Difference of Two Means (Test) - AP Statistics

1. What is the pooled variance formula in a two-sample t-test?

   $s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}$. Weighted average of sample variances.

2. Identify a scenario where a paired t-test is more appropriate.

   When samples are dependent, such as pre-test/post-test designs. Accounts for correlation between paired observations.

3. What is the purpose of a confidence interval in two-sample t-tests?

   To estimate the range for the true difference between population means. Provides range of plausible values for true difference.

4. State the effect of an outlier on the two-sample t-test.

   Can significantly affect the test result, possibly leading to biased conclusions. Can distort mean and increase variability.

5. What is the critical value in hypothesis testing?

   The value that separates the rejection region from the non-rejection region. Determines when to reject the null hypothesis.

6. What is the role of the significance level ($\text{alpha}$) in hypothesis testing?

   Defines the threshold for rejecting the null hypothesis. Sets probability cutoff for statistical significance.

7. Which distribution is used for a two-sample t-test under the null hypothesis?

   Student's t-distribution. Accounts for uncertainty in variance estimates.

8. What is the relationship between sample size and the width of a confidence interval?

   Larger sample size leads to a narrower confidence interval. Increased precision with more data points.

9. What does a significant p-value indicate in a two-sample t-test?

   Evidence against $H_0$, supporting $H_a$. Supports rejecting the null hypothesis.

10. What is a Type II error in hypothesis testing?

    Failing to reject a false null hypothesis. False negative error in hypothesis testing.

11. Identify one method to check for normality in a dataset.

    Use a Q-Q plot or Shapiro-Wilk test. Assesses if data follows normal distribution.

12. What is the formula for calculating the margin of error in a two-sample t-test?

    $t^{*} \times SE$. Critical value multiplied by standard error.

13. Identify the assumption about population variances in a two-sample t-test.

    The populations have equal variances (homogeneity of variance). Allows use of pooled variance formula.

14. What does a 95% confidence interval imply?

    95% of such intervals will contain the true population mean difference. Long-run frequency interpretation of confidence level.

15. Define the power of a test in hypothesis testing.

    Probability of correctly rejecting a false null hypothesis. Ability to detect true differences when they exist.

16. Identify the conditions for using a two-sample t-test.

    Randomness, independence, and normality/large sample size. Ensures valid test assumptions are met.

17. What does homoscedasticity mean in the context of t-tests?

    Equal variances across groups being compared. Required assumption for valid t-test results.

18. What is a Type I error in hypothesis testing?

    Incorrectly rejecting a true null hypothesis. False positive error in hypothesis testing.

19. Which condition checks for independence in a two-sample t-test?

    Samples must be randomly selected from independent populations. Prevents bias from dependent observations.

20. What is the consequence of violating the equal variance assumption?

    May need to use a different test, like Welch's t-test. Adjusts for unequal variances between groups.

21. What is the impact of variance inequality on a two-sample t-test?

    May increase Type I error rate; consider using Welch's test. Welch's test adjusts for unequal variances.

22. Choose an appropriate test for comparing means of more than two groups.

    ANOVA (Analysis of Variance). Extends two-sample comparison to multiple groups.

23. How do you interpret a confidence interval that includes zero?

    There is no significant difference between the means. Zero indicates no significant difference exists.

24. What does the central limit theorem imply for large samples?

    Sample means are approximately normally distributed, regardless of the population distribution. Justifies normality assumption for hypothesis testing.

25. What is the impact of increasing sample size on the test statistic?

    Increases the test statistic, potentially increasing significance. Reduces standard error, making differences more detectable.

26. What is the formula for the standard error (SE) of the difference of means?

    $SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$. Combines variability from both samples.

27. Define the p-value in the context of hypothesis testing.

    Probability of observing data as extreme as the sample, assuming $H_0$ is true. Measures strength of evidence against null hypothesis.

28. How does increasing sample size affect statistical power?

    Increases power, reducing the chance of a Type II error. More data improves ability to detect true effects.

29. State the formula for the test statistic in a two-sample t-test.

    $t = \frac{(\bar{x}_1 - \bar{x}_2) - (\text{Difference})}{\text{SE}}$. Standardizes the observed difference by its standard error.

30. How do you calculate degrees of freedom for a two-sample t-test?

    Use the smaller of $n_1 - 1$ and $n_2 - 1$ or software for more precision. Conservative approach uses minimum; software gives exact calculation.

31. What assumptions are required for a two-sample z-test?

    Normality, known population variances, and random sampling. Uses z-distribution instead of t-distribution.

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