Concluding Tests Population Proportion - AP Statistics
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What is the null hypothesis in a test for a population proportion?
What is the null hypothesis in a test for a population proportion?
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$H_0: p = p_0$. States the population proportion equals the hypothesized value.
$H_0: p = p_0$. States the population proportion equals the hypothesized value.
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What is the z-score formula for testing a population proportion?
What is the z-score formula for testing a population proportion?
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$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$. Standardizes sample proportion using hypothesized proportion.
$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$. Standardizes sample proportion using hypothesized proportion.
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What is the alternative hypothesis in a two-tailed test for a population proportion?
What is the alternative hypothesis in a two-tailed test for a population proportion?
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$H_a: p \neq p_0$. Tests if population proportion differs from hypothesized value.
$H_a: p \neq p_0$. Tests if population proportion differs from hypothesized value.
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Which condition ensures the validity of the normal approximation in proportion tests?
Which condition ensures the validity of the normal approximation in proportion tests?
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Large sample size. Required for central limit theorem to apply.
Large sample size. Required for central limit theorem to apply.
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Find the z-score if $\hat{p} = 0.52$, $p_0 = 0.5$, $n = 100$.
Find the z-score if $\hat{p} = 0.52$, $p_0 = 0.5$, $n = 100$.
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$z = 0.4$. Calculated using $z = \frac{0.52 - 0.5}{\sqrt{\frac{0.5(0.5)}{100}}} = 0.4$.
$z = 0.4$. Calculated using $z = \frac{0.52 - 0.5}{\sqrt{\frac{0.5(0.5)}{100}}} = 0.4$.
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What is the critical value for a 99% confidence level in a two-tailed test?
What is the critical value for a 99% confidence level in a two-tailed test?
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$z^* = 2.576$. Higher critical value for more stringent 99% confidence.
$z^* = 2.576$. Higher critical value for more stringent 99% confidence.
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Choose the correct test statistic for a population proportion test.
Choose the correct test statistic for a population proportion test.
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z-test. Uses normal distribution to test population proportions.
z-test. Uses normal distribution to test population proportions.
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What is the alternative hypothesis in a right-tailed test for a population proportion?
What is the alternative hypothesis in a right-tailed test for a population proportion?
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$H_a: p > p_0$. Tests if population proportion is greater than hypothesized value.
$H_a: p > p_0$. Tests if population proportion is greater than hypothesized value.
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What is the alternative hypothesis in a left-tailed test for a population proportion?
What is the alternative hypothesis in a left-tailed test for a population proportion?
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$H_a: p < p_0$. Tests if population proportion is less than hypothesized value.
$H_a: p < p_0$. Tests if population proportion is less than hypothesized value.
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State the decision rule for rejecting $H_0$ based on p-value and significance level.
State the decision rule for rejecting $H_0$ based on p-value and significance level.
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Reject $H_0$ if $p$-value $< \alpha$. Standard criterion for statistical significance.
Reject $H_0$ if $p$-value $< \alpha$. Standard criterion for statistical significance.
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What is the p-value interpretation in hypothesis testing?
What is the p-value interpretation in hypothesis testing?
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Probability of observing a test statistic as extreme as the sample statistic. Measures strength of evidence against the null hypothesis.
Probability of observing a test statistic as extreme as the sample statistic. Measures strength of evidence against the null hypothesis.
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Identify when to reject the null hypothesis in a one-tailed test.
Identify when to reject the null hypothesis in a one-tailed test.
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If $z > z^*$. Test statistic exceeds critical value in one direction.
If $z > z^*$. Test statistic exceeds critical value in one direction.
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What is the formula for the standard error of a sample proportion?
What is the formula for the standard error of a sample proportion?
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$SE = \sqrt{\frac{p_0(1-p_0)}{n}}$. Measures variability of sample proportion under null hypothesis.
$SE = \sqrt{\frac{p_0(1-p_0)}{n}}$. Measures variability of sample proportion under null hypothesis.
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State the condition for a sample size to be considered large enough for a proportion test.
State the condition for a sample size to be considered large enough for a proportion test.
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$np_0 \geq 10$ and $n(1-p_0) \geq 10$. Ensures normal approximation is valid for proportion test.
$np_0 \geq 10$ and $n(1-p_0) \geq 10$. Ensures normal approximation is valid for proportion test.
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What is an assumption of a z-test for a population proportion?
