Question

SKILL BUILDER 2 In Exercises 3.45 to 3.48 , construct an interval giving a range of plausible values for the given parameter using the given sample statistic and margin of error. For \(\mu_{1}-\mu_{2}\), using \(\bar{x}_{1}-\bar{x}_{2}=5\) with margin of error 8 .

Short answer

The range of plausible values for \(\mu_{1}-\mu_{2}\) is (-3, 13).

Step-by-step solution

Understand the problem

Given two populations with means \(\mu_{1}\) and \(\mu_{2}\), sample mean differences \(\bar{x}_{1}-\bar{x}_{2}\) and the margin of error of 8. The goal is to determine an interval for plausible values of \(\mu_{1} - \mu_{2}\). This will be done using the sample mean difference and the margin of error.

Calculate Lower Bound

First, calculate the lower bound for the possible values for \(\mu_{1}-\mu_{2}\). This is found by subtracting the margin of error from the given difference between the sample means: Lower bound = \(\bar{x}_{1} - \bar{x}_{2} - \text{Margin of Error}\). Substituting the given values, Lower bound = 5 - 8 = -3.

Calculate Upper Bound

Next, calculate the upper bound for the possible values for \(\mu_{1}-\mu_{2}\). This is found by adding the margin of error to the given difference between the sample means: Upper bound = \(\bar{x}_{1} - \bar{x}_{2} + \text{Margin of Error}\). Substituting the given values, Upper bound = 5 + 8 = 13.

Construct the Interval

The final step is to construct the interval from these values. The lower bound is -3 and the upper bound is 13. So, the interval is (-3, 13). This is the range of plausible values for \(\mu_{1}-\mu_{2}\).

Key concepts

Margin of Error

The margin of error is a statistic expressing the amount of random sampling error in the data's result. It provides a range within which we can be confident that the true population parameter lies, based on the sample data. Think of it as a buffer for your sample estimates to account for uncertainty and natural variation in the data.

For example, in the exercise provided, the margin of error is given as 8. This means that the true difference between the population means, \(\mu_{1} - \mu_{2}\), is likely to be within 8 units of the sample mean difference of 5. By incorporating this margin of error, we acknowledge that our sample provides an estimate, but not the precise value of the population parameter. The use of the margin of error is crucial when presenting results in statistics as it conveys the confidence we have in the interval constructed from sample data.

Sample Mean Difference

The sample mean difference, denoted as \(\bar{x}_{1} - \bar{x}_{2}\), represents the difference between the average values in two samples. It can serve as an estimate for the difference between two population means, which in statistical terms are notated as \(\mu_{1} - \mu_{2}\).

In our example, the sample mean difference is calculated to be 5. This number is derived from measuring, observing, or collecting data from our samples and calculating the average (mean) for each one separately, then finding the difference. This difference indicates how much one sample mean is larger or smaller than the other. Understanding the sample mean difference is important in hypothesis testing and in estimating the range of plausible values for a population parameter.

Statistical Inference

Statistical inference encompasses the processes and methods that enable you to make conclusions about a population based on data collected from a sample. The idea is that by observing just a small piece of the whole, we can draw conclusions about the entire population.

The confidence interval calculated in the exercise, which goes from -3 to 13, showcases statistical inference in action. This interval was constructed using the sample mean difference and the margin of error. It infers a range of values for the population mean difference \(\mu_{1} - \mu_{2}\) that we can be reasonably confident includes the true difference. This is a fundamental aspect of inferential statistics, allowing us to base decisions on sample data despite the inherent uncertainty due to sampling variability.