Explain the purpose of the random condition in a two-proportion z-test.
Ensures the samples are representative of the populations, avoiding bias and allowing for generalization of results.
Explain the purpose of the independence condition in a two-proportion z-test.
Ensures that the observations within each sample are independent of one another. Often checked using the 10% condition.
Explain the purpose of the normal condition (large counts) in a two-proportion z-test.
To ensure that the sampling distribution of the difference in sample proportions is approximately normal, allowing us to use the z-distribution for inference.
Explain what a p-value represents in the context of a significance test.
The probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true.
Explain the meaning of rejecting the null hypothesis.
Rejecting the null hypothesis means there is sufficient evidence to support the alternative hypothesis. The observed difference is statistically significant.
What are the differences between the null and alternative hypotheses?
Null Hypothesis: Assumes no difference between population proportions. | Alternative Hypothesis: Claims there is a significant difference between population proportions.
What are the differences between Type I and Type II errors?
Type I error: Rejecting a true null hypothesis. | Type II error: Failing to reject a false null hypothesis.
What are the differences between one-sided and two-sided alternative hypotheses?
One-sided: Tests for difference in a specific direction ($p_1 > p_2$ or $p_1 < p_2$). | Two-sided: Tests for any difference ($p_1 \neq p_2$).
What are the differences between random sampling and random assignment?
Random Sampling: Selecting a sample randomly from a population. | Random Assignment: Assigning participants to treatment groups randomly in an experiment.
What are the differences between the z-test and t-test?
Z-test: Used when the population standard deviation is known or sample size is large. | T-test: Used when the population standard deviation is unknown and sample size is small.
What is the null hypothesis?
Statement of no difference between population proportions: $H_0: p_1 = p_2$
What is the alternative hypothesis?
Statement that there is a difference between population proportions ($H_a: p_1 > p_2$, $H_a: p_1 < p_2$, or $H_a: p_1 \neq p_2$).
Define $p_1$ and $p_2$ in the context of a two-proportion z-test.
$p_1$ and $p_2$ represent the population proportions of two different groups being compared.
What is a pooled proportion?
A weighted average of the sample proportions, used when the null hypothesis assumes equal population proportions.
What is a significance test?
A procedure for measuring the strength of evidence against a null hypothesis.
