c. Determine the critical values for a two-tailed test of a po...

Question

c. Determine the critical values for a two-tailed test of a population standard deviation for a sample of size n = 30 at the α = 0.05 level of significance.

Accepted Answer

Hello and welcome everyone. The next problem says, for a two-tailed Z test with a significance level of alpha equals 0.10, what are the critical values and rejection regions? So, as always, it's useful to start with our graph. We have a two-tailed test, so we know we're looking at Two areas, as emphasized by the fact that we're looking for critical values, plural and rejection regions plural. So we're going to be looking for these outer regions. So, I'm drawing two lines. We will have two Z critical values, one to the left and one to the right, and our Rejection regions will be those regions outside. Those critical points. So There's one further modification we have to think about here, because we recall alpha is an area, it's the area underneath the curve in these two regions. So, we know that it's the alpha here is corresponding to both of these pieces. So if I want to use a P or Z table, excuse me, to look up my Z value, I actually need to think about the fact that this area One of these is alpha divided by 2. So, in each case, alpha divided by 2 describes that area. So, I want to look up that. As my P-value. So, I have alpha divided by 2, is going to equal 0.050. So I'm going to use that on my Z table. Now recall, I'm looking to the area to the left of the critical line on the where the Z critical is negative, and to the right of that critical line where the Z critical is positive. And because my tables refer to the area to the left of that Z line, I will have an easier time or a more straightforward process if I use the negative Z table. So that will end up giving me so you can look on your table. A negative critical Z value of -1.645. And then recall we have two critical values. So, the, on the other side, we'll have the corresponding positive values, so 1.645. So those are two critical Z values. And then we also are looking for rejection regions, so. That will be that area where the probability is less than alpha divided by 2 on either side. So that will correspond to an area where our Z value is less than -1.645 on the left side. And where Z is greater than 1.645 on the right side. So, there we have it. We have our two critical values. And our two rejection regions that. Which is that we have those critical values of negative 1.645 and 1.645, and then our rejection region being where Z is less than 6 excuse me, less than 1.645, or And I put a comma between those two, but I need to change that to or. Z is Greater than 1.645. So again, that is when we're talking about a two-tailed Z test with the significance level of alpha equals 0.10. See you in the next video.

c. Determine the critical values for a two-tailed test of a population standard deviation for a sample of size n = 30 at the α = 0.05 level of significance.

Key Concepts

Chi-Square Distribution

Two-Tailed Test

Critical Values

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