Study on Sampling and CLT

Overview

The lecture discussed sampling distribution, central limit theorem (CLT), Z-test, and confidence interval. Clear examples and computations were provided for each concept.

Review of Hypothesis Testing

• In hypothesis testing, the null hypothesis (H₀) is tested and we want to know if there is a significant difference.
• The p-value is compared to the significance level (alpha); if p < alpha, reject the null hypothesis.
• Commonly used alphas are 0.05 or 0.01.

Sampling Distribution and Central Limit Theorem

• If the sample mean (m) of many individuals is used, it is no longer compared to individual scores (x) but to the sampling distribution.
• The sampling distribution is a histogram of sample means from different samples from the population.
• Mean of the sampling distribution = mean of the population.
• Variance of the sampling distribution = variance of the population ÷ sample size (n).
• Standard deviation of the sampling distribution (standard error) = sd of the population ÷ √n.
• Standard error: estimate of the deviation of the sample mean from the population mean.
• CLT: As the sample size grows (n ≥ 30), the shape of the sampling distribution becomes normal regardless of the population shape.
• If the population is normal, the sampling distribution is always normal regardless of n.
• Law of large numbers: the larger the sample size, the closer the sample mean is to the population mean.

Z-Test and Computation

• Z-test: used when the population mean and standard deviation are known.
• Formula for Z-test: Z = (sample mean - population mean) ÷ standard error.
• Example: If population mean = 200, SD = 48, n = 64, sample mean = 220; standard error = 48/8 = 6; Z = (220-200)/6 = 3.33.
• Decision rule: Reject the null hypothesis if the computed Z exceeds the critical value (±1.96 for 2-tailed, 1.64 for 1-tailed, alpha 0.05).
• In the research example, positive personality qualities increase attractiveness rating.

Confidence Interval (CI)

• CI: used to estimate the range where the true population mean can be found based on the sample mean.
• Formula: CI = sample mean ± (critical Z value * standard error).
• 95% confidence level: critical Z = 1.96.
• Example: CI = 8 ± (1.96 × 0.67) → 6.69 to 9.31.*

Key Terms & Definitions

• Null Hypothesis (H₀) — Statement that there is no significant effect or difference.
• Sampling Distribution — Distribution of means from all possible samples of the same size from the population.
• Standard Error — Measure of the spread of sample means from the population mean.
• Central Limit Theorem (CLT) — States that the sampling distribution will approach a normal curve as n grows.
• Z-Test — Test statistic used when the population mean and variance are known.
• Confidence Interval — Range containing the true population mean based on sample data.

Action Items / Next Steps

• Answer the computation exercises on sampling distribution and Z-test.
• Review the formula for standard error and confidence interval.
• Use Jamovi to further explore CLT and sampling distribution.
• Read advanced materials on power analysis for the next lesson.