# Relationship between z score and distribution critical values

### Critical value (z*) for a given confidence level (video) | Khan Academy

Critical values of the chi-squared distribution Values of z of particular importance: z. A(z). Lower limit of right 5% tail. Lower limit of. Critical Value of Z Find a Critical Value in Any tail: Find a critical value for a. A critical value of z (Z-score) is used when the sampling distribution is normal, . probability that a relationship between two variables (like “taking a statistics class ”. We explain How to Find a Critical Z Value with video tutorials and quizzes, using our part of the distribution where we are not going to attribute the difference in.

Let's look at an example using a 2-tailed hypothesis. Actually, let's just change the last example to save us all a little work. We just don't know if there will be an increase or a decrease which would be unfortunate.

Step 4 will change a bit.

### Hypothesis Testing with the Z score

This is where we make our decision. To make our decision we will again draw a distribution. For the 2-tailed hypothesis test, the calculated z score must still be farther away from the mean than the critical value.

The difference is that the alpha level was split across both tails giving us 2 critical values. So, we consequently have 2 rejection regions with the area in between representing the population. Note that for the same 0.

For this reason you must think carefully about whether to use a one-tailed or two-tailed test as you could come to opposite conclusions with the same data depending on which kind of test you carry out. Chose a one-sided test if you suspect the effect is in only one direction — otherwise choose a two-sided test.

In this situation, we still fail to reject the null hypothesis. Our sample mean still does not differ significantly from what we might expect if the null hypothesis of "no effect" is true. Note that failing to reject the null hypothesis does not mean we can say the null is true.

But then we also wanna construct a confidence interval.

## Z-statistics vs. T-statistics

And the way that we do this is we take the statistic. Let me just write this in general form, even if we're not talking about a proportion. It could be if we're trying to estimate the population mean. So we take our statistic, statistic, and then we go plus or minus around that statistic, plus or minus around that statistic, and then we say okay how many standard deviations for the sampling distribution do we wanna go above or beyond?

So the number of standard deviations we wanna go, that is our critical value, and then we multiply that times the standard deviation of the statistic, of the statistic. Now in this particular situation, our statistic is p hat from this one sample that Elena made.

### What is a Z score What is a p-value

So it's that one sample proportion that she was able to calculate, plus or minus z star, and we're gonna think about which z star because that's essentially the question, the critical value. So plus or minus some critical value times and what we do because in order to actually calculate the true standard deviation of the sampling distribution of the sample proportions, well then you actually have to know the population parameter.

But we don't know that, so we multiply that times the standard error of the statistic. And we've done this in previous videos. But the key question here is, what is our z star?

## What is a Z score What is a p-value

And what we really needed to think about is, assuming that the sampling distribution is roughly normal, and this is the mean of it which would actually be our true population parameter, which we do not know. Now all we have to really do is look it up on a z-table, but even there we have to be careful.

And you should always be careful which type of z-table you're using or if you're using a calculator function what your calculator function does.

Because a lot of z-tables will actually do something like this. For a given z, they'll say, what is the total area going all the way from negative infinity up to including z standard deviations above, above the mean? So when you look up a lot of z-tables, they will give you, they will give you this area.