Sampling for Differences in Sample Proportions - AP Statistics
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What is the critical region for a 5% significance level in a two-tailed test?
What is the critical region for a 5% significance level in a two-tailed test?
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$z \text{ less than } -1.96 \text{ or } z \text{ greater than } 1.96$. Rejection region for two-tailed test at α = 0.05.
$z \text{ less than } -1.96 \text{ or } z \text{ greater than } 1.96$. Rejection region for two-tailed test at α = 0.05.
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What is the interpretation of a 95% confidence interval in the context of proportions?
What is the interpretation of a 95% confidence interval in the context of proportions?
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95% confident interval captures the true difference. Interval contains true parameter 95% of the time.
95% confident interval captures the true difference. Interval contains true parameter 95% of the time.
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Which condition must be met for the proportion difference sampling distribution to be valid?
Which condition must be met for the proportion difference sampling distribution to be valid?
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Both samples must be random and independent. Ensures unbiased sampling distribution properties.
Both samples must be random and independent. Ensures unbiased sampling distribution properties.
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What is the significance level commonly used in hypothesis testing for differences?
What is the significance level commonly used in hypothesis testing for differences?
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0.05. Standard α level for statistical significance.
0.05. Standard α level for statistical significance.
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What is the purpose of a confidence interval in sampling distributions?
What is the purpose of a confidence interval in sampling distributions?
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Estimate the range of the difference in true proportions. Provides interval estimate for true difference.
Estimate the range of the difference in true proportions. Provides interval estimate for true difference.
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What is the requirement for sample size in a sampling distribution of proportions?
What is the requirement for sample size in a sampling distribution of proportions?
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Sample sizes must be sufficiently large. Needed for normal approximation to be valid.
Sample sizes must be sufficiently large. Needed for normal approximation to be valid.
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For a hypothesis test, what does a $p$-value less than $\text{0.05}$ indicate?
For a hypothesis test, what does a $p$-value less than $\text{0.05}$ indicate?
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Reject the null hypothesis. Strong evidence against null hypothesis at α = 0.05.
Reject the null hypothesis. Strong evidence against null hypothesis at α = 0.05.
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Which concept describes the difference between sample proportions from two independent samples?
Which concept describes the difference between sample proportions from two independent samples?
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Sampling distribution for differences in sample proportions. Distribution of $p_1 - p_2$ from repeated sampling.
Sampling distribution for differences in sample proportions. Distribution of $p_1 - p_2$ from repeated sampling.
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Identify the condition required for normal approximation in sample proportions.
Identify the condition required for normal approximation in sample proportions.
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Both $n_1p_1$, $n_1(1-p_1)$, $n_2p_2$, and $n_2(1-p_2)$ must be $\text{≥} 10$. Ensures normal approximation is valid for both samples.
Both $n_1p_1$, $n_1(1-p_1)$, $n_2p_2$, and $n_2(1-p_2)$ must be $\text{≥} 10$. Ensures normal approximation is valid for both samples.
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What is the null hypothesis when testing differences in sample proportions?
What is the null hypothesis when testing differences in sample proportions?
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$H_0: \text{P}_1 - \text{P}_2 = 0$. Tests equality of population proportions.
$H_0: \text{P}_1 - \text{P}_2 = 0$. Tests equality of population proportions.
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State the formula for the test statistic for the difference in sample proportions.
State the formula for the test statistic for the difference in sample proportions.
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$z = \frac{(\text{p}_1 - \text{p}_2) - (\text{P}_1 - \text{P}_2)}{\text{SE}}$. Standardizes the difference using standard error.
$z = \frac{(\text{p}_1 - \text{p}_2) - (\text{P}_1 - \text{P}_2)}{\text{SE}}$. Standardizes the difference using standard error.
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What condition checks the independence of samples in differences of proportions?
What condition checks the independence of samples in differences of proportions?
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Samples must be randomly selected and independent. Random sampling prevents bias and ensures validity.
Samples must be randomly selected and independent. Random sampling prevents bias and ensures validity.
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What is the critical value for a 95% confidence interval using a normal distribution?
What is the critical value for a 95% confidence interval using a normal distribution?
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1.96. Standard normal distribution 95% cutoff value.
1.96. Standard normal distribution 95% cutoff value.
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What is the alternative hypothesis for a two-tailed test on differences in sample proportions?
What is the alternative hypothesis for a two-tailed test on differences in sample proportions?
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$H_a: \text{P}_1 \neq \text{P}_2$. Tests if proportions differ in either direction.
$H_a: \text{P}_1 \neq \text{P}_2$. Tests if proportions differ in either direction.
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Define the pooled sample proportion used in hypothesis testing.
