1. What is the Z-Test?

The z-test is a statistical test used to determine whether two population means are different when the variances are known and the sample size is large (typically n > 30). It compares sample means to population means using the standard normal distribution.

2. How Does the Calculator Work?

The calculator uses the z-test formula:

Where:

• \( \bar{x} \) — Sample mean
• \( \mu \) — Population mean
• \( \sigma \) — Population standard deviation
• \( n \) — Sample size

Explanation: The z-score measures how many standard deviations an element is from the population mean. A higher absolute z-score indicates a more significant difference.

3. Importance of Z-Test

Details: The z-test is crucial for hypothesis testing in statistics, particularly for large sample sizes. It helps determine whether observed differences are statistically significant or due to random chance.

4. Using the Calculator

Tips: Enter the sample mean, population mean, population standard deviation, and sample size. All values must be valid (n > 0, σ ≥ 0).

5. Frequently Asked Questions (FAQ)

Q1: When should I use a z-test vs t-test?
A: Use z-test when population standard deviation is known and sample size is large (n > 30). Use t-test for smaller samples or when population standard deviation is unknown.

Q2: What does the z-score tell me?
A: The z-score indicates how many standard deviations your sample mean is from the population mean. Higher absolute values suggest more significant differences.

Q3: What's a significant z-score?
A: Typically, z-scores beyond ±1.96 (for α=0.05) or ±2.58 (for α=0.01) are considered statistically significant.

Q4: Can I use this for proportions?
A: Yes, with modifications. For proportions, σ = √[p(1-p)] where p is the population proportion.

Q5: What are limitations of the z-test?
A: Requires known population standard deviation and assumes normal distribution. Not suitable for very small samples.