Two-Tailed z-test Hypothesis Test By Hand

Running a Two-Tailed z-test Hypothesis Test by Hand

HOW TO Video z-test Using Excel


Suppose it is up to you to determine if a certain state (Michigan) receives a significantly different amount of public school funding (per student) than the USA average. You know that the USA mean public school yearly funding is $6800 per student per year, with a standard deviation of $400.

Next, suppose you collect a sample (n = 1000) from Michigan and determine that the sample mean for Michigan (per student per year) is $6873

Use the z-test and the correct Ho and Ha to run a hypothesis test to determine if Michigan receives a significantly different amount of funding for public school education (per student per year).


NOTE: This entire example works the same way if you have a dataset. Using the dataset, you would need to first calculate the sample mean. To run a z-test, it is generally expected that you have a larger sample size (30 or more) and that you have information about the population mean and standard deviation. If you do not have this information, it is sometimes best to use the t-test.


Step 1: Set up your hypothesis

Hypothesis: The mean per student per year funding in Michigan is significantly different than the average per student per year funding over the entire USA.


Step 2: Create Ho and Ha

NOTE: There are many ways to write out Ho.

Ho: mean per student per year funding for Michigan = mean per student per year funding for the USA

This can also be written as the following. Ho: Michigan mean – Population mean = 0

Ha: mean per student per year funding for Michigan ≠ mean per student per year funding for the USA


NOTICE1: The Ha in this example is TWO-TAILED because we are interested in seeing if Michigan is significantly different than the population mean. In a two-tailed test, the Ha contains a NOT EQUAL and the test will see if there is a significant difference (greater or smaller).

NOTICE2: The Ho is the null hypothesis and so always contains the equal sign as it is the case for which there is no significant difference between the two groups.


Step 3: Calculate the z-test statistic

Now, calculate the test statistic. In this example, we are using the z-test and are doing this by hand. However, there are many applications that run such tests. This Site has several examples under the Stats Apps link.

z = (sample mean – population mean) / [population standard deviation/sqrt(n)]

z = (6873 – 6800) / [400/sqrt(100)]

z = 73 / [400/10]

z = 73/ [40]

z = 1.825


So, the z-test result, also called the test statistic is 1.825.


Step 4: Using the z-table, determine the rejection regions for you z-test. To do this, you must first select an alpha value. The alpha value is the percentage chance that you will reject the null (choose to go with your Ha research hypothesis as you conclusion) when in fact the Ho really true (and your research Ha should not be selected). This is also called a Type I error (choosing Ha when Ho is actually correct).

The smaller the alpha, the smaller the percentage of error, BUT the smaller the rejection regions and more difficult to reject Ho.

Most research uses alpha at .05, which creates only a 5% chance of Type I error. However, in cases such as medical research, the alpha is set much smaller.

In our case, we will use alpha = .05


This is TWO-TAILED test, therefore the rejection regions are denoted by + or – 1.96.

HOW TO Find Critical Values and Rejection Regions

NOTE: From the z-table, the critical values for a two-tailed z-test at alpha = .05 is +/- 1.96



Step 5: Create a conclusion

Our z-test result is 1.825

Because 1.825 < 1.96 it is NOT inside the rejection region.

Recall that the rejection regions for a two tailed test with alpha set to .05 is any value above 1.96 OR any value below – 1.96. Because 1.825 is not above 1.96 or below -1.96, it is NOT in the rejection region.

Therefore, this result is NOT significant. We CANNOT reject Ho. We CANNOT conclude that there is a significant difference between the funding for Michigan and the average funding for the USA.