In this lesson, I'm going to show you how to find a critical T-value based on a corresponding level of significance. And I'm going to show you with a calculator using the T-table, as well as in Excel. And I'm going to go through a few examples.
Remember, a critical value is a value that corresponds to the number of standard deviations away from the mean that we're willing to attribute to chance. How far from the center of our distribution can a T-test statistic fall? We'll decide to either fail to reject the null hypothesis or reject the null hypothesis.
So the first example that we're going to look at is a left-tailed test. And we're asked to find the critical T-value for a hypothesis test with 8 degrees of freedom that would reject the null at a 2 and 1/2% level of significance. Now, I'd like to say I can show you how to do this on your calculator so that we can find that cutoff for that lower 2 and 1/2% of the distribution. What's the T-score that corresponds to that lower 2 and 1/2%? That would be our rejection region.
But unfortunately, with the TI-83 Plus calculator that I have, I can only do this for inverse norm for a normal distribution or a critical Z-value. If you have the TI-84, the TI-89, the Nspire, or the CAS, you should have the function for doing the inverse T. So all I can do for you is show you on pen and paper what you would insert into the calculator.
So using the calculator function to find this critical T-value-- it would be, again, under the second distribution function. And it would be inverse T. And because we're looking at a left-tailed test, we're looking at the bottom 2 and 1/2% of the distribution. You put in 0.025, comma, always your degrees of freedom. Because remember, your distribution changes shape based on the sample size.
And in this case, pretending we're hitting Enter on our calculator, we would get a corresponding critical T-value of negative 2.306. So that falls about here. And that lower shaded region corresponds to the lower 2 and 1/2% of our distribution. So negative 2.306.
So any T-test statistic that we calculate for this corresponding hypothesis test that is less than negative 2.306 means we would reject the null hypothesis. Anything greater than that critical value is in this safe region. We would just attribute it to chance. And we would fail to reject the null.
Using the T-table to find our T critical value-- remember, this is a lower-tail test. And our significance level is 2 and 1/2%. So we're going to locate the closest thing to 2 and 1/2%, which we actually have exactly. And you can see here it's our tail probability. And we had 8 degrees of freedom. So that is going to correspond to a T critical value of 2.306.
Now, one thing with the T-table-- unlike the Z-table where you have a positive table and a negative table, for the T-table it's all positive. But you can use it for both positive and negative. Therefore, you just have to recognize that it's a lower-tail test. We're lower than the mean of the distribution. Therefore, it should be a negative 2.306. So always be careful of that when using your T-table.
To find a critical T-value in Excel, go under your Formulas tab. You're going to insert a function. And under the Statistical column, we're going to look for T.INV, or inverse of T. And what we're going to do is we're going to put in the corresponding significance level. So 0.025, comma, degrees of freedom, which was 8. Hit Enter. And notice how we get the same value we did on the calculator as well as in the table-- a negative 2.306.
For our second example, we're going to look at a right-tailed test and find the T critical value for a hypothesis test with only 4 degrees of freedom that would reject the null at 5% significance level. So because it's a right-tailed test, where's our cutoff for our T-scores? On the upper part of this distribution that corresponds to the top 5% of our distribution.
Again, I'm sorry. I don't have a calculator to show you. But you do your second distribution-- inverse T. And you wouldn't put 0.05. Because we're looking at the top portion of our distribution. And we always read a distribution from left to right. From 0% to 100%.
So for our right-tailed test, we're looking at the top 5%, which corresponds to the 95th percentile. So you'd actually put in 0.95, comma, our degrees of freedom, which was 4. And your calculator would give you a corresponding T-test statistic of 2.132.
So that is our T critical value. So that falls right about here on my very rough sketch of my distribution. So this corresponds to the top 5%. And it makes sense that this is a positive value, because we're above the mean. So any T-test statistic that is above 2.132 for this particular hypothesis test means you would reject the null. Anything below that value, we'd attribute to chance and you'd fail to reject.
Now we're going to use our T-table to find our critical T value with a significant level of 5% for an upper-tail test. So again, looking at my top row which has my tail probabilities, I have 5%. And I've been looking at 4 degrees of freedom. And that will correspond to my T-test statistic, or in this case my T critical, being 2.132. And because it is an upper-tail test, we're above the center of the distribution. So it should be positive. So we're going to leave it a positive 2.132.
Now let's use Excel to find our critical T-value. So go ahead-- again, go under our Formulas tab and look under the Statistical column for T.INV, or inverse T. And in this case, remember again, because it's an upper-tail test, we have to put in 95% or 0.95. Because that corresponds to the upper 5% from a cutoff value-- comma-- our degrees of freedom was 4. And voila. We get the same critical T-value we did in our calculator and our table-- a positive 2.132.
For our last example, we're going to look at a two-sided test and finding the T critical value for a hypothesis test with 13 degrees of freedom that would reject the null at 1% significance level. Because it's a two-sided test, we have to divide this 1% onto both sides of our distribution. So half of 1% means we're going to be finding that critical value, that cutoff for the lower half percent of the distribution and the upper half percent of our distribution.
So in your calculator, for the lower half percent you would do inverse T 0.005 comma 13. And you would get a corresponding critical T-value of negative 3.012. So we're looking at about right here on our distribution. That would correspond to the lower 0.5% of my distribution. And it should be negative since it's below the mean.
And then we're going to do the same thing-- inverse T. But the upper half of a percent corresponds to 0.995, or 99.5%, of our distribution, comma, 13. And we get a positive 3.012. Remember, in our distribution we always read left to right from 0% to 100%. So the top half percent corresponds to the 99.5th percentile. So we're looking at about right here-- so the top 5% of our distribution for positive 3.012.
So for this particular hypothesis test, if we got a T-test statistic that was above a positive 3.012 or below a negative 3.012, we would reject the null. Anything in between, we'd attribute to chance and we would fail to reject the null.
Now we're going to use our table to find our T critical value. So we had 13 degrees of freedom. And our significance level was 1%. But this was a two-tailed test. So we're not going to use the 0.01. We have to divide that in two to do the 0.5%, or 0.005. So then my 13 degrees of freedom. And I get a corresponding T critical value of 3.012. But because it's a two-sided test, it's both the positive and the negative 3.012.
Now we're going to use Excel to find our T critical value for our two-sided test. This one's going to be a little bit different than the previous two, which were one-tailed tests. And I'm going to show you. Again, go under your Statistical column. And in this case, we're going to do the inverse of T for 2T, which means the two-tailed test.
In this case, we do not have to divide our significant level into the two halves. So because we were doing a 1% significant level, I can just put 0.01. And Excel knows, because we indicated it's a two-tailed test, to automatically divide that 1% We were at 13 degrees of freedom. And therefore, it gives us the positive corresponding T critical value. But we also know that it's not only a positive 3.012, but it's a negative 3.012.
I hope this tutorial was helpful in showing you how to calculate the critical T-value. At the end of this lesson, I've attached a PDF where you can try some examples for yourself.
A value (associated with a level of significance) that can be compared to the test statistic to decide the outcome of a hypothesis test.