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Chapter 7: The Central Limit Theorem

Introduction

  • The Central Limit Theorem (CLT) is a fundamental concept in statistics that describes how sample means behave.

Sampling Distribution

  • Definition: The sampling distribution refers to the distribution of sample statistics, such as the mean, derived from multiple samples taken from a population.

Central Limit Theorem (CLT)

  • The CLT states that:

    • For large samples (size n), the means of these samples will approximately follow a normal distribution, regardless of the population's distribution.

  • Key elements:

    • Drawn from a population with known mean (μ) and standard deviation (σ).

    • As the sample size n increases, the histogram of sample means trends toward a normal bell shape.

  • Important Note:

    • The population distribution does not need to be known.

    • A sample size of at least 30 is typically seen as "large enough."

Sample Size Considerations

  • The adequacy of sample size (n) for applying the CLT depends on the underlying population distribution:

    • If original population is normal, smaller n may suffice.

    • If unknown or non-normal, n should be at least 30.

Practice Problems

Practice 1

  1. Sample size n=50, μ=45, σ=8: Can CLT be applied? Yes.

  2. Sample size n=10: Can CLT be applied? No.

  3. Sample size n=50 (normal distribution): Can CLT be applied? Yes.

Central Limit Theorem for Sample Means (Averages)

Section 7.1 Learning Objectives

  • Students will use CLT properties to estimate the means and standard deviations of sampling distributions from sample means.

The Central Limit Theorem for Sample Means

  • If X is a random variable, its mean (μX) and standard deviation (σX) apply:

    • As n increases, the distribution of sample means becomes normally distributed.

    • Normal distribution symbol: ~ N(μX, σX/√n).

    • σX/√n is termed the Standard Error of the Mean (SEM).

Sampling Error

  • Definition: Variability observed in sample statistics due to random sampling.

    • "Error" denotes variability, not mistakes.

Examples of Sampling Error

Sampling Error Example 1

  • When studying behavioral issues in children, variability occurs between different samples due to randomness in selected subjects.

    • One sample may contain predominantly well-behaved children, while another may show higher instances of behavior problems.

Sampling Error Example 2

  • Conducting 10,000 samples and recording means produces a distribution of means with variability, showing a range of sample averages due to chance.

Sampling Error Example 3

  • Majority of sample means will cluster around the true population mean (45-55), indicating consistent representation.

Practice Problems

Practice 2

  • Scenario: Researching game strategies for 29-35 year-olds based on average gamer age.

  • Given mean age of strategy players is 28 (SD = 4.8), with a sample of 100 players showing a probability of 0.0186 for ages 29-35.

  • Question: Is the development strategy viable? Needs analysis of probability outcome.

Practice 3

  • Scenario: Cola beverage claims 16 ounces.

    • Sample n=34, sample mean = 16.01, μ = 16.00, σ = 0.143.

  • Questions:

    1. Do results indicate cans are filled over 16 ounces?

    2. Feelings from consumer and manufacturer perspectives?

Practice 4

  • Data: Females aged 18-24 have average systolic BP of 114.8 (SD = 13.1).

    • Sample of 40 females, probability mean BP > 120 is 0.3457.

  • Questions:

    1. Interpret the probability outcome.

    2. If using a sample of 4 females and distribution is unknown, can CLT be applied?

    • Answer: No, insufficient sample size.