lecture 8 Hypothesis Testing – Single Population

What is a hypothesis in statistics?

A statement about a population parameter that we test using sample data.

What is the null hypothesis (H₀)?

The default claim — no difference, no effect (e.g. μ = 50).

What is the alternative hypothesis (H₁)?

The claim we test for — a difference or effect exists (e.g. μ ≠ 50).

What is a one-tailed test?

A test where the direction matters (e.g. H₁: μ > 50).

What is a two-tailed test?

A test where any difference matters (e.g. H₁: μ ≠ 50).

What is a Type I error?

Rejecting H₀ when it’s actually true (false positive).

What is a Type II error?

Failing to reject H₀ when it’s actually false (false negative).

What is the significance level (α)?

The probability of making a Type I error; commonly 0.05 or 5%.

What is a p-value?

The probability of observing the sample result (or more extreme) if H₀ is true.

What is the decision rule for p-value?

If p < α, reject H₀; if p > α, fail to reject H₀.

What is the critical value method?

Compare test statistic to critical value from Z or t distribution.

When is a Z-test used?

When σ is known and population is normal or sample size is large (n ≥ 30).

What is the Z-test formula?

Z = (x̄ - μ₀) / (σ / √n)

When is a t-test used?

When σ is unknown and sample is small (n < 30).

What is the t-test formula?

t = (x̄ - μ₀) / (s / √n)

What does μ₀ represent?

The value of the population mean under the null hypothesis.

What is the test statistic for variance?

χ² = (n - 1)s² / σ₀²

When is a chi-square test used?

To test hypotheses about population variance.

What does it mean to “fail to reject H₀”?

You don’t have enough evidence to say H₀ is false.

What does “reject H₀” mean?

You have enough evidence to support the alternative hypothesis.