Central Limit Theorem

Definition and Importance

The Central Limit Theorem (CLT) is a fundamental concept in statistics that describes the behavior of the sampling distribution of the sample mean. It states that for all samples of the same size n with n > 30, the sampling distribution of the mean (\( \bar{x} \)) can be approximated by a normal distribution, regardless of the shape of the original population distribution.

• Population Distribution: Can be any shape (normal, skewed, uniform, etc.)

• Sample Size Requirement: The sample size must be greater than 30, or the population itself must be normally distributed.

• Sampling Distribution: The distribution of sample means from repeated samples of size n.

Key Point: As sample size increases, the distribution of sample means approaches a normal distribution, even if the original population is not normal.

Mathematical Properties

• Mean of Sample Means: The mean of all values of \( \bar{x} \) is equal to the population mean \( \mu \).

• Standard Deviation of Sample Means (Standard Error): The standard deviation of all values of \( \bar{x} \) is called the standard error of the mean (SEM).

• Z-score for Sample Mean: To find probabilities related to sample means, use:

• Z-score for Individual Value: For a single value from a normal population:

Requirements for Using the CLT

• The population must be normally distributed or the sample size must be greater than 30.

• If the population is not normal and n ≤ 30, the CLT does not apply, and normal approximation is not valid.

Practical Problem Solving with the CLT

Steps and Considerations

• Check Requirements: Confirm that the sample size is large enough or the population is normal.

• Distinguish Between Individual and Sample Mean: Use the appropriate formula for z-score depending on whether you are working with a single value or a sample mean.

• Standard Error: Always use \( \sigma / \sqrt{n} \) for the standard deviation when working with sample means.

Notation for Sampling Distribution

Symbols and Terminology

• \( \mu_{\bar{x}} \): Mean of all sample means (equals population mean \( \mu \)).

• \( \sigma_{\bar{x}} \): Standard deviation of all sample means (standard error).

• SEM: Standard Error of the Mean, another term for \( \sigma_{\bar{x}} \).

Examples and Applications

Example 1: Pulse Rates

• Population: Female pulse rates, \( \mu = 77 \) bpm, \( \sigma = 12.0 \) bpm.

• a) Individual Probability: Probability that a randomly selected female has a pulse rate less than 83 bpm. Use

• b) Sample Mean Probability: Probability that the mean pulse rate of 9 females is less than 83 bpm. Use

Example 2: Elevator Safety

• Scenario: 16 adult male passengers, mean weight limit 160 lb.

• Population: Male weights, \( \mu = 165 \) lb, \( \sigma = 31 \) lb.

• Question: Probability that the mean weight of 16 passengers exceeds 160 lb.

• Application: Use CLT to assess safety based on sample mean.

Example 3: Overhead Reach Distances

• Population: Female overhead reach, \( \mu = 204 \) cm, \( \sigma = 8.5 \) cm.

• a) Individual Probability: Probability that a single distance is greater than 216 cm.

• b) Sample Mean Probability: Probability that the mean of 23 distances is greater than 216 cm.

• c) CLT Applicability: Normal distribution can be used for sample mean because the population is normal, even though n does not exceed 30.

Example 4: Aircraft Seat Design

• Population: Male hip widths, \( \mu = 14.3 \) in., \( \sigma = 0.9 \) in.

• a) Individual Probability: Probability that a randomly selected male has a hip width greater than 16.0 in.

• b) Sample Mean Probability: Probability that the mean hip width of 126 males exceeds 16.0 in.

• c) Design Relevance: The sample mean is more relevant for seat design, as it reflects the average space needed for all passengers. Results suggest caution in reducing seat width.

Summary Table: CLT Formulas and Requirements