And our hypotheses are making some statement about a parameter in that population. And to test it, we take a sample of a certain size. We calculate a statistic, in this case, it would be the sample mean, and we say if we assume that our null hypothesis is true, what is the probability of getting that sample statistic?
And if that's below a threshold, which we call a significance level, we reject the null hypothesis. And so that world that we have been living in, one way to think about it. In a world where you assume the null hypothesis is true, you might have a sampling distribution that looks something like this.
If the null hypothesis is true, then the center of your sampling distribution would be right over here at mu one, and given your sample size, you would get a certain sampling distribution for the sample means. If your sample size increases, this will be narrower. If it decreases, this thing is going to be wider. And you set a significance level, which is essentially your probability of rejecting the null hypothesis even if it is true. You could even view it as, and we've talked about it, you can view your significance level as the probability of making a Type I error.
So your significance level is some area. And so let's say it's this area that I'm shading in orange right over here. That would be your significance level. So if you took a sample right over here and you calculated its sample mean and you happened to fall in this area, this area, or this area right over here, then you would reject your null hypothesis.
Now if the null hypothesis actually was true, you would be committing a Type I error without knowing about it. But for power, we are concerned with a Type II error. So in this one, it's a conditional probability that our null hypothesis is false. And so let's construct another sampling distribution in the case where our null hypothesis is false.
So let me just continue this line right over here, and I'll do that. So let's imagine a world where our null hypothesis is false and it's actually the case that our mean is mu two, and let's say that mu two is right over here, and in this reality, our sampling distribution might look something like this.
Once again, it will be for a given sample size. The larger the sample size, the narrower this bell curve would be. And so it might look something like this. So in which situation, so in this world, we should be rejecting the null hypothesis. But what are the samples in which case we are not rejecting the null hypothesis even though we should? Well, we're not going to reject the null hypothesis if we get samples in, if we get a sample here, or a sample here, or a sample here, a sample, where if you assume the null hypothesis is true, the probability isn't that unlikely.
And so the probability of making a Type II error when we should reject the null hypothesis but we don't is actually this area right over here. Hence, discussion of power should be included in an introductory course.
Doug Rush provides a refresher on Type I and Type II errors including power and effect size in the Spring issue of the Statistics Teacher Network , but, briefly, a Type I Error is rejecting the null hypothesis in favor of a false alternative hypothesis, and a Type II Error is failing to reject a false null hypothesis in favor of a true alternative hypothesis.
Now on to power. Many learners need to be exposed to a variety of perspectives on the definition of power. Bullard describes multiple ways to interpret power correctly:. Mathematically, power is 1 — beta. The power of a hypothesis test is between 0 and 1; if the power is close to 1, the hypothesis test is very good at detecting a false null hypothesis.
Beta is commonly set at 0. Consequently, power may be as low as 0. Powers lower than 0. Power is increased when a researcher increases sample size, as well as when a researcher increases effect sizes and significance levels. In terms of significance level and power, Weiss says this means we want a small significance level close to 0 and a large power close to 1. Having stated a little bit about the concept of power, the authors have found it is most important for students to understand the importance of power as related to sample size when analyzing a study or research article versus actually calculating power.
We have found students generally understand the concepts of sampling, study design, and basic statistical tests, but sometimes struggle with the importance of power and necessary sample size.
Therefore, the chart in Figure 1 is a tool that can be useful when introducing the concept of power to an audience learning statistics or needing to further its understanding of research methodology. Figure 1 A tool that can be useful when introducing the concept of power to an audience learning statistics or needing to further its understanding of research methodology.
This concept is important for teachers to develop in their own understanding of statistics, as well. This tool can help a student critically analyze whether the research study or article they are reading and interpreting has acceptable power and sample size to minimize error. Rather than concentrate on only the p -value result, which has so often traditionally been the focus, this chart and the examples below help students understand how to look at power, sample size, and effect size in conjunction with p -value when analyzing results of a study.
We encourage the use of this chart in helping your students understand and interpret results as they study various research studies or methodologies. Imagine six fictitious example studies that each examine whether a new app called StatMaster can help students learn statistical concepts better than traditional methods. Each of the six studies were run with high-school students, comparing the morning AP Statistics class 35 students that incorporated the StatMaster app to the afternoon AP Statistics class 35 students that did not use the StatMaster app.
Unfortunately, these calculations are not easy to do by hand, so unless you are a statistics whiz, you will want the help of a software program.
Several software programs are available for free on the Internet and are described below. When these values are entered, a power value between 0 and 1 will be generated. If the power is less than 0. Testing for statistical significance helps you learn how likely it is that these changes occurred randomly and do not represent differences due to the program. To learn whether the difference is statistically significant, you will have to compare the probability number you get from your test the p-value to the critical probability value you determined ahead of time the alpha level.
If the p-value is less than the alpha value, you can conclude that the difference you observed is statistically significant. P-Value : the probability that the results were due to chance and not based on your program. P-values range from 0 to 1. The lower the p-value, the more likely it is that a difference occurred as a result of your program.
Alpha is often set at. The alpha level is also known as the Type I error rate. An alpha of. An alpha level of less than. Statistical Significance Creative Research Systems, Beginner This page provides an introduction to what statistical significance means in easy-to-understand language, including descriptions and examples of p-values and alpha values, and several common errors in statistical significance testing.
Part 2 provides a more advanced discussion of the meaning of statistical significance numbers. Statistical Significance Statpac, Beginner This page introduces statistical significance and explains the difference between one-tailed and two-tailed significance tests. The site also describes the procedure used to test for significance including the p value. When a difference is statistically significant, it does not necessarily mean that it is big, important, or helpful in decision-making.
It simply means you can be confident that there is a difference.
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