Sampling for Differences in Sample Means - AP Statistics

1. What is the formula for calculating the confidence interval for the difference in means?

   $CI = (\bar{x}_1 - \bar{x}_2) \pm t^* \times \text{SE}$. Margin of error added and subtracted from difference.

2. Identify the assumption regarding the samples for the sampling distribution of differences in means.

   Samples should be independent. No relationship between the two samples.

3. What does a high p-value suggest in the context of a hypothesis test for difference in means?

   Fail to reject the null hypothesis. Insufficient evidence to conclude difference exists.

4. Identify the test used for comparing the means when population variances are known.

   Z-test. Known population parameters allow normal distribution use.

5. What is the primary purpose of a two-sample t-test?

   To compare the means of two independent samples. Tests whether two groups have different means.

6. Which test statistic is used when population standard deviations are known?

   Z-score. Population parameters known, normal distribution applies.

7. What condition should be met if the populations are not normally distributed?

   Sample sizes should be large. Central Limit Theorem compensates for non-normality.

8. What does the Central Limit Theorem ensure for the sampling distribution of the difference in means?

   It will be approximately normal if sample sizes are large. Large samples ensure normal distribution shape.

9. Which test statistic is used when population standard deviations are unknown?

   T-score. Population parameters unknown, t-distribution needed.

10. What is the effect of a larger sample size on the power of a hypothesis test?

    Increases the power. Better chance of detecting true differences.

11. What assumption is necessary for using pooled variance?

    The population variances are equal. Required for pooled variance calculation validity.

12. What is a critical assumption when interpreting a two-sample t-test?

    The samples are randomly selected. Ensures valid probability calculations and inference.

13. What effect does increasing the sample size have on the variability of the sampling distribution?

    Variability decreases. Standard error decreases with larger samples.

14. What is the degree of freedom used for testing the difference in means with equal variances?

    $df = n_1 + n_2 - 2$. Total observations minus parameters estimated.

15. Identify the hypothesis test used when sample sizes are small and variances are unknown.

    T-test with adjusted degrees of freedom. Small samples need t-distribution for unknown variances.

16. Define the parameter $\text{SE}$ in the context of sampling distributions.

    Standard Error. Standard deviation of sampling distribution.

17. Define the symbol $\bar{x}_1 - \bar{x}_2$ in the context of sampling distributions.

    Difference in sample means. Point estimate for comparing two population means.

18. Identify the condition under which you would use Welch's t-test.

    When variances are not equal. Accounts for unequal population variances.

19. What is the condition for using a two-sample t-test?

    Samples are independent and normally distributed or large. Independence and normality or large sample requirements.

20. Identify the test used for comparing the means when population variances are unknown.

    T-test. Unknown parameters require t-distribution.

21. What is the implication of a confidence interval that includes zero in a difference of means test?

    There may be no significant difference. Zero difference is plausible value.

22. What is the formula for pooled variance when variances are assumed equal?

    $S_p^2 = \frac{(n_1-1)S_1^2 + (n_2-1)S_2^2}{n_1+n_2-2}$. Weighted average of sample variances.

23. Identify the term for the difference between sample means and population means.

    Sampling error. Random variation between sample and population.

24. What does a significant p-value indicate in the context of a hypothesis test for the difference in means?

    Reject the null hypothesis. Evidence against null hypothesis of no difference.

25. What is the impact of increasing sample size on the standard error of the difference in means?

    Standard error decreases. Larger samples reduce sampling variability.

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