Testing Proportions with Large Samples: Hypotheses, CI & Critical Values

Null and Alternative Hypotheses

Both the critical‑value method and the p‑value method can be used to test hypotheses about a population proportion p . For a specific proportion p₀ between 0 and 1, the null hypothesis is expressed as p = p₀ . The alternative hypothesis takes one of three forms: p > p₀ , p < p₀ , or p \neq p₀ .

Confidence Interval

For a large‑sample estimate of a proportion, an approximate 99% confidence interval is constructed (the factor 3 represents three standard errors). The interval is based on the Central Limit Theorem, so the quantiles used are those of the standard normal distribution.

Standardized Test Statistic

The standardized test statistic for a large‑sample test of a single population proportion is

z = \frac{\hat{p} - p₀}{\sqrt{p₀(1-p₀)/n}}

Rejection Region

For each form of the alternative hypothesis (left‑tailed, right‑tailed, or two‑tailed), the distribution of the standardized test statistic and the corresponding rejection region are shown in the figure below.

Example

A soft‑drink manufacturer claims that most adults prefer its beverage. To verify this, a journalist surveyed 500 randomly selected adults, assigning them to taste two drinks. 270 chose the manufacturer’s brand, 211 chose the competitor’s brand, and 19 were undecided. Is there sufficient evidence at the 5% significance level to support the manufacturer’s claim?

Solution: We use the critical‑value method.

Step 1: Verify that the sample size is large enough – it is.

Step 2: Formulate the null and alternative hypotheses. Let p be the proportion of all adults who prefer the manufacturer’s drink. Use a significance level of 0.05.

Step 3: Compute the standardized test statistic.

Step 4: Make a decision. This is a right‑tailed test, so the critical value from the standard normal table is 1.645. The calculated statistic (z ≈ 1.789) falls in the rejection region, providing sufficient evidence at the 5% level to conclude that a majority of adults prefer the manufacturer’s beverage.

Reference

https://saylordotorg.github.io/text_introductory-statistics/s12-05-large-sample-tests-for-a-popul.html