1. What is the Z-Score for Sample Mean?

The Z-score for sample mean measures how many standard errors a sample mean is from the population mean. It's used in hypothesis testing and confidence interval estimation.

2. How Does the Calculator Work?

The calculator uses the Z-score formula:

Where:

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

Explanation: The numerator measures how far the sample mean is from the population mean, while the denominator is the standard error of the mean.

3. Importance of Z-Score Calculation

Details: Z-scores are crucial for statistical hypothesis testing, determining how unusual a sample result is, and calculating confidence intervals for population parameters.

4. Using the Calculator

Tips: Enter all required values. Population standard deviation must be positive and sample size must be at least 1. The result is dimensionless.

5. Frequently Asked Questions (FAQ)

Q1: When should I use this Z-score formula?
A: Use when you know the population standard deviation and have a sample size typically ≥30 (or normally distributed population).

Q2: What does the Z-score value mean?
A: A Z-score of 0 means the sample mean equals the population mean. Positive values are above the mean, negative below. Higher absolute values are more unusual.

Q3: What if I don't know the population standard deviation?
A: Use the t-statistic instead, which uses sample standard deviation, especially for small sample sizes.

Q4: How is this different from regular Z-score?
A: This version accounts for sample size through the standard error term (σ/√n), while regular Z-score uses just σ.

Q5: What's a significant Z-score value?
A: Typically, |Z| > 1.96 indicates statistical significance at α=0.05 level (two-tailed test).