# Single Sample Z-Test

### Comparing a single sample/population/group/mean with a standard value

Single sample z-test is applied when (1) the data is normally distributed, (2) population standard deviation is known, and (3) the sample size is reasonably high (over 30).

Assume that someone wants to check/verify the height of the U.S. male population. Currently, the population mean is about 70 inches with a standard deviation of 3 inches. A total of 30 representative samples are taken and the height is measured (Table 1).

Table 1. U.S. Male Height (inches) Sample Data

# Step 2. Method

Single sample z-test as the description of the problem matches all the criteria for it.

MS Excel can be used to find the z value using the Equation 1. And the p-value can be calculated by typing =NORMSDIST(calculated z-value).

Equation 1

Any software, such as, Minitab can be used to analyze the data.

# Step 3. Result

Results using MS Excel and Minitab are provided in Figure 2 and Figure 3, respectively.

Figure 2. Manual Analysis Results for a Single Sample z-test Using MS Excel

Figure 3. Software Analysis Results for a Single Sample z-test Using Minitab

## Statistical Interpretation of the Results

We do not reject the null hypothesis because the p-value (0.543) is larger than the level of significance (0.05).

[p-value = the observed probability for the null hypothesis to happen, which is calculated from the sample data]

# Step 4. Contextual Conclusion

Statistically, the height of U.S. male is equal to 70 inches. [rewrite the accepted hypothesis for an eighth grader without using any statistical jargon such as p-value, level of significance, statistical method use, etc. In this case, the null hypothesis is accepted]