Testing of Means

Testing a Single Mean:

Here we are testing mu, we are not testing x bar. When we knew the value of mu, then there is nothing to test.

All the testing of hypothesis is done under the supposition that the null hypothesis is true!

It is not enough to emphasize this. The value for all population parameters in test statistics come from the null hypothesis. It is true not only for means; however for all the testing we are going to do.

Population Standard Deviation Known:

When the population standard deviation sigma, is known, then the population mean consists of a normal distribution, and we will be using z-score formula for the sample means. The test statistic is a standard formula we have seen before.


The normal table or the bottom line from the t-table is used to obtain the critical value.

Population Standard Deviation Unknown:

When the population standard deviation sigma, is unknown, then the population mean consists of a student's t-distribution and we will be employing the t-score formula for sample means. The test statistic is much similar to that for z-score, apart from that sigma has been substituted by s and z has been substituted by t.


The critical value is obtained from t-table. The degree of freedom for this test is (n-1).

If you are executing a t-test where you found the statistics on calculator (as opposed to being provided them in problem), then employ the VARS key to pull up the statistics in the computation of test statistic. It will we helpful in saving data entry and avoid round off errors.

General Pattern:

Note the general pattern of such test statistics is (Observed - Expected)/Standard deviation.

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