Start by explaining the Central Limit Theorem (CLT) and its relevance to sampling distributions of the mean. Then, walk through the definitions of expected value, population vs. sample standard deviation, and finally construct a 95% confidence interval using the sample mean and standard error. Use a concrete example to illustrate each concept.
Pro tip: Emphasize that in practice, the population standard deviation is often unknown, so we use the sample standard deviation and the t-distribution for confidence intervals, especially with small samples. Also, mention that the CLT holds regardless of the underlying distribution as long as the sample size is large enough (typically n ≥ 30).
Describe how the CLT states that the sampling distribution of the sample mean approaches a normal distribution as sample size increases, regardless of the population distribution. Apply this to the average comments per user from a sample.
Define the expected value (mean) of comments across all users as the population mean (μ). If given a sample, calculate the sample mean (x̄) as an estimate of μ.
Explain that population standard deviation (σ) measures variability in the entire population, while sample standard deviation (s) estimates σ from a sample, with a denominator of n-1 for unbiasedness.
Use the formula: x̄ ± (critical value) * (s / √n). For large samples, use z* = 1.96; for small samples, use t* from the t-distribution with n-1 degrees of freedom. Interpret the interval in context.
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