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Define null hypothesis (H₀) in the context of two proportions.
Statement that there is no difference between the two population proportions being compared (p₁ = p₂).
Define alternative hypothesis (Hₐ) in the context of two proportions.
Statement that there is a difference between the two population proportions (p₁ ≠ p₂ , p₁ > p₂, or p₁ < p₂).
What is a z-score in hypothesis testing for two proportions?
A test statistic that measures how many standard deviations the observed difference in sample proportions is from the hypothesized difference (usually zero).
What is a p-value?
The probability of observing a sample difference as extreme as, or more extreme than, the one observed, assuming the null hypothesis is true.
Define significance level (α).
The probability of rejecting the null hypothesis when it is actually true (Type I error). Common values are 0.05 or 0.01.
What does it mean to 'reject the null hypothesis'?
It means there is sufficient evidence to conclude that there is a statistically significant difference between the two population proportions.
What are the differences between a one-sample z-test for proportions and a two-sample z-test for proportions?
One-sample: Compares a sample proportion to a hypothesized population proportion. Two-sample: Compares the proportions of two different samples to see if there's a difference between the populations.
What are the key differences between using the p-value and using the z-score to conclude a test?
P-value: Compare directly to the significance level (alpha). | Z-score: Assesses extremity using empirical rule (68-95-99.7 rule).
What are the differences between Type I and Type II errors?
Type I: Rejecting a true null hypothesis (false positive). | Type II: Failing to reject a false null hypothesis (false negative).
What is the formula for the z-score when testing the difference of two population proportions?
z = \frac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}}
What do and represent in the z-score formula?
: Sample proportion for group 1. : Sample proportion for group 2.
What do and represent in the z-score formula?
: Sample size for group 1. : Sample size for group 2.
In the z-score formula, what is the typical value for (p₁ - p₂)?
Usually 0, representing the null hypothesis that there is no difference between the population proportions.
What conditions must be met to use the z-score formula for two proportions?
Random samples, independence, and large enough sample sizes (np ≥ 10 and n(1-p) ≥ 10 for each group).