Question

Confidence interval and hypothesis testing are commonly used in business (and other fields) to make decision(s) from data. For this assignment, find a journal or article that have used confidence interval(s) and/or hypothesis test(s) to make decision(s) or conclusion(s). In your post, include: (a) a brief summary of the journal/article; (b) how confidence interval(s) or hypothesis test(s) were used; and (c) decision(s) and/or conclusion(s). Be sure to cite the journal/article you've used. For peer responses, discuss the similarities and/or differences between your article and your peer's article.

Answer 1

A confidence interval is a range of values that is likely to contain an unknown population parameter. If you draw a random sample many times, a certain percentage of the confidence intervals will contain the population mean. This percentage is the confidence level.

Most frequently, you’ll use confidence intervals to bound the mean or standard deviation, but you can also obtain them for regression coefficients, proportions, rates of occurrence (Poisson), and for the differences between populations.

The confidence level represents the theoretical ability of the analysis to produce accurate intervals if you are able to assess many intervals and you know the value of the population parameter. For a specific confidence interval from one study, the interval either contains the population value or it does not—there’s no room for probabilities other than 0 or 1. And you can't choose between these two possibilities because you don’t know the value of the population parameter.

"The parameter is an unknown constant and no probability statement concerning its value may be made."
—Jerzy Neyman, original developer of confidence intervals.

Confidence intervals serve as good estimates of the population parameter because the procedure tends to produce intervals that contain the parameter. Confidence intervals are comprised of the point estimate (the most likely value) and a margin of error around that point estimate. The margin of error indicates the amount of uncertainty that surrounds the sample estimate of the population parameter.

In this vein, you can use confidence intervals to assess the precision of the sample estimate. For a specific variable, a narrower confidence interval [90 110] suggests a more precise estimate of the population parameter than a wider confidence interval [50 150].

Let’s move on to see how confidence intervals account for that margin of error. To do this, we’ll use the same tools that we’ve been using to understand hypothesis tests. I’ll create a sampling distribution using probability distribution plots, the t-distribution, and the variability in our data. We'll base our confidence interval on the energy cost data set that we've been using.

When we looked at significance levels, the graphs displayed a sampling distribution centered on the null hypothesis value, and the outer 5% of the distribution was shaded. For confidence intervals, we need to shift the sampling distribution so that it is centered on the sample mean and shade the middle 95%.

95% Confidence interval for our sample mean of 330.6 T, df = 24 0.95 0.4 0.3 Density 0.2 0.1 0.0 267 394 330.6 Energy Cost

The shaded area shows the range of sample means that you’d obtain 95% of the time using our sample mean as the point estimate of the population mean. This range [267 394] is our 95% confidence interval.

Using the graph, it’s easier to understand how a specific confidence interval represents the margin of error, or the amount of uncertainty, around the point estimate. The sample mean is the most likely value for the population mean given the information that we have. However, the graph shows it would not be unusual at all for other random samples drawn from the same population to obtain different sample means within the shaded area. These other likely sample means all suggest different values for the population mean. Hence, the interval represents the inherent uncertainty that comes with using sample data.