1. What is the Central Limit Theorem?

The Central Limit Theorem (CLT) states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough. This holds regardless of the shape of the population distribution.

2. How Does the Calculator Work?

The calculator uses the CLT formula:

Where:

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

Explanation: The calculator approximates the sampling distribution as normal and computes probabilities based on this approximation.

3. Importance of CLT

Details: CLT is fundamental in statistics as it allows us to make inferences about population parameters using sample statistics, even when the population distribution is unknown.

4. Using the Calculator

Tips: Enter the population mean and standard deviation, sample size, and the sample mean you want to evaluate. The calculator will show the standard error, z-score, and cumulative probability.

5. Frequently Asked Questions (FAQ)

Q1: How large should n be for CLT to apply?
A: Typically n ≥ 30 is considered sufficient, but for very non-normal distributions, larger samples may be needed.

Q2: Can CLT be used for proportions?
A: Yes, for proportions with np ≥ 10 and n(1-p) ≥ 10, the sampling distribution of the proportion is approximately normal.

Q3: What if my population standard deviation is unknown?
A: For large samples, you can use the sample standard deviation as an estimate. For small samples, consider using the t-distribution.

Q4: Does CLT apply to medians?
A: No, CLT specifically applies to means. Different theorems describe the sampling distribution of medians.

Q5: How accurate is the normal approximation?
A: The approximation improves with larger sample sizes. For small samples from non-normal populations, consider non-parametric methods.