8.1

Hypothesis Testing

Introduction to Hypothesis Testing

Hypothesis testing is a fundamental procedure in inferential statistics, used to make decisions or inferences about population parameters based on sample data. This chapter introduces the basic concepts, steps, and methods involved in hypothesis testing, applicable to proportions, means, and variances.

8-1 Basics of Hypothesis Testing

Key Concepts

• Hypothesis: A claim or statement about a property of a population (e.g., mean, proportion, variance).

• Hypothesis Test (Test of Significance): A formal procedure for testing a claim about a population property using sample data.

Formulating Hypotheses

• Null Hypothesis (H0): A statement that the value of a population parameter is equal to a specified value. It represents the status quo or no effect.

• Alternative Hypothesis (H1 or Ha): A statement that the parameter differs from the null hypothesis value. It uses the symbols <, >, or ≠.

Example:

Claim: "Most Internet users utilize two-factor authentication." Let p = proportion of users who use two-factor authentication. Symbolic claim: p > 0.5 (majority means more than half).

Significance and Decision Making

• Results are evaluated as significantly high, significantly low, or not significant based on the probability of observing such results under the null hypothesis.

Procedure for Hypothesis Tests

1. Step 1: Identify the claim and express it symbolically.

2. Step 2: State the symbolic form that must be true if the original claim is false.

3. Step 3: Assign the null and alternative hypotheses:

   • Alternative hypothesis (H1): Symbol does not include equality (<, >, ≠).

   • Null hypothesis (H0): Symbol includes equality (=).

4. Step 4: Select the significance level (α).

5. Step 5: Identify the relevant test statistic and its sampling distribution (e.g., normal, t, χ2).

6. Step 6: Calculate the value of the test statistic and find the P-value or critical value(s).

7. Step 7: Make a decision to reject or fail to reject H0.

8. Step 8: Restate the decision in simple, nontechnical terms.

Practice Examples

• Example 1: Cereal boxes labeled as 20 oz. Inspector suspects mean weight is less than 20 oz. H0: μ = 20 H1: μ < 20

• Example 2: Inspector thinks mean weight is the same as the label. H0: μ = 20 H1: μ ≠ 20

Test Procedure Analogy

• Hypothesis testing is like a criminal trial: H0 is the defendant (assumed innocent/true until evidence suggests otherwise).

• If evidence (data) strongly contradicts H0, we reject H0 in favor of H1.

Type I and Type II Errors

• Type I Error (α): Rejecting H0 when it is actually true. $\alpha = P(\text{Type I error}) = P(\text{rejecting } H_0 \text{ when } H_0 \text{ is true})$

• Type II Error (β): Failing to reject H0 when it is actually false. $\beta = P(\text{Type II error}) = P(\text{failing to reject } H_0 \text{ when } H_0 \text{ is false})$

Preliminary Conclusion	True State of Nature Null hypothesis is true	True State of Nature Null hypothesis is false
Reject H0	Type I error P(Type I error) = α	Correct decision
Fail to reject H0	Correct decision	Type II error P(Type II error) = β

Controlling Errors

• Cannot minimize both Type I and Type II errors simultaneously.

• Priority is given to controlling Type I error rate (α), typically set at 0.05 or 0.01.

Step 4: Select the Significance Level (α)

• Significance Level (α): The probability threshold for rejecting H0. It is the probability of making a Type I error. $\alpha = P(\text{rejecting } H_0 \text{ when } H_0 \text{ is true})$

Step 5: Identify the Test Statistic and Sampling Distribution

• Choose the appropriate test statistic (z, t, χ2, etc.) based on the parameter and sample characteristics.

Parameter	Sampling Distribution	Requirements	Test Statistic
Proportion p	Normal (z)	np ≥ 5 and nq ≥ 5	$z = \frac{\hat{p} - p}{\sqrt{\frac{pq}{n}}}$
Mean μ (unknown σ)	t	σ not known and normally distributed population or n > 30	$t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$
Mean μ (known σ)	Normal (z)	σ known and normally distributed population or n > 30	$z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$
St. dev. σ or variance σ2	χ2	Strict requirement: normally distributed population	$\chi^2 = \frac{(n-1)s^2}{\sigma^2}$

Step 6: Find the Value of the Test Statistic and P-value or Critical Value(s)

• Calculate the test statistic using sample data.

• Find the corresponding P-value or compare to critical value(s).

Test Statistic

• The test statistic is a standardized value (z, t, χ2, etc.) used to decide whether to reject H0.

• Calculated by converting the sample statistic under the assumption that H0 is true.

Example Calculation:

Given n = 926, &hat;p; = 0.52, H0: p = 0.5 $z = \frac{0.52 - 0.5}{\sqrt{(0.5)(0.5)/926}} = 1.22$ (More accurate calculation yields z = 1.25)

Critical Region

• The critical region (rejection region) consists of all values of the test statistic that lead to rejection of H0.

Types of Tests

• Two-tailed test: Critical region in both tails; H1 uses ≠.

• Left-tailed test: Critical region in left tail; H1 uses <.

• Right-tailed test: Critical region in right tail; H1 uses >.

Critical Value Method

• Compare the test statistic to the critical value(s).

• If the test statistic falls in the critical region, reject H0.

P-Value Method

• Compare the P-value to the significance level (α).

• P-value: Probability of obtaining a test statistic as extreme as the observed, assuming H0 is true.

• If P-value ≤ α, reject H0; if P-value > α, fail to reject H0.

Example:

Test statistic z = 1.25, right-tailed test. P-value = 0.1056. Since 0.1056 > 0.05, fail to reject H0.

Step 7: Make a Decision

• Reject H0 if evidence is strong (test statistic in critical region or P-value ≤ α).

• Fail to reject H0 if evidence is not strong enough.

Step 8: Restate the Decision in Simple Terms

• Express the conclusion in nontechnical language, addressing the original claim.

• Avoid saying "accept H0"; instead, say "fail to reject H0" as we can never prove H0.

Example Conclusion:

"There is not sufficient evidence to support the claim that most Internet users utilize two-factor authentication."

Power of a Hypothesis Test

• Power: Probability of correctly rejecting a false null hypothesis. $\text{Power} = 1 - \beta$

• Higher power means a greater chance of detecting a true effect.

Practical Considerations

• Null hypothesis often represents "no difference" or "no change."

• Type I error is controlled at a low level to avoid making serious mistakes.

• Type II error is less critical if failing to reject H0 does not have severe consequences.