1. What Is The Critical Z-Score?

The critical Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. It's used in hypothesis testing to determine whether to reject the null hypothesis.

2. How Does The Calculator Work?

The calculator uses the Z-score formula:

Where:

• \( Z \) — Critical Z-score (dimensionless)
• \( X \) — Sample mean (units vary)
• \( \mu \) — Population mean (units vary)
• \( \sigma \) — Standard deviation (units vary)
• \( n \) — Sample size (dimensionless)

Explanation: The formula calculates how many standard errors the sample mean is from the population mean, helping determine statistical significance.

3. Importance Of Z-Score Calculation

Details: Critical Z-scores are essential for hypothesis testing, confidence interval estimation, and determining statistical significance in research studies and data analysis.

4. Using The Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a statistically significant Z-score?
A: Typically, Z-scores beyond ±1.96 (for 95% confidence) or ±2.58 (for 99% confidence) are considered statistically significant.

Q2: 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).

Q3: Can Z-scores be negative?
A: Yes, negative Z-scores indicate the sample mean is below the population mean.

Q4: What are the limitations of Z-scores?
A: Z-tests assume normally distributed data and known population standard deviation, which may not always be realistic.

Q5: How is the critical Z-score related to p-values?
A: The critical Z-score corresponds to specific p-values. For example, Z = 1.96 corresponds to p = 0.05 in a two-tailed test.