Inference using normal and t-distributions

Need to know

What you need to know

• **Population Variance $\sigma^2$ Known**: If you know the true variance of the population, you should always use the normal distribution for inference on the mean, provided the underlying population is normal or the sample size is large (Central Limit Theorem).
• **Population Variance $\sigma^2$ Unknown & Sample Size Large ($n > 30$)**: If you don't know $\sigma^2$ but have a large sample, the Central Limit Theorem allows you to use the normal distribution. You use the unbiased sample variance $s^2$ as an estimate for $\sigma^2$.
• **Population Variance $\sigma^2$ Unknown & Sample Size Small ($n \le 30$)**: If you don't know $\sigma^2$ and have a small sample, you must use the t-distribution. This is conditional on the assumption that the underlying population from which the sample is drawn is normally distributed.

Explanation

Guessing with Confidence

1. Identify the parameter of interest (population mean, μ) and the sample statistics (mean x̄, variance s², size n).
2. Choose the distribution: Use the Normal distribution if the population variance σ² is known, or if the sample size n is large. Use the t-distribution if σ² is unknown and n is small.
3. For a hypothesis test, state the null (H₀) and alternative (H₁) hypotheses, then calculate the test statistic (z or t).
4. For a confidence interval, find the appropriate critical value (z* or t*) and calculate the interval: sample mean ± (critical value) × (standard error).