A z-test, also called a standard normalvariate hypothesis test, is another statistical test used to determine if there's a significant difference between a sample mean and a known population mean, or between a sample proportion and a hypothesized population proportion. Unlike the t-test, the z-test assumes you know the population standard deviation (not just the sample standard deviation).

Here's a breakdown of the key characteristics and applications of z-tests:

When to Use a Z-Test:

• Known Population Standard Deviation: If you have the population standard deviation or a very large sample size (generally greater than 30), a z-test is appropriate.
• Normally Distributed Data: The data for your test statistic (usually the sample mean or proportion) should be approximately normally distributed.

Types of Z-Tests:

• One-Sample Z-Test: This compares the mean of a single sample to a known population mean. You can use this to test if the sample mean is statistically different from a hypothesized value.
• Two-Sample Z-Test for Proportions: This compares the proportion of a specific characteristic in one sample to a hypothesized population proportion. It's useful for analyzing categorical data (e.g., percentage of people who prefer option A).

Steps Involved in a Z-Test:

1. Formulate Hypotheses:

   • Null Hypothesis (H0): There is no significant difference between the sample mean (or proportion) and the hypothesized population value.
   • Alternative Hypothesis (Ha): There is a significant difference between the sample mean (or proportion) and the hypothesized population value. You can specify a one-tailed or two-tailed alternative hypothesis depending on your prediction.

2. Calculate the Z-Statistic: This statistic represents how many standard deviations the sample mean (or proportion) is away from the hypothesized population value.

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4. Determine the P-Value: This is the probability of observing your data (or more extreme data) assuming the null hypothesis is true. A low p-value (less than your chosen significance level, typically 0.05) suggests you can reject the null hypothesis.

5. Interpret the Results:

   • P-Value: Similar to other tests, a low p-value indicates you can reject the null hypothesis and provides evidence for your alternative hypothesis.
   • Confidence Intervals:  These can be calculated to estimate the range within which the true population mean (or proportion) is likely to fall with a certain level of confidence (usually 95%).

6. Draw a Conclusion: Based on the p-value and confidence intervals, you can decide whether to reject the null hypothesis or not. Rejecting the null hypothesis suggests a statistically significant difference between the sample and the hypothesized population value.

Comparison Between Z-Tests and T-Tests:

• Population Standard Deviation: Z-tests require knowing the population standard deviation, while t-tests estimate it from the sample data (assuming normality).
• Sample Size: Z-tests are generally used with larger samples (where the sample standard deviation is a good approximation of the population standard deviation). For smaller samples, t-tests are preferred.

In Conclusion:

Z-tests are a valuable tool for hypothesis testing when you know the population standard deviation or have a large sample size with a normal distribution. By understanding their applications and interpreting the results correctly, you can draw meaningful conclusions about the relationship between your sample and the population.