In this hypothesis testing video we discuss how to find rejection regions and critical values using a z test, when the standard deviation is known. We cover right-tailed, left-tailed and two-tailed rejection regions and critical values.
Transcript/notes (partial)
In determining whether or not to reject the null hypothesis, one method you can use is to find rejection regions. A rejection region is an area or range of values under a standard normal distribution, where the null hypothesis is not probable.
As a pure visual of this, here are graphs for a left tailed, a right tailed and a 2 tailed test. And these shaded areas would be rejection regions, and these red lines are critical values. The shaded areas are the levels of significance, noted as alpha in one tailed tests, and one half alpha for 2 tailed tests, and the critical values are often noted as z naught for a one tailed test, and negative z naught and positive z naught for 2 tailed tests.
So, lets say that you have a random sample with a sample size greater than 30 from a population. And from this sample you calculate a sample mean, x bar.
So, if you found a standardized test statistic, the green line, which is often noted as z, whose value fell in a shaded area, you would reject the null hypothesis, and if the standardized test statistic, the green line did not fall in a shaded area, you would fail to reject the null hypothesis.
Lets say that you have a job offer in a city located across the country, and you are concerned about the cost of living in that city. The mean cost of living per month in that city is $2800. You believe this information is incorrect, you claim that the mean cost of living is less than $2800. So, you go out and gather data about the cost of living and find that in a random sample of 35 the mean is $2690 for the cost of living per month.
Assume the standard deviation is $400, at a level of confidence, alpha, of 0.05, test your claim.
Step 1 is to make sure 2 conditions are met. First the sample must be a random sample, and it is as that was stated in the information given, and second, the population must be normally distributed or n, the sample size must be greater than or equal to 30, and our sample size is 35, so that condition is met.
Step 2 is to write the claim out and identify the null and alternative hypotheses. The claim is that the mean, mew is less than 2800. And we know the alternative hypothesis contains a statement of inequality, so h sub a is mew is less than 2800. The null hypothesis is the complement of the alternative hypothesis and contains a statement of equality, so h sub 0 is mew greater than or equal to 2800.
Step 3 is to identify the level of significance, which was given, alpha = 0.05.
Step 4, is to determine the test to use, and because the alternative hypothesis contains the less than inequality, this will be a left tailed test.
Step 5 is to determine the critical value or values and since this is a one tailed test, a left tailed test there will be only 1 critical value. Since the level of significance, alpha equals 0.05, we need to find the value for z naught, in a z distribution table where the area to the left equals 0.05. And in the table that value is -1.645.
Step 6 is to identify the rejection region, and our rejection region is any standardized test statistic value that falls in the shaded area, that is any value that is less than z naught, which is any value less than -1.645. This is often written as z less than z naught, so z less than -1.645.
Step 7 is to use the formula and calculate the z value, or the value of the standardized test statistic. And the formula for this is z equals x bar minus mew, divided by, sigma over the square root of n. And in our example, we have x bar, the sample mean = 3000, mew, the hypothesized population mean equals 2800, sigma, the population standard deviation equals 400, and n, the sample size equals 35. So, we can plug these into the formula and we get z equals -1.627.
Step 8 is to make a decision to reject or fail to reject the null hypothesis. On our graph, you can see that the standardized test statistic does not fall in the rejection region, as z, the standardized test statistic is greater than z naught, the critical value. So, in this case we would fail to reject the null hypothesis.
Step 9 is to interpret the decision. There is not enough evidence at the 5% level of significance to support the claim that the mean cost of living in the city is less than $2800 per month.
Timestamps
0:00 What Are Rejection Regions And Critical Values?
1:05 Example Problem 1 For Hypothesis Testing
2:44 How To Find A Critical Value For Example 1
3:10 How To Find A Rejection Region For Example 1
3:32 How To Calculate The Z Value For Example 1
4:34 Example Problem 2 For Hypothesis Testing
5:35 How To Find Critical Values For Example 2
6:12 How To Find Rejection Regions For Example 2
6:32 How To Calculate The Z Value For Example 2