Module 5 · Inference · Topic 1

Point vs Interval Estimates

Point Estimate. A point estimate is a single value used to estimate a population parameter. For example, x̄ is a point estimate for μ, and p̂ is a point estimate for p.

Interval Estimate. An interval estimate gives a range of plausible values for the parameter, along with a confidence level. It acknowledges sampling variability rather than pretending we know the parameter exactly.

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Key formulas

General Form

Confidence Interval = point estimate ± margin of error

From the Inference formula sheet

Confidence Interval for a Mean: xbar +/- critical value * standard error — Estimate plus or minus a margin of error.
One-Sample Z Test: z = (xbar - mu_0) / (sigma / sqrt(n)) — Tests a sample mean against a null mean when population sigma is known.
Power: Power = 1 - beta — Probability of correctly rejecting a false null hypothesis.

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Worked example

Example

A sample of 50 students has x̄ = 72. The point estimate of μ is 72. A 95% CI might be (69, 75), meaning we are 95% confident the true population mean is between 69 and 75.

Related glossary terms

P-value: The probability of results this extreme or more extreme, assuming the null hypothesis is true. Example: A p-value of 0.03 is evidence against H0 at alpha = 0.05.
Confidence Level: The long-run success rate of a confidence interval procedure. Example: A 95% method captures the true parameter in about 95% of repeated samples.
Type I Error: Rejecting the null hypothesis when it is actually true. Example: The significance level alpha is the Type I error rate.
Type II Error: Failing to reject the null hypothesis when the alternative hypothesis is true. Example: A false negative in a hypothesis test is a Type II error.
Power: The probability of rejecting the null hypothesis when it is false. Example: Power equals 1 - beta, where beta is the Type II error probability.

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