Sampling for Differences in Sample Proportions - AP Statistics

1. What condition checks the independence of samples in differences of proportions?

   Samples must be randomly selected and independent. Random sampling prevents bias and ensures validity.

2. Which concept describes the difference between sample proportions from two independent samples?

   Sampling distribution for differences in sample proportions. Distribution of $p_1 - p_2$ from repeated sampling.

3. What is the requirement for sample size in a sampling distribution of proportions?

   Sample sizes must be sufficiently large. Needed for normal approximation to be valid.

4. Define the pooled sample proportion used in hypothesis testing.

   $\text{p}_{\text{pooled}} = \frac{x_1 + x_2}{n_1 + n_2}$. Combines successes from both samples.

5. Which type of error occurs if the null hypothesis is incorrectly rejected?

   Type I error. Rejecting true null hypothesis incorrectly.

6. What is the critical region for a 5% significance level in a two-tailed test?

   $z \text{ less than } -1.96 \text{ or } z \text{ greater than } 1.96$. Rejection region for two-tailed test at α = 0.05.

7. What is the alternative hypothesis for a two-tailed test on differences in sample proportions?

   $H_a: \text{P}_1 \neq \text{P}_2$. Tests if proportions differ in either direction.

8. What is the interpretation of a 95% confidence interval in the context of proportions?

   95% confident interval captures the true difference. Interval contains true parameter 95% of the time.

9. Which method is used to test the hypothesis of equal sample proportions?

   Two-proportion $z$-test. Hypothesis test for equality of proportions.

10. Find the pooled proportion if $x_1=40$, $x_2=50$, $n_1=100$, $n_2=120$.

    $p_{\text{pooled}} = 0.45$. Total successes divided by total sample size.

11. Which condition must be met for the proportion difference sampling distribution to be valid?

    Both samples must be random and independent. Ensures unbiased sampling distribution properties.

12. What is the significance level commonly used in hypothesis testing for differences?

    0.05. Standard α level for statistical significance.

13. What is the critical value for a 95% confidence interval using a normal distribution?

    1.96. Standard normal distribution 95% cutoff value.

14. In hypothesis testing, what does a Type II error represent?

    Failing to reject a false null hypothesis. Missing a real difference when it exists.

15. Which statistical concept allows comparison between two independent sample proportions?

    Sampling distribution for differences in sample proportions. Enables hypothesis testing between two groups.

16. What is the parameter of interest in a difference of proportions test?

    Difference in true population proportions. What we're testing: $P_1 - P_2$.

17. What is the symbol for the sample proportion in a sample?

    $\text{p}$. Sample statistic estimating population proportion.

18. State the sampling distribution shape for large sample sizes in differences in proportions.

    Approximately normal. Central Limit Theorem applies to proportion differences.

19. What is the purpose of the pooled proportion in hypothesis testing?

    Estimate the common proportion in two samples. Used when assuming equal population proportions.

20. For a hypothesis test, what does a $p$-value less than $\text{0.05}$ indicate?

    Reject the null hypothesis. Strong evidence against null hypothesis at α = 0.05.

21. Which hypothesis test is used for comparing two sample proportions?

    $z$-test for proportions. Appropriate test for comparing two proportions.

22. Identify the condition for using a normal model to approximate binomial distribution.

    Both $np \text{ and } n(1-p) \text{ ≥ } 10$. Success-failure condition for normal approximation.

23. Identify the formula for calculating the $z$-score in a hypothesis test for proportions.

    $z = \frac{(\text{p}_1 - \text{p}_2) - \text{0}}{\text{SE}}$. Under null hypothesis, difference equals zero.

24. What does a $p$-value represent in hypothesis testing?

    Probability of observing data given $H_0$ is true. Strength of evidence against null hypothesis.

25. Identify the condition required for normal approximation in sample proportions.

    Both $n_1p_1$, $n_1(1-p_1)$, $n_2p_2$, and $n_2(1-p_2)$ must be $\text{≥} 10$. Ensures normal approximation is valid for both samples.

26. What is the null hypothesis when testing differences in sample proportions?

    $H_0: \text{P}_1 - \text{P}_2 = 0$. Tests equality of population proportions.

27. Identify the formula for calculating the confidence interval for differences in proportions.

    $(p_1 - p_2) \text{±} z^*\text{SE}$ . Margin of error added to point estimate.

28. State the formula for the test statistic for the difference in sample proportions.

    $z = \frac{(\text{p}_1 - \text{p}_2) - (\text{P}_1 - \text{P}_2)}{\text{SE}}$. Standardizes the difference using standard error.

29. What is the purpose of a confidence interval in sampling distributions?

    Estimate the range of the difference in true proportions. Provides interval estimate for true difference.

30. Which symbol represents the true proportion in the population for a single sample?

    $P$. Population parameter for true proportion.

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