*84*

AÂ **one proportion z-testÂ **is used to compare an observed proportion to a theoretical one.

For example, suppose a phone company claims that 90% of its customers are satisfied with their service. To test this claim, an independent researcher gathered a simple random sample of 200 customers and asked them if they are satisfied with their service, to which 85% responded yes.

We can use a one proportion z-test to test whether or not the true percentage of customers who are satisfied with their service is actually 90%.

**Steps to Perform a One Sample Z-Test**

We can use the following steps to perform the one proportion z-test:

**Step 1. State the hypotheses.Â **

The null hypothesis (H0):Â P = 0.90

The alternative hypothesis: (Ha):Â P â‰ 0.90

**Step 2. Find the test statistic and the corresponding p-value.**

Test statistic *z*Â =Â (p-P) / (âˆšP(1-P) / n)

where p is the sample proportion, P is the hypothesized population proportion, and n is the sample size.

z = (.85-.90) / (âˆš.90(1-.90) / 200) = (-.05) / (.0212) =Â **-2.358**

Use theÂ Z Score to P Value CalculatorÂ with a z score of -2.358 and a two-tailed test to find that the p-value =Â **0.018**.

**Step 3. Reject or fail to reject the null hypothesis.**

First, we need to choose a significance level to use for the test. Common choices are 0.01, 0.05, and 0.10. For this example, letâ€™s use 0.05. Since the p-value is less than our significance level of .05, we reject the null hypothesis.

Since we rejected the null hypothesis, we have sufficient evidence to say thatÂ itâ€™s not true that 90% of customers are satisfied with their service.

**How to Perform a One Sample Z-Test in Excel**

The following examples illustrate how to perform a one sample z-test in Excel.

**One Sample Z Test (Two-tailed)**

A phone company claims that 90% of its customers are satisfied with their service. To test this claim, an independent researcher gathered a simple random sample of 200 customers and asked them if they are satisfied with their service, to which 190 responded yes.

**Test the null hypothesis that 90% of customers are satisfied with their service against the alternative hypothesis that not 90% of customers are satisfied with their service. Use a 0.05 level of significance.**

The following screenshot shows how to perform a two-tailed one sample z test in Excel, along with the formulas used:

You need to fill in the values for cellsÂ **B1:B3**. Then, the values for cellsÂ **B5:B7Â **are automatically calculated using the formulas shown in cellsÂ **C5:C7**.

Note that the formulas shown do the following:

- Formula in cellÂ
**C5**: This calculates the sample proportion using the formula**Frequency / Sample size** - Formula in cellÂ
**C6**: This calculates the test statistic using the formula**(p-P) / (âˆšP(1-P) / n)Â**where p is the sample proportion, P is the hypothesized population proportion, and n is the sample size. - Formula in cellÂ
**C6**: This calculates the p-value associated with the test statistic calculated in cell**B6**Â using the Excel function**NORM.S.DIST**, which returns the cumulative probability for the normal distribution with mean = 0 and standard deviation = 1. We multiply this value by two since this is a two-tailed test.

Since the p-valueÂ (**0.018**) is less than our chosen significance level ofÂ Â **0.05**, we reject the null hypothesis and conclude that the true percentage of customers who are satisfied with their service is not equal to 90%.

**One Sample Z Test (One-tailed)**

A phone company claims that *at leastÂ *90% of its customers are satisfied with their service. To test this claim, an independent researcher gathered a simple random sample of 200 customers and asked them if they are satisfied with their service, to which 176 responded yes.

**Test the null hypothesis that****Â at leastÂ **

**90%**

**Â of customers are satisfied with their service against the alternative hypothesis that less than 90% of customers are satisfied with their service. Use a 0.1 level of significance.**The following screenshot shows how to perform a one-tailed one sample z test in Excel, along with the formulas used:

You need to fill in the values for cellsÂ **B1:B3**. Then, the values for cellsÂ **B5:B7Â **are automatically calculated using the formulas shown in cellsÂ **C5:C7**.

Note that the formulas shown do the following:

- Formula in cellÂ
**C5**: This calculates the sample proportion using the formula**Frequency / Sample size** - Formula in cellÂ
**C6**: This calculates the test statistic using the formula**(p-P) / (âˆšP(1-P) / n)Â**where p is the sample proportion, P is the hypothesized population proportion, and n is the sample size. - Formula in cellÂ
**C6**: This calculates the p-value associated with the test statistic calculated in cell**B6**Â using the Excel function**NORM.S.DIST**, which returns the cumulative probability for the normal distribution with mean = 0 and standard deviation = 1.

Since the p-valueÂ (**0.17**) is greater than our chosen significance level ofÂ Â **0.1**, we fail to reject the null hypothesis. We do not have sufficient evidence to say that the true percentage of customers who are satisfied with their service is less than 90%.