This lesson will explain how to find a critical z value by using either a z-table or technology.
So, the first example we're going to look at is for left-tailed test. And we need to find the critical z value for a hypothesis test that would reject the null at a 2 and 1/2% significance level. So, we want to find on our normal distribution, where is that cut off on the left tail that corresponds to the lower 2 and 1/2% of our distribution?
The first thing I'm going to show you is how to do this on your calculator. So, going ahead, turn on your calculator. And you're going to hit second, stat-- no, you're not, I lied. You're going to hit second, distribution-- I'm sorry-- and you're going to scroll down to this third function, which is inverse norm. So, the inverse of the normal distribution. You're going to hit enter.
And we are going to input 0.025, because we're looking at the lower 2 and 1/2% of our distribution because this is a left-tailed test. Hit enter, and we get a z-test statistic of negative 1.96. So, that's the z-score. So, at about right here. So, negative 1.96. That is the cutoff for the lower 2 and 1/2% of our data.
So, basically any z-score that is below a 1.96, or, I'm sorry, a negative 1.96 means we are going to reject the null hypothesis. Any z-score that is above a negative 1.96 is going to fall in this unshaded region of our distribution, and that means that we're willing to accept the variation in our sample from the center of our distribution due to chance, and we're going to fail to reject the null.
When using the z-table, what you do is you look for your significance level in the table. So, in this case, remember we were looking at a left-tailed test. That's why I'm using the negative z-table with my negative z-scores, not my positive, because I'm looking at the lower half of my distribution. And remember, our significance level is 0.025, or 2.5%, so I'm going to look for that value or the closest thing to it. And here it is on our z-table. And that corresponds to a z-score of negative 1.96. Therefore, our z critical value is negative 1.96.
Find our critical z-value in Excel that corresponds to a 2 and 1/2% significance level for a left-tailed test. All we have to do is go to our formulas tab. We're going to insert under the statistical column. We're looking for norm dot s dot inverse, which is right here. So, the inverse of the normal distribution, and because it's a left-tailed test, we're looking at the lower half of our distribution. So, we're just going to put in the 0.025 for the lower 2 and 1/2%. Hit Enter, and notice how we get the same z critical value that we did using the calculator and table.
For our second example, we're going to be looking at a right-tailed test. And we're asked to find the critical z-value for a hypothesis tests that would reject the null at a 5% significance level. So, where on this distribution is my cut off on the upper part of my distribution where I am not going to attribute the difference in proportion due to chance? So, again, let's go ahead and take out our calculator.
We're going to go to second, distribution, go down to the function of inverse norm and our significance level is 5%, but we're not going to put in 0.05 like we did with the left-tailed test where the significance level is 2 and 1/2%. So, we got to put in 0.025. In our normal distribution, we always read left to right, and it always goes from 0 percent to 100 percent. And we're looking at a right-tailed test, so the upper portion of our distribution. So, that cutoff is the top 5% of our distribution.
So 100% minus 5% is going to be 95%. So, you're going to put in the inverse norm of 0.95, and that's going to get us a corresponding z critical value of 1.644. So, about right here, a positive 1.6. Actually, let's round that up to 1.645. So, any t-test statistic that is greater than 1.645 falls in the upper 5% of my distribution, and therefore we would reject the null.
Using the z-table. Because I'm looking at a right-tailed test, I'm going to have positive z-scores because I'm looking at the upper half of my distribution. I'm using the positive z-table that corresponds with my positive z-scores. So, our significance level was 5%, but it was the upper 5%. So, remember that corresponds to the 95th percentile on my distribution. So, in my table, I need to look for the closest thing to 95%. And it actually falls in between these two values, and that corresponds to a z-score of 1.6, and it falls between the 0.04 and the 0.05. So, when I take that average, we get a z critical value of 1.645.
We're going to find our critical z value that corresponds to 5% significance level for an upper tail test, or a right-tailed test. Again, go to formulas. You're going to insert under the statistical column our norm dot s dot inverse, but we're not going to put in 0.05 for the 5% just like we did in our calculator, because we're looking at the upper part of our distribution. That's going to correspond to the 95th percentile. So, we're going to put a 0.95. And notice, again, how we get the same critical value we did from our table and our calculator, which is a positive 1.645.
For our last example, we're going to look at a two-sided test, and again, find the critical z-score for a hypothesis test that would reject the null at a 1% significance level. Because it's a two-sided test, we have to divide that 1% into each tail. Therefore, 1% divided by 2 means we're going to be looking for the cutoff at the lower 1/2% of the distribution, and the upper 1/2% of our distribution. So, in our calculators, let's go ahead and first find the corresponding critical z-score for the lower part of our distribution. So hit second, distribution, inverse norm, and it's 1/2%, so you're going to put 0.005. And that gives us a corresponding z-score of negative 2.576.
So, that falls right about here, negative 2.576. So, this shaded region corresponds to the lower 1/2% of my distribution. And, if we do this correctly, we should get the same z-score but a positive value for the upper portion of our distribution for that 1/2% cut off. So, go ahead again and do inverse norm. But we can't put in 0.005, because remember, our distribution reads from 0% to 100%. So, 100% minus 1/2% is 99.5%. So, we're going to put in 0.995, and voila. We get a positive 2.576. So, we get a positive 2.576, and that corresponds to the upper 1/2% of our distribution.
So, any z-score that we would calculate that would be greater than a positive 2.576 or less than negative 2.576 means we would reject the null hypothesis.
Using our table. First, we're going to look for the corresponding critical value for the lower half of our distribution, since it's our two-sided test. So, we're not going to look for the closest thing to 1%, but we're going to look for the closest value to 1/2%. So, let's see. The closest thing is going to be between these two values, and that corresponds to my negative 2.5, in between the 0.07 and the 0.08. So, if you're using the table, you're going to get a z critical value of negative 2.575, which is really close to what the calculator gave us. Remember, sometimes the table can just give us an estimate.
Using the table to find the upper critical value for our two-sided test in a 1% significance level, we are going to try to find a 99.5%, the closest value that will correspond to our z-score. So, the closest thing to 99.5% is in between these two values. And that corresponds to a positive 2.5, falling between the 0.07 and the 0.08. So, if you were using the table, you would get a z critical value of a positive 2.575, taking the average between those two values.
So, in Excel, we're going to find our critical z values that correspond to the 1% significance level for our two-sided test. So, we're going to find two critical z-values. Again, go under your formulas tab. We're going to insert under the statistical column our norm dot s dot inverse. And let's go ahead and first find the lower critical value that corresponds to the lower 1/2%, so go ahead and put in 0.005. And you can see, we get our first z critical value of negative 2.576. Now, if we do this correctly, we should get a positive 2.576. So, again, we're going to insert to get the second critical value for the upper part of my distribution. So, the upper percentage that correspond to the top 1 and 1/2% is going to be your 99.5%, so 0.995. Hit Enter, and voila. We get the positive z critical value.
I hope you found this lesson helpful in finding the z critical value. I've attached a PDF at the bottom of this lesson, so you can try a few examples for yourself.
Key Terms
Critical Value
A value (associated with a level of significance) that can be compared to the test statistic to decide the outcome of a hypothesis test.
A value that can be compared to the test statistic to decide the outcome of a hypothesis test