What is an assumption of a z-test for a population proportion?
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Random sampling. Ensures representative and unbiased sample.
Random sampling. Ensures representative and unbiased sample.
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Determine the outcome if $p$-value is 0.07 and $\alpha$ is 0.05.
Determine the outcome if $p$-value is 0.07 and $\alpha$ is 0.05.
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Fail to reject $H_0$. P-value (0.07) exceeds significance level (0.05).
Fail to reject $H_0$. P-value (0.07) exceeds significance level (0.05).
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Determine the conclusion if $p$-value is 0.01 in a test with $\alpha = 0.05$.
Determine the conclusion if $p$-value is 0.01 in a test with $\alpha = 0.05$.
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Reject $H_0$. P-value (0.01) is less than significance level (0.05).
Reject $H_0$. P-value (0.01) is less than significance level (0.05).
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What is the primary goal of a hypothesis test for a population proportion?
What is the primary goal of a hypothesis test for a population proportion?
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To determine if $p \neq p_0$. Tests whether population proportion differs from hypothesized value.
To determine if $p \neq p_0$. Tests whether population proportion differs from hypothesized value.
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State the rule for concluding a test with a non-significant result.
State the rule for concluding a test with a non-significant result.
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Fail to reject $H_0$. Insufficient evidence to support alternative hypothesis.
Fail to reject $H_0$. Insufficient evidence to support alternative hypothesis.
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What is the relationship between power and Type II Error?
What is the relationship between power and Type II Error?
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Power = 1 - P(Type II Error). Power equals one minus beta (Type II error rate).
Power = 1 - P(Type II Error). Power equals one minus beta (Type II error rate).
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What does the z-score measure in a population proportion test?
What does the z-score measure in a population proportion test?
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Number of standard deviations from the mean. Distance from hypothesized value in standard error units.
Number of standard deviations from the mean. Distance from hypothesized value in standard error units.
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What is the effect of a higher significance level on the probability of a Type I Error?
What is the effect of a higher significance level on the probability of a Type I Error?
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Increases probability. Higher $\alpha$ means more liberal rejection criterion.
Increases probability. Higher $\alpha$ means more liberal rejection criterion.
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State the effect of increasing sample size on the power of a test.
State the effect of increasing sample size on the power of a test.
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Increases power. Larger samples provide more precision and stronger tests.
Increases power. Larger samples provide more precision and stronger tests.
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What does the power of a test represent?
What does the power of a test represent?
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Probability of correctly rejecting a false $H_0$. Ability to detect a false null hypothesis.
Probability of correctly rejecting a false $H_0$. Ability to detect a false null hypothesis.
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What is the formula for the standard error of a sample proportion?
What is the formula for the standard error of a sample proportion?
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$SE = \sqrt{\frac{p_0(1-p_0)}{n}}$. Measures variability of sample proportion under null hypothesis.
$SE = \sqrt{\frac{p_0(1-p_0)}{n}}$. Measures variability of sample proportion under null hypothesis.
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Which condition ensures the validity of the normal approximation in proportion tests?
Which condition ensures the validity of the normal approximation in proportion tests?
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Large sample size. Required for central limit theorem to apply.
Large sample size. Required for central limit theorem to apply.
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What is the relationship between power and Type II Error?
What is the relationship between power and Type II Error?
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Power = 1 - P(Type II Error). Power equals one minus beta (Type II error rate).
Power = 1 - P(Type II Error). Power equals one minus beta (Type II error rate).
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What is the formula for calculating the sample proportion?
What is the formula for calculating the sample proportion?
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$\hat{p} = \frac{x}{n}$. Successes divided by total sample size.
$\hat{p} = \frac{x}{n}$. Successes divided by total sample size.
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What is the alternative hypothesis in a two-tailed test for a population proportion?
What is the alternative hypothesis in a two-tailed test for a population proportion?
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$H_a: p \neq p_0$. Tests if population proportion differs from hypothesized value.
$H_a: p \neq p_0$. Tests if population proportion differs from hypothesized value.
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What is the p-value interpretation in hypothesis testing?
What is the p-value interpretation in hypothesis testing?
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Probability of observing a test statistic as extreme as the sample statistic. Measures strength of evidence against the null hypothesis.
Probability of observing a test statistic as extreme as the sample statistic. Measures strength of evidence against the null hypothesis.
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