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Test Statistic

Test Statistic

Author: Ryan Backman
Description:

Determine whether to reject a null hypothesis from a given p-value and significance level. 

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Hi, this tutorial covers the test statistic. All right, so let's start by defining that. So a test statistic is a measurement in standardized units of how far a sample statistic is from the assumed parameter if the null hypothesis is true.

All right, so kind of as a formula, your test statistic is equal to your statistic minus your parameter. Now, this parameter is usually what's going to come from your null hypothesis-- that's your assumed value-- divided by the standard deviation of the statistic. All right, so let's take a look now.

Let's say, suppose you are using the statistic x bar equals 6. So remember, a statistic is a measurement based on a sample. So we use x bar here, so a sample mean of 6 to test the following hypothesis, your null hypothesis is that mu is equal to 5, and your alternative hypothesis is that mu is greater than 5.

All right, so one type of test statistic is a z-score. A z-score that is far from 0 in the direction of the-- the direction specified by the alternative hypothesis provides evidence against the null hypothesis. So basically, what we'd want to do is we'd want to calculate a z-score for this statistic. In order to do that, we would need the standard deviation of this statistic. If that standard deviation was low, then we would get a larger value of the z-score. So these scores that are far from 0 will give you evidence against the null hypothesis and evidence for your alternative hypothesis.

So if we were to take a look at kind of what the picture of this situation would look like, so let's say we have-- this is just our x bar axis. We know that our null hypothesis, we're going to assume that that value's true, so that's going to be centered at 5. If we're talking about a z-score, we're talking about a normal distribution, so I'm going to draw normal distribution here. And let's say that based on the standard deviation of my statistic, let's say that z-- that my z value is going to be about there, and that my sample mean was about 6. So let's say that's my x bar value there. So x bar equals 6.

Again, if this test statistic is far away from 0, then we're going to have evidence against the null hypothesis. And then further, a p-value is the probability that the test statistic is that value or more extreme in the direction of the alternative hypothesis. So remember, our alternative hypothesis is that that mu is greater than 5.

So if we're talking about probability, we're talking about the area under this curve. So this shaded region here, the area of that shaded region would represent p-value. And again, that's going to represent the probability of a test statistic of that much or greater.

When we're talking about hypothesis tests like this, we can also think about our critical value. So critical value's a value associated with the level of significance that can be compared to the test statistic to decide the outcome of a hypothesis test. So if we go back to the picture here, somewhere in here-- so let's say that we had a level of significance of about 5%.

So what that would mean is that the upper 5% of our values would be considered unlikely. So we're trying to-- what we would do is figure out what that critical value would be, and then compare that to the z-score for 6. If the z-score for 6 is less than that critical value, we're not going to have evidence against the null.

But if that z-score for 6 is greater than that critical value, then we will have evidence against the null. So to summarize that, if the test statistic is beyond the critical value, the null hypothesis should be rejected. And then, obviously, if it's not beyond that critical value, then the null should not be rejected.

And then for two-tailed tests there are symmetric critical values. So if we're talking about a two-tailed test, we'd want to know, well, is it extreme on the upper end of the distribution or extreme on the lower end of the distribution? So what we're going to have is kind of symmetric critical value, so we're going to have a critical value on both sides.

So let's say my critical value was at positive 2, then I'd have another critical value down at negative 2, OK? So again, for two-tailed tests there are symmetric critical values. All right, this has been the tutorial on the test statistic. Thanks for watching.

Terms to Know
Critical Value

A value that can be compared to the test statistic to decide the outcome of a hypothesis test

P-value

The probability that the test statistic is that value or more extreme in the direction of the alternative hypothesis

Test Statistic

A measurement, in standardized units, of how far a sample statistic is from the assumed parameter if the null hypothesis is true

Formulas to Know
Test Statistic

fraction numerator s t a t i s t i c space minus space p a r a m e t e r over denominator s tan d a r d space d e v i a t i o n space o f space s t a t i s t i c end fraction

z-statistic of Means

z space equals space fraction numerator x with bar on top space minus space mu over denominator begin display style bevelled fraction numerator sigma over denominator square root of n end fraction end style end fraction

z-statistic of Proportions

z space equals space fraction numerator p with hat on top space minus space p over denominator square root of begin display style bevelled fraction numerator p q over denominator n end fraction end style end root end fraction