This lesson introduces the comparison of two independent population proportions. It explains how sample proportions estimate population proportions and why a difference between two sample results might reflect either a real population difference or ordinary sampling variation.
A population proportion, such as pA, is the fraction of all individuals in population A who have a specified characteristic. Usually, the population is too large to measure completely, so a sample is used to estimate it.
All flashcards
Sample proportion
The fraction of sample observations with the counted outcome: p^=x/n.
Population and sample proportions
p describes a population; p^ describes a sample and estimates p.
Pooled proportion
Under the equal-proportions null hypothesis, combine successes and sample sizes: p^c=(xA+xB)/(nA+nB).
Pooled standard error
For testing equal population proportions, SE0=p^c(1−p^c)(1/nA+1/nB).
Failing to reject the null
It means the data do not provide sufficient evidence for the specified alternative; it does not prove equal population proportions.
Quiz questions
If a research question asks whether population A has a higher proportion than population B, which alternative hypothesis matches it?Answer: Ha:pA>pB
For a two-proportion z test with group A first, how is the test statistic calculated?Answer: z=(p^A−p^B)/SE0
Under H0:pA=pB, which expression gives the pooled standard error for a two-proportion z test?Answer: p^c(1−p^c)(1/nA+1/nB)
What does a p-value represent in a hypothesis test?Answer: Assuming the null is true, the probability of a result at least as extreme in the alternative's direction.
At significance level α, what decision rule applies when the p-value is less than or equal to α?Answer: Reject the null hypothesis.
Based on Introductory Statistics 2e by Barbara Illowsky, Susan Dean (OpenStax), licensed CC BY 4.0. Adapted into a summary, flashcards and quizzes.