Hypothesis Testing

8-2 Testing a Claim about a Proportion

This section introduces the formal procedure for testing a claim about a population proportion p. The methods covered include the P-value method, critical value method, and the use of confidence intervals. These approaches are foundational in statistics for evaluating claims about proportions, probabilities, or percentages in a population.

Key Concepts

• Population Proportion (p): The proportion of individuals in a population with a certain characteristic.

• Sample Proportion (\(\hat{p}\)): The proportion of individuals in a sample with that characteristic, calculated as \(\hat{p} = \frac{x}{n}\), where x is the number of successes and n is the sample size.

• Normal Approximation: For large samples, the binomial distribution of sample proportions can be approximated by a normal distribution.

Finding the Number of Successes (x)

• When only the sample proportion \(\hat{p}\) is given, calculate the number of successes as \(x = n\hat{p}\).

• Always round \(x\) to the nearest whole number, as it represents a count.

• Caution: Slightly different results may occur if you use the sample proportion directly versus the rounded value of \(x\).

• Example: If 52% of 926 respondents answered "yes," then \(x = 926 \times 0.52 = 481.52\), rounded to 482.

Equivalent Methods

• The P-value method and critical value method use the same standard deviation based on the claimed proportion p and are equivalent.

• The confidence interval method uses the sample proportion \(\hat{p}\) for standard deviation and may yield different conclusions.

• Recommendation: Use a confidence interval to estimate a population proportion, but use the P-value or critical value method to test a claim about a proportion.

Testing a Claim About a Population Proportion (Normal Approximation Method)

The objective is to conduct a formal hypothesis test of a claim about a population proportion p using the normal approximation to the binomial distribution.

Notation

• n: Sample size or number of trials

• p: Population proportion (value in the null hypothesis)

• \(\hat{p}\): Sample proportion, \(\hat{p} = \frac{x}{n}\)

• q: \(1 - p\)

Requirements

• The sample is a simple random sample.

• The conditions for a binomial distribution are satisfied:

  • Fixed number of trials

  • Trials are independent

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

  • Probability of success remains the same in all trials

• Both \(np \geq 5\) and \(nq \geq 5\) are satisfied, so the binomial distribution of sample proportions can be approximated by a normal distribution with mean \(\mu = np\) and standard deviation \(\sigma = \sqrt{npq}\).

• Note: The value of p used here is the assumed proportion from the claim, not the sample proportion \(\hat{p}\).

Test Statistic for Testing a Claim about a Proportion

The test statistic for hypothesis testing about a proportion is:

• P-values: Usually provided by technology. If not, use the standard normal distribution table.

• Critical values: Use the standard normal distribution table.

Worked Example: Testing a Claim about Internet Users and Two-Factor Authentication

Suppose 926 Internet users are surveyed, and 52% (or 482) report using two-factor authentication. Test the claim that "most" Internet users use two-factor authentication (interpreted as "more than half," or \(p > 0.5\)).

Step-by-Step Solution

1. Check Requirements:

   • Random sample: 926 users randomly selected.

   • Fixed number of independent trials (926), two categories (uses or does not use two-factor authentication).

   • \(np = 926 \times 0.5 = 463\), \(nq = 926 \times 0.5 = 463\); both \(\geq 5\).

2. State Hypotheses:

   • Null hypothesis:

   • Alternative hypothesis: (original claim)

3. Significance Level:

4. Calculate Test Statistic:

   • Sample proportion:

5. Find P-value:

   • P-value is the area to the right of in the standard normal distribution.

   • Cumulative area to the left of is 0.8944, so P-value =

6. Decision:

   • P-value (0.1056) > (0.05): Fail to reject the null hypothesis.

7. Conclusion:

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

Critical Value Method

• Critical value for right-tailed test at is .

• Test statistic does not exceed the critical value, so fail to reject the null hypothesis.

Confidence Interval Method

• Construct a 90% confidence interval for using the sample data.

• Result:

• Since 0.5 is within the interval, we cannot conclude .

Summary Table: Methods for Testing a Claim about a Proportion

Key Points for Exam Preparation

• Always check the requirements for using the normal approximation.

• State hypotheses clearly, using correct symbolic notation.

• Calculate the test statistic using the correct formula.

• Interpret the P-value in relation to the significance level.

• Understand the differences between the P-value, critical value, and confidence interval methods.

Additional info: These notes are based on textbook slides and cover the essential steps and concepts for hypothesis testing about a population proportion, suitable for college-level statistics students.