Concluding Tests Population Proportion - AP Statistics

1. Identify the decision if $z$-score is 2.5 in a right-tailed test at 0.05 significance.

   Reject $H_0$. Z-score (2.5) exceeds critical value (1.645) for right tail.

2. What is the effect of a higher significance level on the probability of a Type I Error?

   Increases probability. Higher $\alpha$ means more liberal rejection criterion.

3. State the decision rule for rejecting $H_0$ based on p-value and significance level.

   Reject $H_0$ if $p$-value $< \alpha$. Standard criterion for statistical significance.

4. Find the z-score if $\hat{p} = 0.52$, $p_0 = 0.5$, $n = 100$.

   $z = 0.4$. Calculated using $z = \frac{0.52 - 0.5}{\sqrt{\frac{0.5(0.5)}{100}}} = 0.4$.

5. What is the effect of a Type I Error?

   Rejecting a true null hypothesis. Concludes effect exists when it actually doesn't.

6. Determine the conclusion if $p$-value is 0.01 in a test with $\alpha = 0.05$.

   Reject $H_0$. P-value (0.01) is less than significance level (0.05).

7. Identify the decision if $z$-score is 1.8 in a two-tailed test at 0.05 significance.

   Fail to reject $H_0$. Z-score (1.8) doesn't exceed critical value (1.96).

8. Identify the decision if $z$-score is -1.5 in a left-tailed test at 0.05 significance.

   Fail to reject $H_0$. Z-score (-1.5) doesn't exceed critical value (-1.645).

9. What is the alternative hypothesis in a right-tailed test for a population proportion?

   $H_a: p > p_0$. Tests if population proportion is greater than hypothesized value.

10. State the condition for a sample size to be considered large enough for a proportion test.

    $np_0 \geq 10$ and $n(1-p_0) \geq 10$. Ensures normal approximation is valid for proportion test.

11. What is the formula for the standard error of a sample proportion?

    $SE = \sqrt{\frac{p_0(1-p_0)}{n}}$. Measures variability of sample proportion under null hypothesis.

12. What is the alternative hypothesis in a left-tailed test for a population proportion?

    $H_a: p < p_0$. Tests if population proportion is less than hypothesized value.

13. What is the critical value for a 99% confidence level in a two-tailed test?

    $z^* = 2.576$. Higher critical value for more stringent 99% confidence.

14. Determine the decision if $p$-value is 0.03 and $\alpha$ is 0.05.

    Reject $H_0$. P-value (0.03) is less than significance level (0.05).

15. What is the null hypothesis in a test for a population proportion?

    $H_0: p = p_0$. States the population proportion equals the hypothesized value.

16. Determine the outcome if $p$-value is 0.07 and $\alpha$ is 0.05.

    Fail to reject $H_0$. P-value (0.07) exceeds significance level (0.05).

17. Identify when to reject the null hypothesis in a one-tailed test.

    If $z > z^*$. Test statistic exceeds critical value in one direction.

18. State the impact of a smaller $\alpha$ on the critical value in a two-tailed test.

    Increases critical value. Smaller $\alpha$ requires more extreme evidence.

19. State the effect of increasing sample size on the power of a test.

    Increases power. Larger samples provide more precision and stronger tests.

20. What does the z-score measure in a population proportion test?

    Number of standard deviations from the mean. Distance from hypothesized value in standard error units.

21. What is the purpose of hypothesis testing in statistics?

    To make inferences about a population. Tests claims about population parameters using sample data.

22. What is an assumption of a z-test for a population proportion?

    Random sampling. Ensures representative and unbiased sample.

23. State the rule for concluding a test with a non-significant result.

    Fail to reject $H_0$. Insufficient evidence to support alternative hypothesis.

24. Identify the critical value for a 95% confidence level in a two-tailed test.

    $z^* = 1.96$. Standard critical value for 95% confidence in two-tailed tests.

25. Which condition ensures the validity of the normal approximation in proportion tests?

    Large sample size. Required for central limit theorem to apply.

26. Choose the correct test statistic for a population proportion test.

    z-test. Uses normal distribution to test population proportions.

27. Identify the type of error made if $H_0$ is rejected when it is true.

    Type I Error. False positive - rejecting a true null hypothesis.

28. What is the alternative hypothesis in a two-tailed test for a population proportion?

    $H_a: p \neq p_0$. Tests if population proportion differs from hypothesized value.

29. What is the primary goal of a hypothesis test for a population proportion?

    To determine if $p \neq p_0$. Tests whether population proportion differs from hypothesized value.

30. What is the formula for calculating the sample proportion?

    $\hat{p} = \frac{x}{n}$. Successes divided by total sample size.

31. What is the effect of a Type II Error?

    Failing to reject a false null hypothesis. Misses detecting a real effect that exists.

32. What is the z-score formula for testing a population proportion?

    $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$. Standardizes sample proportion using hypothesized proportion.

33. What is the role of the significance level in hypothesis testing?

    Threshold for rejecting $H_0$. Sets the bar for statistical significance.

34. What is the p-value interpretation in hypothesis testing?

    Probability of observing a test statistic as extreme as the sample statistic. Measures strength of evidence against the null hypothesis.

35. Identify the type of error made if $H_0$ is not rejected when it is false.

    Type II Error. False negative - failing to reject a false null hypothesis.

36. Explain what a critical region is in hypothesis testing.

    Range of values leading to rejection of $H_0$. Test statistic values that support rejecting null hypothesis.

37. What is the significance level commonly used in hypothesis testing?

    $\alpha = 0.05$. Common threshold for statistical significance.

38. What is the relationship between power and Type II Error?

    Power = 1 - P(Type II Error). Power equals one minus beta (Type II error rate).

39. What does the power of a test represent?

    Probability of correctly rejecting a false $H_0$. Ability to detect a false null hypothesis.

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