interval

To calculate the confidence interval, we need to use the given difference between the sample means and the margin of error. The lower limit of the confidence interval can be calculated by subtracting the margin of error from the difference of the sample means, i.e., \(\bar{x}_{1}-\bar{x}_{2} - E = 3.0 - 1.2 = 1.8\). Similarly, the upper limit of the confidence interval can be calculated by adding the margin of error to the difference of the sample means, i.e., \(\bar{x}_{1}-\bar{x}_{2} + E = 3.0 + 1.2 = 4.2\). Therefore, the 95% confidence interval for the difference between the population means is \((1.8, 4.2)\).