Chapter 8: Hypothesis Testing for Population Proportions

Chapter 8: Hypothesis Testing for Population Proportions

Learning Objectives

• Know how to test hypotheses concerning a population proportion.

• Understand the meaning of p-value and how it is used.

• Understand the meaning of significance level and how it is used.

• Know the conditions required for calculating a p-value and significance level.

8.1 The Essential Ingredients of Hypothesis Testing

Definition and Purpose

Hypothesis Testing is a statistical procedure that enables us to choose between two competing claims about a population parameter when there is variability in the outcomes.

General Four Steps of Hypothesis Testing

1. Hypothesize: State a hypothesis (claim) that will be weighed against a neutral “skeptical” claim (the null hypothesis).

2. Prepare: Determine how you will use data to make your decision and ensure you have enough data to minimize the probability of making mistakes.

3. Compute to Compare: Collect data and compare them to your expectations under the null hypothesis.

4. Interpret: State your conclusion. Decide whether the evidence supports the claim or if there is insufficient evidence.

Example: Coin Spinning

• You hypothesize a coin is more likely to come up heads because one side bulges out.

• After spinning the coin 20 times, it lands heads all 20 times. If the coin were fair, this would be extremely unlikely.

• This leads you to conclude that the coin is more likely to come up heads when spun.

Example: Surface Cracks in Ingots

• Historically, 20% of ingots have surface cracks. After a process change, only 17% of 400 ingots cracked.

• Null Hypothesis (H0): p = 0.20 (no change)

• Alternative Hypothesis (Ha): p < 0.20 (crack rate decreased)

Null and Alternative Hypotheses

• Null Hypothesis (H0): The status quo, no change, always contains an equals sign (e.g., p = 0.5).

• Alternative Hypothesis (Ha): The claim we seek to support, always contains <, >, or ≠ (e.g., p ≠ 0.5).

• Hypothesis testing is always about the population parameter (p), not the sample statistic (p̂).

One-Sided vs. Two-Sided Hypotheses

The sign in the alternative hypothesis determines the test type:

Type	Null Hypothesis	Alternative Hypothesis
Two-Sided	H0: p = p0	Ha: p ≠ p0
One-Sided (Left)	H0: p = p0	Ha: p < p0
One-Sided (Right)	H0: p = p0	Ha: p > p0

Examples of Hypotheses

• Marriage Rates: H0: p = 0.70, Ha: p < 0.70 (testing if marriage rates have declined)

• Internet Sales: H0: p = 0.30, Ha: p > 0.30 (testing if online sales have increased)

8.2 Characterizing p-values

Definition of p-value

• The p-value is the probability, assuming the null hypothesis is true, of obtaining a test statistic as extreme as or more extreme than the observed value.

Types of p-values

Test Type	Null Hypothesis	Alternative Hypothesis
Left-Tailed	H0: p = p0	Ha: p < p0
Right-Tailed	H0: p = p0	Ha: p > p0
Two-Tailed	H0: p = p0	Ha: p ≠ p0

Calculating the Test Statistic

• The one-proportion z-test statistic is calculated as:

$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

• Where $\hat{p}$ is the sample proportion, $p_0$ is the hypothesized population proportion, and $n$ is the sample size.

Interpreting the Test Statistic

• A positive value means the observed outcome was greater than expected; a negative value means it was less.

• The farther the test statistic is from 0, the more evidence against the null hypothesis.

Example Calculation

• Suppose $p_0 = 0.10$, $\hat{p} = 0.19$, $n = 1377$:

$z = \frac{0.19 - 0.10}{\sqrt{\frac{0.10 \times 0.90}{1377}}} = 11.13$

• This large z-value suggests strong evidence against the null hypothesis.

Making Decisions with p-values

• If p-value < significance level ($\alpha$), reject H0.

• If p-value > significance level, fail to reject H0.

• Common significance levels: $\alpha = 0.05$, $0.01$, $0.10$.

Conditions for Hypothesis Testing

• Random sample: Data must be collected randomly.

• Independence: Observations must be independent.

• Large sample size: $np_0 \geq 10$ and $n(1-p_0) \geq 10$.

• Large population: If sampling without replacement, population size should be at least 10 times the sample size.

Statistical vs. Practical Significance

• Statistical significance means the result is unlikely to have occurred by chance (p-value is small).

• Practical significance means the result is large enough to be meaningful in real-world terms.

Summary Table: Steps in Hypothesis Testing for Proportions

Step	Description
1. Hypothesize	State H0 and Ha about the population proportion p.
2. Prepare	Choose significance level $\alpha$, check conditions, select test statistic.
3. Compute to Compare	Calculate test statistic and p-value.
4. Interpret	Compare p-value to $\alpha$ and state conclusion in context.

Key Formulas

• Standard Error: $SE = \sqrt{\frac{p_0(1-p_0)}{n}}$

• z-Test Statistic: $z = \frac{\hat{p} - p_0}{SE}$

Example Table: Types of Hypothesis Tests

Test Type	Null Hypothesis	Alternative Hypothesis	When to Use
Left-Tailed	p = p0	p < p0	Testing for a decrease
Right-Tailed	p = p0	p > p0	Testing for an increase
Two-Tailed	p = p0	p ≠ p0	Testing for any change

Additional info:

• All examples and explanations are based on standard introductory statistics curriculum and are suitable for college-level statistics students.