Lessons 5 & 6 BUS 310

Null hypothesis (H₀)

Represents the status quo, historical value, or equality-based claim. Contains =, ≤, or ≥. Assumed true unless the sample gives strong evidence against it.

Alternative hypothesis (H₁ or Hₐ)

What the researcher is trying to find evidence for. Contains ≠, <, or >.

What symbol goes in H₀?

=, ≤, or ≥ (the equality part always belongs in H₀, never in H₁)

What symbol goes in H₁?

≠, <, or > (never =)

How do you determine the type of test (two-tailed, left-tailed, right-tailed)?

Look at H₁, not H₀. "≠" = two-tailed, "

Two-tailed test

Used when H₁ contains ≠; the rejection region is split between both tails of the distribution.

Left-tailed test

Used when H₁ contains <; the rejection region is in the left tail.

Right-tailed test

Used when H₁ contains >; the rejection region is in the right tail.

Courtroom analogy for hypothesis testing

H₀ is "innocent until proven guilty." Rejecting H₀ is like saying the evidence is strong enough to convict. Failing to reject H₀ means the evidence wasn't strong enough — not that H₀ was proven true.

Significance level (α)

The maximum probability of a Type I error you are willing to tolerate. Common values: 0.10, 0.05, 0.01. Smaller α means you demand stronger evidence to reject H₀.

Type I error

Rejecting a true H₀ (a "false alarm"). Probability = α.

Type II error

Failing to reject a false H₀ (missing a real effect). Probability = β.

Memory trick for Type I vs Type II error

Type I = reject something true. Type II = fail to reject something false.

Test statistic (what it measures)

A standardized distance between the sample result and what H₀ predicts. Near 0 = close to H₀; large positive/negative = far from H₀.

z-test formula for a mean (σ known)

z = (x̄ − μ₀) / (σ/√n)

t-test formula for a mean (σ unknown)

t = (x̄ − μ₀) / (s/√n)

z-test formula for a proportion

z = (p̂ − p₀) / √[p₀(1−p₀)/n]

When do you use a z-test vs. a t-test for a mean?

σ known → z-test. σ unknown (only sample s given) → t-test.

Degrees of freedom for a one-sample t-test

df = n − 1

What test is used for a proportion hypothesis test?

Always a z-test (never a t-test)

Rejection region

The tail(s) of the distribution containing rare, extreme results — if the test statistic lands here, you reject H₀.

Non-rejection region

The center of the sampling distribution containing results that are reasonably believable under H₀ — if the test statistic lands here, you fail to reject H₀.

Critical value

The boundary between the rejection region and the non-rejection region.

For a two-tailed test at α = 0.05, how is alpha split between the tails?

Equally — 0.025 in each tail.

Critical value approach — decision rule

Compare the test statistic to the critical boundary. If it falls in the rejection region → reject H₀; otherwise fail to reject H₀.

p-value — plain English definition

Assuming H₀ is true, the probability of getting a sample result at least as extreme as the one observed.

p-value approach — decision rule

If p-value ≤ α → reject H₀. If p-value > α → fail to reject H₀.

What does a small p-value mean?

The sample is unusual under H₀ — evidence against H₀ (reject H₀).

What does a large p-value mean?

The sample is not very unusual under H₀ — not enough evidence to reject H₀.

Acceptable conclusion phrases in hypothesis testing

Only "reject H₀" or "fail to reject H₀." Never say "accept H₀."

Why can't you say "accept H₀"?

Failing to reject H₀ does not prove H₀ is true — it just means there wasn't enough evidence against it.

Conclusion template when you reject H₀

"There is sufficient evidence to conclude that [state H₁ in words]."

Conclusion template when you fail to reject H₀

"There is insufficient evidence to conclude that [state H₁ in words]."

For a confidence interval for a proportion, what value is used in the standard error?

p (the sample proportion)

For a hypothesis test for a proportion, what value is used in the standard error?

p₀ (the null/claimed proportion), not p̂

Common mistake: choosing the tail of the test

Choosing the tail based on H₀ instead of H₁ (always use H₁'s symbol)

Common mistake: z vs. t

Using a z-test when σ is unknown and only s is given (should use t-test)

Common mistake: degrees of freedom

Forgetting df = n − 1 for a one-sample t-test

Common mistake: p-value rule

Reversing the rule — remember p ≤ α → reject H₀

Common mistake: proportion standard error

Using p̂ instead of p₀ in the null-standard-error formula for a hypothesis test

Steps of hypothesis testing (in order)

1) Set up H₀ and H₁. 2) Choose significance level α. 3) Compute the test statistic. 4) Make the decision and write the conclusion.

Bolt diameter example — hypotheses

Claim: bolt diameter is 30 mm → H₀: μ = 30, H₁: μ ≠ 30 (two-tailed)

Cell phone bill example — hypotheses

Claim: average bill increased above $52 → H₀: μ ≤ 52, H₁: μ > 52 (right-tailed)

Response time example — hypotheses

Claim: response time is less than 25 minutes → H₀: μ ≥ 25, H₁: μ < 25 (left-tailed)

Mailing response rate example — hypotheses

Claim: response rate is 8% → H₀: p = 0.08, H₁: p ≠ 0.08 (two-tailed)

Worked example: right-tailed t-test result

n=25, x̄=53.1, s=10, testing μ>52 at α=0.05 → t=(53.1−52)/(10/√25)=0.55 → fail to reject H₀ (insufficient evidence mean bill is greater than $52)

Worked example: two-tailed z-test result

σ=0.8, n=100, x̄=29.84, testing μ=30 at α=0.05 → z=(29.84−30)/0.08=−2.00, critical values ±1.96 → reject H₀ (sufficient evidence mean diameter differs from 30 mm)

Quick recognition checklist — first three questions to ask

1) Am I testing μ or p? 2) What exactly is the claim? 3) Write H₀ with the equality part.