1. What is the Single Sample Z-Test?

The Single Sample Z-Test is a statistical test used to determine whether the mean of a single sample differs significantly from a known or hypothesized population mean when the population standard deviation is known.

2. How Does the Calculator Work?

The calculator uses the Z-test formula:

Where:

• \( x \) — Sample mean
• \( \mu \) — Hypothesized population mean
• \( \sigma \) — Population standard deviation
• \( n \) — Sample size

Explanation: The Z-score measures how many standard errors the sample mean is away from the hypothesized population mean.

3. Importance of Z-Test Calculation

Details: The Z-test is crucial for hypothesis testing in research and quality control, helping determine if observed differences are statistically significant or due to random chance.

4. Using the Calculator

Tips: Enter the sample mean, hypothesized population mean, population standard deviation, and sample size. All values must be valid (σ > 0, n ≥ 1).

5. Frequently Asked Questions (FAQ)

Q1: When should I use a Z-test instead of a t-test?
A: Use a Z-test when the population standard deviation is known and the sample size is large (typically n > 30). Use a t-test when the population standard deviation is unknown.

Q2: What does the Z-score represent?
A: The Z-score represents how many standard errors the sample mean is above or below the hypothesized population mean.

Q3: How do I interpret the Z-score?
A: Typically, a Z-score beyond ±1.96 (for α=0.05) indicates statistical significance, meaning the sample mean differs significantly from the population mean.

Q4: What are the assumptions of the Z-test?
A: The test assumes normally distributed data, known population standard deviation, and independent observations.

Q5: Can I use this test for small samples?
A: For small samples (n < 30), a t-test is generally more appropriate unless the population is normally distributed and σ is known.