Chapter 5: Hypothesis Testing of Mean (Part 1)

What is hypothesis testing?

A statistical method used to determine whether there is enough evidence from a sample to support a claim about a population.

What is the null hypothesis (H₀)?

The statement assumed to be true unless there is sufficient evidence against it.

What is the alternative hypothesis (H₁ or Ha)?

The statement that the researcher wants to find evidence to support.

What does H₀ usually contain?

Equality (=, ≤ or ≥).

What does H₁ usually contain?

<, > or ≠.

What is the population mean represented by?

μ

What is the significance level (α)?

The probability of making a Type I error.

Common significance levels?

0.10, 0.05 and 0.01.

What is the test statistic?

A standardized value calculated from sample data used to decide whether to reject H₀.

What is the p-value?

The probability of obtaining the observed result, or something more extreme, assuming H₀ is true.

Decision rule using p-value?

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

Decision rule using critical value?

If the test statistic falls in the rejection region, reject H₀.

What does "reject H₀" mean?

There is sufficient evidence to support H₁.

What does "fail to reject H₀" mean?

There is insufficient evidence to support H₁.

Does "fail to reject H₀" mean H₀ is true?

No. It only means there is insufficient evidence against it.

What is a one-tailed test?

A hypothesis test where H₁ specifies a direction (> or <).

What is a two-tailed test?

A hypothesis test where H₁ states the parameter is different (≠).

When is a right-tailed test used?

When H₁: μ > μ₀.

When is a left-tailed test used?

When H₁: μ < μ₀.

When is a two-tailed test used?

When H₁: μ ≠ μ₀.

H₁: μ > 50. Which test?

Right-tailed test.

H₁: μ < 50. Which test?

Left-tailed test.

H₁: μ ≠ 50. Which test?

Two-tailed test.

A company claims its batteries last at least 10 hours. What is H₀?

H₀: μ ≥ 10.

What is H₁?

H₁: μ < 10.

A manufacturer claims the average weight is exactly 500g. What are the hypotheses?

H₀: μ = 500, H₁: μ ≠ 500.

A gym claims its members lose more than 5kg on average. What are the hypotheses?

H₀: μ ≤ 5, H₁: μ > 5.

A restaurant claims waiting time is less than 8 minutes. What are the hypotheses?

H₀: μ ≥ 8, H₁: μ < 8.

The p-value is 0.03 and α = 0.05. Decision?

Reject H₀.

The p-value is 0.12 and α = 0.05. Decision?

Fail to reject H₀.

The p-value is 0.008 and α = 0.01. Decision?

Reject H₀.

The p-value is 0.08 and α = 0.01. Decision?

Fail to reject H₀.

The test statistic falls inside the rejection region. Decision?

Reject H₀.

The test statistic falls outside the rejection region. Decision?

Fail to reject H₀.

Smaller p-value means...?

Stronger evidence against H₀.

Larger p-value means...?

Weaker evidence against H₀.

If α increases, rejecting H₀ becomes...?

Easier.

If α decreases, rejecting H₀ becomes...?

Harder.

What is a Type I error?

Rejecting H₀ when H₀ is actually true.

What is a Type II error?

Failing to reject H₀ when H₀ is actually false.

Probability of Type I error?

α.

Probability of Type II error?

β.

Increasing α increases which error?

Type I error.

Increasing α decreases which error?

Type II error.

A researcher concludes a medicine works when it actually does not. Which error?

Type I error.

A researcher concludes a medicine does not work when it actually does. Which error?

Type II error.

A company claims average delivery time is less than 3 days. Which test?

Left-tailed.

A company claims average customer spending exceeds $100. Which test?

Right-tailed.

A company wants to know whether customer satisfaction has changed. Which test?

Two-tailed.

A question mentions "greater than." Immediately think...

Right-tailed test.

A question mentions "less than." Immediately think...

Left-tailed test.

A question mentions "different from." Immediately think...

Two-tailed test.

Point estimate vs Hypothesis test

Point estimate estimates a parameter; hypothesis testing evaluates a claim.

Confidence interval vs Hypothesis test

Confidence intervals estimate a range; hypothesis tests decide whether evidence supports a claim.

One-tailed vs Two-tailed

One-tailed tests look for change in one direction; two-tailed tests look for change in either direction.

Reject H₀ vs Fail to reject H₀

Reject means evidence supports H₁; fail to reject means insufficient evidence.

Significance level vs p-value

α is chosen before the test; p-value is calculated from the sample.

Which hypothesis contains equality?

H₀.

Which hypothesis contains inequality?

H₁.

Which error is controlled by α?

Type I error.

Which value is compared directly with α?

The p-value.

Which hypothesis represents the research claim?

H₁.

True or False: Rejecting H₀ proves H₁ is true.

False.

True or False: Failing to reject H₀ proves H₀ is true.

False.

True or False: Smaller p-values provide stronger evidence against H₀.

True.

True or False: H₀ usually contains an equal sign.

True.

True or False: Type I error means rejecting a true H₀.

True.

True or False: Type II error means failing to reject a false H₀.

True.

The sample suggests strong evidence against H₀. What is the decision?

Reject H₀.

The sample does not provide sufficient evidence against H₀. What is the decision?

Fail to reject H₀.