Interpreting p-Values - AP Statistics

1. What is the effect on power when the significance level is increased?

   Power increases. Higher $\alpha$ makes it easier to reject false $H_0$.

2. State the p-value threshold for a 5% significance level.

   $0.05$. The cutoff value for determining statistical significance.

3. What does a small p-value indicate about the null hypothesis?

   Evidence against the null hypothesis. Small p-values suggest data is unlikely under $H_0$.

4. What is the typical confidence level corresponding to a 5% significance level?

   95\text{%}. Standard confidence level when $\alpha = 0.05$.

5. What is the complement of the significance level?

   Confidence level. Confidence level $= 1 - \alpha$.

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

   The probability of observing data as extreme as the sample, assuming the null hypothesis is true. It quantifies how unlikely the observed data would be under $H_0$.

7. Which hypothesis is tested against which in hypothesis testing?

   Null hypothesis against the alternative hypothesis. The null is tested to see if alternative is supported.

8. What happens to the p-value if the test statistic increases?

   The p-value decreases. Larger test statistics move further into rejection region.

9. Identify the term for the probability of a Type I error.

   Significance level ($\text{alpha}$). The probability of incorrectly rejecting $H_0$.

10. What is the relationship between p-value and statistical significance?

    Lower p-values indicate higher statistical significance. Smaller p-values indicate stronger evidence against $H_0$.

11. What is the relationship between confidence level and Type I error?

    Confidence level is $1 - \text{Type I error rate}$. Confidence level and Type I error rate are complements.

12. What does a large p-value indicate about the null hypothesis?

    Insufficient evidence against the null hypothesis. Large p-values suggest data is consistent with $H_0$.

13. What is the relationship between p-value and test statistic?

    The smaller the p-value, the larger the test statistic. Larger test statistics yield smaller p-values.

14. Identify the significance level commonly used in hypothesis testing.

    $\text{0.05}$. Most commonly used threshold for statistical significance.

15. What does a two-tailed test examine regarding p-values?

    Both ends of the distribution for extreme values. Tests for differences in either direction from $H_0$.

16. Which error increases when the significance level is reduced?

    Type II error. Smaller $\alpha$ makes it harder to reject $H_0$.

17. What is considered a statistically significant p-value?

    A p-value less than the significance level. When $p < \alpha$, results are statistically significant.

18. What happens if the p-value is less than the significance level?

    Reject the null hypothesis. When $p < \alpha$, evidence against $H_0$ is strong enough.

19. What is the term for the probability of correctly rejecting a false null hypothesis?

    Power of the test. Power $= 1 - \beta$; probability of detecting false $H_0$.

20. What conclusion is drawn if the p-value is greater than the significance level?

    Fail to reject the null hypothesis. When $p > \alpha$, insufficient evidence to reject $H_0$.

21. What is the inference when p-value equals significance level?

    Test is inconclusive. Borderline case requires careful interpretation or more data.

22. What type of error occurs when the null hypothesis is incorrectly rejected?

    Type I error. Rejecting a true null hypothesis; probability equals $\alpha$.

23. What does a p-value of 0.03 indicate at a 5% significance level?

    Reject the null hypothesis. Since $0.03 < 0.05$, evidence supports rejecting $H_0$.

24. State the typical significance level for a stringent test.

    $\text{0.01}$. Used when requiring stronger evidence against $H_0$.

25. What is the alternative hypothesis symbol in statistics?

    $H_a$. Standard notation for the competing hypothesis.

26. What conclusion is drawn with a p-value of 0.20 at a 5% significance level?

    Fail to reject the null hypothesis. Since $0.20 > 0.05$, insufficient evidence against $H_0$.

27. What does a p-value of 0.001 suggest about the null hypothesis?

    Strong evidence to reject the null hypothesis. Very small p-value provides strong evidence against $H_0$.

28. What is the typical interpretation of a p-value close to 1?

    Very weak evidence against the null hypothesis. P-values near 1 suggest data strongly supports $H_0$.

29. What is the primary use of p-values in statistics?

    To decide whether to reject the null hypothesis. P-values provide evidence for or against $H_0$.

30. What is the decision if p-value is 0.045 at a 5% significance level?

    Reject the null hypothesis. Since $0.045 < 0.05$, evidence supports rejecting $H_0$.

31. What type of error occurs when the null hypothesis is incorrectly not rejected?

    Type II error. Failing to reject a false null hypothesis; probability equals $\beta$.

32. If p-value is 0.07 at a 5% significance level, what is the decision?

    Fail to reject the null hypothesis. Since $0.07 > 0.05$, insufficient evidence against $H_0$.

33. What does a p-value of exactly 0.05 suggest at a 5% significance level?

    Borderline decision; often interpreted as fail to reject. Exactly at threshold; convention varies by context.

34. What is the null hypothesis symbol in statistics?

    $H_0$. Standard notation for the hypothesis being tested.

35. Identify the term for the probability of a Type II error.

    Beta ($\beta$). The probability of incorrectly failing to reject $H_0$.

36. What is the p-value if the test statistic equals the critical value?

    Equal to the significance level. Critical value corresponds exactly to the significance level.

37. Which hypothesis is assumed true when calculating the p-value?

    Null hypothesis ($H_0$). P-values are calculated assuming $H_0$ is true.

38. What is the effect of a larger sample size on p-values?

    P-values tend to decrease. Larger samples provide more precise evidence about $H_0$.

39. How is the p-value related to the significance level in hypothesis testing?

    It determines if the null hypothesis should be rejected. Compare p-value to $\alpha$ to make rejection decision.

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