A Z-test tells you how likely it is that a sample mean came from a specific population based on your assumed population mean and standard deviation.
Here's a breakdown of what a Z-test reveals:
- Statistical Significance: The Z-test calculates a Z-score, which represents how many standard deviations your sample mean is away from the assumed population mean. A higher absolute value of the Z-score indicates a greater difference between your sample and the assumed population. The test then uses this score to determine the p-value, which is the probability of observing your sample mean if the assumed population mean were true. A low p-value (typically less than 0.05) suggests that your sample mean is statistically different from the assumed population mean.
- Hypothesis Testing: Z-tests are commonly used for hypothesis testing, where you test a specific claim about a population based on a sample. For example, you might want to test the hypothesis that the average height of women in a certain city is 5'4" (the assumed population mean). You could then collect a sample of women's heights and use a Z-test to see if your sample data supports or rejects this hypothesis.
- One-tailed vs. Two-tailed Tests: You can use a Z-test to test for differences in either direction (two-tailed test) or only in one direction (one-tailed test). For instance, you might want to test if the average weight of a product is greater than a specific value (one-tailed test) or if it's different from that value (two-tailed test).
Example:
Let's say you want to test if the average height of adult males in a country is 5'10". You collect a random sample of 100 adult males and find their average height to be 5'9". You conduct a Z-test and obtain a p-value of 0.02. This suggests that there is a 2% chance of observing an average height of 5'9" in your sample if the true average height of adult males in the country is actually 5'10". Since the p-value is less than 0.05, you would reject the null hypothesis and conclude that the average height of adult males in the country is likely not 5'10".
In summary, a Z-test helps you understand the likelihood of your sample mean being representative of the population mean, providing insights into whether there is statistically significant evidence to support or reject your hypotheses.
