AP Statistics Flashcards: Concluding Tests Population Proportion

What this deck covers

This deck focuses on Concluding Tests Population Proportion, giving you a quick way to review the definitions, rules, and examples that matter most for AP Statistics.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

Flashcard 2: Determine the decision if $p$-value is 0.03 and $\alpha$ is 0.05.

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

Flashcard 3: What is the effect of a higher significance level on the probability of a Type I Error?

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

Flashcard 4: State the decision rule for rejecting $H_0$ based on p-value and significance level.

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

Flashcard 5: State the decision rule for rejecting $H_0$ based on p-value and significance level.

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

Flashcard 6: State the rule for concluding a test with a non-significant result.

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

Flashcard 7: Identify the type of error made if $H_0$ is rejected when it is true.

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

Flashcard 8: What is the effect of a higher significance level on the probability of a Type I Error?

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

Flashcard 9: What does the power of a test represent?

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

Flashcard 10: Find the z-score if $\hat{p} = 0.52$, $p_0 = 0.5$, $n = 100$.

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

Flashcard 11: What is the effect of a Type I Error?

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

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

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

Flashcard 13: Determine the conclusion if $p$-value is 0.01 in a test with $\alpha = 0.05$.

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

Flashcard 14: Identify the decision if $z$-score is 1.8 in a two-tailed test at 0.05 significance.

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

Flashcard 15: What is the primary goal of a hypothesis test for a population proportion?

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

Flashcard 16: Identify when to reject the null hypothesis in a one-tailed test.

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

Flashcard 17: What is the null hypothesis in a test for a population proportion?

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

Flashcard 18: Identify the decision if $z$-score is -1.5 in a left-tailed test at 0.05 significance.

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

Flashcard 19: What is the formula for calculating the sample proportion?

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

Flashcard 20: What is the alternative hypothesis in a right-tailed test for a population proportion?

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

Flashcard 21: Identify the decision if $z$-score is 2.5 in a right-tailed test at 0.05 significance.

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

Flashcard 22: Identify the type of error made if $H_0$ is not rejected when it is false.

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

Flashcard 23: State the condition for a sample size to be considered large enough for a proportion test.

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

Flashcard 24: What is the formula for the standard error of a sample proportion?

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

Flashcard 25: What is the purpose of hypothesis testing in statistics?

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

Flashcard 26: What is an assumption of a z-test for a population proportion?

Answer: Random sampling. Ensures representative and unbiased sample.

Flashcard 27: Identify the decision if $z$-score is -1.5 in a left-tailed test at 0.05 significance.

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

Flashcard 28: What does the z-score measure in a population proportion test?

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

Flashcard 29: What is the alternative hypothesis in a left-tailed test for a population proportion?

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

Flashcard 30: State the condition for a sample size to be considered large enough for a proportion test.

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

Flashcard 31: What is the critical value for a 99% confidence level in a two-tailed test?

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

Flashcard 32: State the impact of a smaller $\alpha$ on the critical value in a two-tailed test.

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

Flashcard 33: Which condition ensures the validity of the normal approximation in proportion tests?

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

Flashcard 34: Determine the decision if $p$-value is 0.03 and $\alpha$ is 0.05.

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

Flashcard 35: What is the significance level commonly used in hypothesis testing?

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

Flashcard 36: What is the null hypothesis in a test for a population proportion?

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

Flashcard 37: What is the effect of a Type II Error?

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

Flashcard 38: Determine the outcome if $p$-value is 0.07 and $\alpha$ is 0.05.

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

Flashcard 39: Identify when to reject the null hypothesis in a one-tailed test.

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

Flashcard 40: State the impact of a smaller $\alpha$ on the critical value in a two-tailed test.

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

Flashcard 41: State the effect of increasing sample size on the power of a test.

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

Flashcard 42: What does the z-score measure in a population proportion test?

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

Flashcard 43: What is the alternative hypothesis in a right-tailed test for a population proportion?

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

Flashcard 44: Identify the critical value for a 95% confidence level in a two-tailed test.

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

Flashcard 45: What is the purpose of hypothesis testing in statistics?

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

Flashcard 46: What is the effect of a Type I Error?

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

Flashcard 47: What is an assumption of a z-test for a population proportion?

Answer: Random sampling. Ensures representative and unbiased sample.

Flashcard 48: Explain what a critical region is in hypothesis testing.

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

Flashcard 49: What is the z-score formula for testing a population proportion?

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

Flashcard 50: State the rule for concluding a test with a non-significant result.

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

Flashcard 51: Identify the critical value for a 95% confidence level in a two-tailed test.

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

Flashcard 52: Which condition ensures the validity of the normal approximation in proportion tests?

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

Flashcard 53: Choose the correct test statistic for a population proportion test.

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

Flashcard 54: Identify the type of error made if $H_0$ is rejected when it is true.

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

Flashcard 55: What is the alternative hypothesis in a two-tailed test for a population proportion?

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

Flashcard 56: What is the primary goal of a hypothesis test for a population proportion?

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

Flashcard 57: What is the alternative hypothesis in a two-tailed test for a population proportion?

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

Flashcard 58: Identify the decision if $z$-score is 1.8 in a two-tailed test at 0.05 significance.

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

Flashcard 59: What is the formula for calculating the sample proportion?

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

Flashcard 60: What is the effect of a Type II Error?

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

Flashcard 61: What is the z-score formula for testing a population proportion?

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

Flashcard 62: Find the z-score if $\hat{p} = 0.52$, $p_0 = 0.5$, $n = 100$.

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

Flashcard 63: What is the p-value interpretation in hypothesis testing?

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

Flashcard 64: What is the role of the significance level in hypothesis testing?

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

Flashcard 65: Determine the conclusion if $p$-value is 0.01 in a test with $\alpha = 0.05$.

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

Flashcard 66: What is the p-value interpretation in hypothesis testing?

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

Flashcard 67: Identify the type of error made if $H_0$ is not rejected when it is false.

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

Flashcard 68: What is the relationship between power and Type II Error?

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

Flashcard 69: Choose the correct test statistic for a population proportion test.

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

Flashcard 70: What is the formula for the standard error of a sample proportion?

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

Flashcard 71: State the effect of increasing sample size on the power of a test.

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

Flashcard 72: Determine the outcome if $p$-value is 0.07 and $\alpha$ is 0.05.

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

Flashcard 73: Explain what a critical region is in hypothesis testing.

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

Flashcard 74: What is the critical value for a 99% confidence level in a two-tailed test?

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

Flashcard 75: What is the role of the significance level in hypothesis testing?

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

Flashcard 76: What is the significance level commonly used in hypothesis testing?

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

Flashcard 77: What is the relationship between power and Type II Error?

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

Flashcard 78: What does the power of a test represent?

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