Define the pooled sample proportion used in hypothesis testing.
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$\text{p}_{\text{pooled}} = \frac{x_1 + x_2}{n_1 + n_2}$. Combines successes from both samples.
$\text{p}_{\text{pooled}} = \frac{x_1 + x_2}{n_1 + n_2}$. Combines successes from both samples.
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State the sampling distribution shape for large sample sizes in differences in proportions.
State the sampling distribution shape for large sample sizes in differences in proportions.
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Approximately normal. Central Limit Theorem applies to proportion differences.
Approximately normal. Central Limit Theorem applies to proportion differences.
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Identify the formula for calculating the confidence interval for differences in proportions.
Identify the formula for calculating the confidence interval for differences in proportions.
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$(p_1 - p_2) \text{±} z^*\text{SE}$ . Margin of error added to point estimate.
$(p_1 - p_2) \text{±} z^*\text{SE}$ . Margin of error added to point estimate.
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Find the pooled proportion if $x_1=40$, $x_2=50$, $n_1=100$, $n_2=120$.
Find the pooled proportion if $x_1=40$, $x_2=50$, $n_1=100$, $n_2=120$.
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$p_{\text{pooled}} = 0.45$. Total successes divided by total sample size.
$p_{\text{pooled}} = 0.45$. Total successes divided by total sample size.
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In hypothesis testing, what does a Type II error represent?
In hypothesis testing, what does a Type II error represent?
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Failing to reject a false null hypothesis. Missing a real difference when it exists.
Failing to reject a false null hypothesis. Missing a real difference when it exists.
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Identify the formula for calculating the $z$-score in a hypothesis test for proportions.
Identify the formula for calculating the $z$-score in a hypothesis test for proportions.
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$z = \frac{(\text{p}_1 - \text{p}_2) - \text{0}}{\text{SE}}$. Under null hypothesis, difference equals zero.
$z = \frac{(\text{p}_1 - \text{p}_2) - \text{0}}{\text{SE}}$. Under null hypothesis, difference equals zero.
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Which statistical concept allows comparison between two independent sample proportions?
Which statistical concept allows comparison between two independent sample proportions?
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Sampling distribution for differences in sample proportions. Enables hypothesis testing between two groups.
Sampling distribution for differences in sample proportions. Enables hypothesis testing between two groups.
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Identify the condition for using a normal model to approximate binomial distribution.
Identify the condition for using a normal model to approximate binomial distribution.
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Both $np \text{ and } n(1-p) \text{ ≥ } 10$. Success-failure condition for normal approximation.
Both $np \text{ and } n(1-p) \text{ ≥ } 10$. Success-failure condition for normal approximation.
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What does a $p$-value represent in hypothesis testing?
What does a $p$-value represent in hypothesis testing?
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Probability of observing data given $H_0$ is true. Strength of evidence against null hypothesis.
Probability of observing data given $H_0$ is true. Strength of evidence against null hypothesis.
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Which method is used to test the hypothesis of equal sample proportions?
Which method is used to test the hypothesis of equal sample proportions?
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Two-proportion $z$-test. Hypothesis test for equality of proportions.
Two-proportion $z$-test. Hypothesis test for equality of proportions.
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What is the parameter of interest in a difference of proportions test?
What is the parameter of interest in a difference of proportions test?
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Difference in true population proportions. What we're testing: $P_1 - P_2$.
Difference in true population proportions. What we're testing: $P_1 - P_2$.
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What is the purpose of the pooled proportion in hypothesis testing?
What is the purpose of the pooled proportion in hypothesis testing?
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Estimate the common proportion in two samples. Used when assuming equal population proportions.
Estimate the common proportion in two samples. Used when assuming equal population proportions.
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Which hypothesis test is used for comparing two sample proportions?
Which hypothesis test is used for comparing two sample proportions?
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$z$-test for proportions. Appropriate test for comparing two proportions.
$z$-test for proportions. Appropriate test for comparing two proportions.
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What is the symbol for the sample proportion in a sample?
What is the symbol for the sample proportion in a sample?
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$\text{p}$. Sample statistic estimating population proportion.
$\text{p}$. Sample statistic estimating population proportion.
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Which type of error occurs if the null hypothesis is incorrectly rejected?
Which type of error occurs if the null hypothesis is incorrectly rejected?
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Type I error. Rejecting true null hypothesis incorrectly.
Type I error. Rejecting true null hypothesis incorrectly.
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Which symbol represents the true proportion in the population for a single sample?
Which symbol represents the true proportion in the population for a single sample?
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$P$. Population parameter for true proportion.
$P$. Population parameter for true proportion.
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