## 1-Sample z-test for Mean, Sigma Known

This is the summary statistics version. The Stats Excel Calculators package also has a version for raw data.

This is a simple calculator for conducting a hypothesis test for a single-sample z-test, sigma known. [Note: some authors refer to this as the Large Sample test.]

**Important Note:** BUS 233 Requirements for using the z-test

- The population standard deviation, sigma (σ), is known.
- The population is normally distributed or the random sample size is greater than or equal to (>=) 30.

**If sigma is known, use the z-test. If sigma is not known, use the t-test** [here].

Note: for large samples, if you do not know the population standard deviation, sigma, use the sample standard deviation, s, in the calculator below.

Enter your data in the blue cells. Then, select the math operator in the Alternative Hypothesis (which tells you the tail of the test) in the orange cell. To get the operator drop-down menu, left-click in the orange cell, e.g. on the current operator. All the yellow answer cells will be updated accordingly.

**Example:**

A random sample of 83 eight grade students’ scores on a national math assessment test has a mean score of 262. The test results prompts a school administrator to declare that the mean score for the state’s eight grade students is more than the national average of 260. Assume the population standard deviation is 38. At a significance level of 0.06, is there enough evidence to support the administrator’s claim?

**Solution**

- The data is entered in the light blue cells.
- Because the school administrator’s claim is that the state’s eight graders’ average was more than the national average, the claim contains a greater than, >, symbol. Thus the claim must be the Alternative Hypothesis because the Null must always be a form of equality.
- Section 3 shows the Null and Alternative hypotheses.
- This is a right tail test. You can remember this because the greater than symbol, >, points to the right.
- The Standard Error is equal to the Population Standard Deviation, sigma, divided by the square root of the sample size, n. The test statistic, z, is equal to the sample mean minus the population mean divided by the Standard Error. The p-value of 0.3158 is the probability of getting as large or even larger test statistic.
- The critical value of z, 1.555, is found by placing all of the 0.06 significance level in the right tail. The rejection region thus is to the right of the critical value of 1.555.
- Both the rejection region method and the p-value method fail to reject the Null hypothesis. Note that these two methods must always agree.
- Thus the conclusion is that there is not sufficient evidence at the 6% level of significance to support the claim the state’s eight graders’ score is greater than the national average of 260.

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