Difference of Two Means (Setup) - AP Statistics

1. What is the effect of increasing sample size on standard error?

   Standard error decreases. Larger samples provide more precise estimates of parameters.

2. What does a confidence interval represent in hypothesis testing?

   Range of plausible values for parameter. Contains likely values for the true population parameter.

3. What determines the critical value in a hypothesis test for two means?

   Significance level and degrees of freedom. These parameters define the rejection region boundaries.

4. What is the formula for the test statistic in a Z-test for two means?

   $Z = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{SE}$. Uses standard normal distribution for large sample hypothesis testing.

5. What is the purpose of the F-test in the context of two means?

   Tests equality of variances. Determines if equal variance assumption is reasonable.

6. What does $SE$ stand for in hypothesis testing?

   Standard Error. Measures variability of the sampling distribution.

7. Which option describes a Type II error in hypothesis testing?

   Failing to reject $H_0$ when it is false. False negative error, missing a real difference between means.

8. How do you denote sample means in statistical tests?

   $\bar{x}_1, \bar{x}_2$. Common notation for observed sample means in statistical analysis.

9. In hypothesis testing, what does $H_0$ represent?

   Null hypothesis. Statement of no effect or no difference to be tested.

10. What is an assumption regarding the sampling distribution in hypothesis testing?

    It is approximately normal. Central Limit Theorem ensures normality for hypothesis testing.

11. What is the impact of unequal sample sizes on hypothesis testing?

    Affects standard error calculation. Unequal sizes require adjusted standard error calculations.

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

    Samples are independent and normally distributed. Required assumptions to ensure valid test results.

13. Identify the primary use of a t-test comparing two samples.

    Assess mean differences. Determines if observed differences are statistically significant.

14. Which test is used for comparing two population means?

    Two-sample t-test. Tests whether two population means are significantly different.

15. What is the significance level typically used in hypothesis testing?

    $\text{alpha} = 0.05$. Common threshold for statistical significance in hypothesis testing.

16. In hypothesis testing, what does $H_a$ represent?

    Alternative hypothesis. Statement of the effect or difference we're testing for.

17. What is the significance of a p-value less than 0.05?

    Reject the null hypothesis. Evidence against null hypothesis is statistically significant.

18. Find the degrees of freedom for a two-sample t-test with $n_1=10$ and $n_2=12$.

    $df = 20$. Calculated as $n_1 + n_2 - 2 = 10 + 12 - 2 = 20$.

19. How is the difference of two means denoted in hypothesis testing?

    $( \bar{x}_1 - \bar{x}_2 )$. Standard notation for the difference between two sample means.

20. Explain the impact of using a 99% confidence level instead of 95%.

    Wider confidence interval. Higher confidence level requires wider interval to capture parameter.

21. What statistical test is used when variances are assumed unequal?

    Welch's t-test. Modified t-test that doesn't assume equal population variances.

22. What is the pooled variance formula for two samples?

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

23. What is the primary assumption for the Z-test with two samples?

    Large sample size or known population variance. Central Limit Theorem makes normal approximation valid.

24. Identify the appropriate test for large sample sizes comparing two means.

    Z-test. Normal distribution applies when sample sizes are large.

25. What is the name of the distribution used for small sample sizes?

    t-distribution. Used when population standard deviation is unknown.

26. What is the pooled standard deviation used for?

    Comparing means with equal variances. Assumes both populations have the same variance.

27. What is the assumption about variances in a two-sample t-test?

    Variances are assumed equal or unequal. Determines whether to use pooled or separate variance formulas.

28. Which option describes a Type I error in hypothesis testing?

    Rejecting $H_0$ when it is true. False positive error, concluding difference when none exists.

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