Testing Claims about a Proportion (Normal Approximation to Binomial)

Introduction

This section explains how to use a normal approximation to the binomial distribution for hypothesis testing about a population proportion. When certain conditions are met, the binomial distribution can be approximated by a normal distribution, simplifying calculations and interpretations in statistical inference.

Requirements for Normal Approximation

• Simple Random Sample: The sample must be randomly selected from the population.

• Binomial Distribution Conditions:

  • Fixed number of trials (n).

  • Trials are independent.

  • Each trial has only two possible outcomes (success/failure).

  • Probability of success (p) is constant for each trial.

• Sample Size: Both np and nq should be at least 5, where q = 1 - p.

Notation

• n: sample size

• p: population proportion

• q:

• p̂: sample proportion

Test Statistic Formula

The test statistic for a hypothesis test about a proportion is:

Methods for Hypothesis Testing

• P-value Method (Recommended): Calculate the probability of observing a test statistic as extreme as the one computed, given the null hypothesis is true.

• Critical Value Method: Compare the test statistic to a critical value determined by the significance level (α).

• Confidence Interval Method: Construct a confidence interval for the population proportion and check if the hypothesized value falls within the interval.

Steps for Hypothesis Testing

1. State the null hypothesis () and alternative hypothesis ().

2. Graph the normal distribution and mark the test statistic value.

3. Calculate and state the P-value or critical value.

4. Decide whether to reject or fail to reject .

5. State the conclusion in non-technical language.

Example 1: Right-Tailed Test

In a survey of 1000 adults, 29% said they are comfortable in self-driving cars. Test the claim that more than 1/4 adults feel comfortable in a self-driving car.

Both and are greater than 5, so normal approximation is appropriate.

Calculate the test statistic:

For a right-tailed test at , find the area to the right of using a calculator or statistical table.

P-value = 0.0017. Since 0.0017 < 0.05, we reject the null hypothesis. There is sufficient evidence to support the claim that more than 1/4 adults feel comfortable in self-driving cars.

Calculator Instructions: 1-PropZTest

• Press STAT, then select TESTS in the menu.

• Select 1-PropZTest.

• Enter the claimed population proportion, number of successes, and sample size.

• Select Calculate and press ENTER.

Critical Value Method

Instead of computing a P-value, compare the test statistic to the critical value for . For a right-tailed test, the critical value is . Since , reject .

Confidence Interval Method

For a right-tailed test at , construct a 90% confidence interval for the population proportion:

Using calculator, interval is (0.2666, 0.3136). Since all values are above 0.25, this supports the alternative hypothesis.

Note: The confidence interval method may not always yield the same result as the P-value method.

Example 2: Survey on Biometric Security

In a survey of 510 respondents, 270 said "yes" to a question about biometric security. Find the number of respondents that said "yes":

Example 3: Fast Food Order Accuracy

McDonald's had 33 inaccurate orders among 362 observed. Test the claim that the rate of inaccurate orders is equal to 10% using .

Test statistic: . P-value = 0.565. Since 0.565 > 0.05, fail to reject .

Confidence interval: (0.0615, 0.1201). Since 0.10 is in the interval, fail to reject .

Exact Methods

With modern technology, we can compute the binomial distribution exactly, without requiring and . This is called the exact method.

• Left-tailed test: P-value = P(X or fewer successes in n trials)

• Right-tailed test: P-value = P(X or more successes in n trials)

• Two-tailed test: P-value = Twice the smaller of the preceding left or right-tailed tests

Example 4: Comparing Methods

Fill in the table below with corresponding P-values for normal approximation and exact methods:

Exact methods are more accurate, especially for small sample sizes.

Simple Continuity Correction

• Left-tailed test: P-value = P(X or fewer) - (1/2)P(exactly X)

• Right-tailed test: P-value = P(X or more) - (1/2)P(exactly X)

• Two-tailed test: P-value = Twice the smaller of the preceding left or right-tailed tests

Summary Table: Methods for Testing a Proportion

Additional info: These notes expand on the original content by providing definitions, formulas, and structured examples for clarity and completeness. The tables and step-by-step instructions are inferred and organized for exam preparation.