8.2

Chapter 8: Hypothesis Testing

8-2 Testing a Claim About a Proportion

This section provides a comprehensive procedure for testing a statistical claim about a population proportion, using both the P-value method and the critical value method. The normal approximation method is applied when certain requirements are met.

Key Concepts

• Hypothesis testing is a formal statistical procedure used to evaluate claims about a population parameter based on sample data.

• For population proportions, the test is based on the sampling distribution of sample proportions, which can be approximated by a normal distribution under certain conditions.

Notation

• n: sample size or number of trials

• p: population proportion (the value used in the null hypothesis)

• \( \hat{p} = \frac{x}{n} \): sample proportion, where x is the number of successes in the sample

• q = 1 - p: probability of failure

Requirements for the Normal Approximation Method

1. The sample observations must be a simple random sample.

2. The conditions for a binomial distribution must be satisfied:

   • There is a fixed number of trials (n).

   • The trials are independent.

   • Each trial has two categories: "success" and "failure".

   • The probability of success (p) remains the same for all trials.

3. The conditions np ≥ 5 and nq ≥ 5 must both be satisfied, so the binomial distribution of sample proportions can be approximated by a normal distribution.

Note: The value of p used in these calculations is the assumed proportion from the null hypothesis, not the sample proportion.

Test Statistic for a Population Proportion

The test statistic used is the z-score, calculated as:

$z = \frac{\hat{p} - p}{\sqrt{\frac{pq}{n}}}$

• P-values: Use the standard normal distribution to find the area from the calculated z to the extreme tails, depending on the alternative hypothesis.

• Critical values: Use the standard normal distribution to find the critical value that separates the rejection region of area \( \alpha \).

Step-by-Step Procedure for Hypothesis Testing (Population Proportion)

1. State the claim in symbolic form (e.g., \( p > 0.5 \)).

2. State the opposite of the claim (e.g., \( p \leq 0.5 \)).

3. Identify the null and alternative hypotheses:

   • Null hypothesis (\( H_0 \)): always contains equality (e.g., \( p = 0.5 \)).

   • Alternative hypothesis (\( H_1 \)): reflects the claim (e.g., \( p > 0.5 \)).

4. Check requirements for using the normal approximation method.

5. Select the significance level (\( \alpha \)), commonly 0.05.

6. Calculate the test statistic (z-score) using the formula above.

7. Find the P-value (or compare the test statistic to the critical value):

   • For right-tailed tests: P-value is the area to the right of z.

   • For left-tailed tests: P-value is the area to the left of z.

   • For two-tailed tests: P-value is twice the area in the tail beyond |z|.

8. Make a decision:

   • If P-value ≤ \( \alpha \), reject the null hypothesis.

   • If P-value > \( \alpha \), fail to reject the null hypothesis.

9. State the conclusion in the context of the original claim, using correct statistical language.

Examples

Example 1: Most Internet Users Utilize Two-Factor Authentication

• Claim: Most Internet users utilize two-factor authentication (interpreted as \( p > 0.5 \)).

• Sample: 926 users, 52% (482) responded "yes".

• Hypotheses:

  • \( H_0: p = 0.5 \)

  • \( H_1: p > 0.5 \) (original claim)

• Requirements: Random sample, fixed number of independent trials, two categories, \( np = nq = 463 \geq 5 \).

• Significance level: \( \alpha = 0.05 \)

• Test statistic:

  • \( \hat{p} = \frac{482}{926} = 0.52 \)

  • \( z = \frac{0.52 - 0.5}{\sqrt{(0.5)(0.5)/926}} = 1.25 \)

• P-value: Area to the right of z = 1.25 is 0.1056.

• Decision: 0.1056 > 0.05, so fail to reject \( H_0 \).

• Conclusion: There is not sufficient sample evidence to support the claim that most Internet users utilize two-factor authentication.

Example 2: Fewer Than 30% of Adults Have Sleepwalked

• Claim: Fewer than 30% of adults have sleepwalked (\( p < 0.30 \)).

• Sample: 19,136 adults, 29.2% have sleepwalked.

• Hypotheses:

  • \( H_0: p = 0.30 \)

  • \( H_1: p < 0.30 \) (original claim)

• Requirements: Random sample, fixed number of independent trials, two categories, \( np, nq \geq 5 \).

• Significance level: \( \alpha = 0.05 \)

• Test statistic:

  • \( \hat{p} = 0.292 \)

  • \( z = \frac{0.292 - 0.3}{\sqrt{(0.3)(0.7)/19136}} = -2.41 \)

• P-value: Area to the left of z = -2.41 is 0.0080.

• Decision: 0.0080 < 0.05, so reject \( H_0 \).

• Conclusion: There is sufficient evidence to support the claim that fewer than 30% of adults have sleepwalked.

Example 3: 30% of Blue Candy?

• Claim: The percentage of blue candies is equal to 30% (\( p = 0.3 \)).

• Sample: 100 candies, 22% are blue.

• Hypotheses:

  • \( H_0: p = 0.3 \)

  • \( H_1: p \neq 0.3 \)

• Requirements: Random sample, fixed number of independent trials, two categories, \( np, nq \geq 5 \).

• Test statistic:

  • \( \hat{p} = 0.22 \)

  • \( z = \frac{0.22 - 0.3}{\sqrt{(0.3)(0.7)/100}} = -1.746 \)

• Critical values: For two-tailed test at \( \alpha = 0.05 \), critical values are \( \pm 1.96 \).

• P-value: \( 2 \times P(Z < -1.75) = 2 \times 0.0401 = 0.0802 \)

• Decision: 0.0802 > 0.05, so fail to reject \( H_0 \).

• Conclusion: There is not sufficient evidence to reject the claim that the percentage of blue candies is 30%.

Summary Table: Steps for Hypothesis Testing for a Proportion

Step	Description
1	State the claim and express it symbolically
2	State the opposite of the claim
3	Identify null and alternative hypotheses
4	Check requirements for normal approximation
5	Select significance level (\( \alpha \))
6	Calculate test statistic (z-score)
7	Find P-value or compare to critical value
8	Make a decision (reject or fail to reject \( H_0 \))
9	State the conclusion in context

Additional info: The examples provided illustrate both right-tailed, left-tailed, and two-tailed tests, and emphasize the importance of correct statistical language in conclusions.