Using the z table (Table E), find the critical value (or values) for each. a. α=0.10 , t

Using the z table (Table E), find the critical value (or...

Key Concepts

Z Table

A z table is a standard normal distribution table that displays the cumulative probability values associated with z-scores. It is commonly used in statistics to determine the probability that a standard normal variable falls below a given z value, which in turn helps in finding critical values for hypothesis testing.

Critical Value

A critical value is the threshold value that determines the boundary of the rejection region in a hypothesis test. It is derived from the significance level and the nature of the test (one-tailed or two-tailed). If the test statistic exceeds the critical value in absolute terms, the null hypothesis is rejected.

Significance Level

The significance level, often denoted as alpha (?), represents the probability of making a Type I error, which is the error of rejecting a true null hypothesis. It is used to set the threshold for statistical significance in hypothesis testing and directly influences the critical value.

Tail Type Testing

Tail type testing refers to the directionality of the hypothesis test: one-tailed tests consider deviation in only one direction (either right-tailed or left-tailed), while two-tailed tests evaluate deviations in both directions. The choice of tail affects the placement of the rejection region and the corresponding critical value(s) obtained from the z table.

Transcript

00:01 We are asked to find critical values or z values for each of the five situations that they've given.

00:08 The first situation is an alpha level of 0 .1 in a two -tailed test.

00:16 So first of all, i'm going to draw this out so you can see what they're trying to tell us.

00:23 A two -tailed test means that our critical region is going to be split between the left and the right tails.

00:30 So these two regions combined should add up to 0 .1, or you could think of that as 10%.

00:39 So that means we're going to need 5 % on each side.

00:45 And our goal with looking up values in the table is to find where this cutoff point right here is.

00:54 So what z value gives us this cutoff point? well, this cutoff point would be the 95 % mark.

01:05 So this area adds up to 95%.

01:09 Because whenever you look up values in the table, it tells you what percent is...

01:14 I'll draw a new graph so you can see it better.

01:17 So if this is the 5 % on the right, the value in the table tells you what percent of the normal curve is represented by what i just shaded in.

01:28 So we are looking for 95 % in the z table.

01:34 So we're going to look for a number somewhere in this chart that is as close to 0 .950 as possible because 95 would be 95%.

01:53 As it turns out, that is going to be at right in between 1 .6 .6.

02:00 And 1 .65.

02:03 So the way you read this chart is you're going to find the whole number and the first decimal over here on the left.

02:14 So 1 .6 will be somewhere in here.

02:18 And then you're going to go across to the second decimal.

02:23 So when we ended up with 1 .64, that means the number would have been right here.

02:31 And if we had 0 .05 right here, then right here would have been the other number.

02:46 So when we look this up, we find 0 .945, right here at the entry with 1 .64.

03:02 And then right next to it, at the entry with 1 .65, we find 0 .9505.

03:13 Now, this number that we are looking for right here is exactly halfway between these two entries.

03:21 This is pretty rare that this happens, but it does happen.

03:25 And so technically our answer would be exactly halfway in between 1 .64 and 1 .65.

03:35 So halfway in between would be 1 .645.

03:41 Most people will go ahead and round this, and i would also suggest to just round this up to 1 .65.

03:48 It can't hurt to give a more generous estimate of your z value, so i would put the answer as 1 .65, but 1 .645 would also work.

04:03 So that's this first one.

04:07 Now i'll go ahead and erase all of this for the next part.

04:15 So the next part asks us to find an alpha level of 0 .01 in a right -tailed test.

04:27 All right, so drawing that out would look something more like this, where the right tail is the only one that we are considering.

04:41 And again, our goal is to find where this cutoff point is for our z value is.

04:47 For what our z value is.

04:49 And the chart is going to tell us the area over here in red.

04:56 So we need to find what the area leading up to that critical value would be.

05:01 So we know that with an alpha level of 0 .01, that this area would be 0 .01, or in other words, 1%, which means that this would be 99%.

05:17 So this time we are going to be looking for something in the chart.

05:21 That gives us 0 .99.

05:29 Go ahead and erase this real quick.

05:38 So this one in the chart is going to appear at 2 .33.

05:46 So we'll be looking for 2 .3 on this axis and a final digit of three, which would be this one here, for the other axis.

06:01 So where these two meet will turn out to be 0 .9901, and that is the closest number to 0 .9901, and that is the closest number to 0 .999 that we can get.

06:19 So that's the number we'll use.

06:21 So part b, the z value is 2 .33.

06:32 All right, moving on to the next one.

06:35 They ask us for part c to do an alpha level of 0 .005, and that will be for a left -tailed test.

06:49 So drawing that one out will look more like this...

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