Confidence Interval for a Population Proportion - AP Statistics
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What is the typical confidence level used in constructing confidence intervals?
What is the typical confidence level used in constructing confidence intervals?
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95%. Most commonly used standard in statistical practice.
95%. Most commonly used standard in statistical practice.
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What is the formula for the confidence interval of a population proportion?
What is the formula for the confidence interval of a population proportion?
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$\bar{p} , \pm , z^* , \sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$. Standard formula using sample proportion, critical value, and standard error.
$\bar{p} , \pm , z^* , \sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$. Standard formula using sample proportion, critical value, and standard error.
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Define the term 'population proportion'.
Define the term 'population proportion'.
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The fraction or percentage of the population with a particular characteristic. Represents the parameter $p$ we're trying to estimate with confidence intervals.
The fraction or percentage of the population with a particular characteristic. Represents the parameter $p$ we're trying to estimate with confidence intervals.
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What does $\bar{p}$ represent in a confidence interval formula?
What does $\bar{p}$ represent in a confidence interval formula?
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Sample proportion. The point estimator for the population proportion $p$.
Sample proportion. The point estimator for the population proportion $p$.
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What does $z^*$ represent in the confidence interval formula?
What does $z^*$ represent in the confidence interval formula?
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The critical value from the standard normal distribution. Corresponds to the desired confidence level (e.g., 1.96 for 95%).
The critical value from the standard normal distribution. Corresponds to the desired confidence level (e.g., 1.96 for 95%).
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Find and correct the error: $CI = \bar{p} , \pm , z^* , \sqrt{\frac{1-\bar{p}}{n}}$.
Find and correct the error: $CI = \bar{p} , \pm , z^* , \sqrt{\frac{1-\bar{p}}{n}}$.
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$CI = \bar{p} , \pm , z^* , \sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$. Missing $\bar{p}$ term in the numerator of the standard error formula.
$CI = \bar{p} , \pm , z^* , \sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$. Missing $\bar{p}$ term in the numerator of the standard error formula.
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What is the minimum sample size needed for a valid confidence interval?
What is the minimum sample size needed for a valid confidence interval?
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Both $np , \geq , 10$ and $n(1-p) , \geq , 10$. Ensures normal approximation is valid for both successes and failures.
Both $np , \geq , 10$ and $n(1-p) , \geq , 10$. Ensures normal approximation is valid for both successes and failures.
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Which statistical distribution is used for the critical value in proportion CI?
Which statistical distribution is used for the critical value in proportion CI?
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Standard normal distribution. Used because sample proportions are approximately normally distributed.
Standard normal distribution. Used because sample proportions are approximately normally distributed.
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State the margin of error formula for a population proportion CI.
State the margin of error formula for a population proportion CI.
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$z^* , \sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$. Half-width of the confidence interval around $\bar{p}$.
$z^* , \sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$. Half-width of the confidence interval around $\bar{p}$.
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Calculate the margin of error for $\bar{p} = 0.6$, $n = 100$, $z^* = 1.96$.
Calculate the margin of error for $\bar{p} = 0.6$, $n = 100$, $z^* = 1.96$.
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0.096. Using formula: $1.96 \times \sqrt{\frac{0.6(0.4)}{100}} = 0.096$.
0.096. Using formula: $1.96 \times \sqrt{\frac{0.6(0.4)}{100}} = 0.096$.
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What is the interpretation of a 95% confidence interval?
What is the interpretation of a 95% confidence interval?
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95% confident the true proportion is within the interval. Describes the long-run capture rate of the true parameter.
95% confident the true proportion is within the interval. Describes the long-run capture rate of the true parameter.
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What assumptions must be met to construct a CI for a proportion?
What assumptions must be met to construct a CI for a proportion?
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Random sample and $np, n(1-p) , \geq , 10$. Required for valid normal approximation to the sampling distribution.
Random sample and $np, n(1-p) , \geq , 10$. Required for valid normal approximation to the sampling distribution.
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State the effect of a higher confidence level on CI width.
State the effect of a higher confidence level on CI width.
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Increases the width of the CI. Higher confidence requires larger critical value and margin of error.
Increases the width of the CI. Higher confidence requires larger critical value and margin of error.
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What is the critical value for an 80% confidence level?
What is the critical value for an 80% confidence level?
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$z^* = 1.282$. Captures the middle 80% of the standard normal distribution.
$z^* = 1.282$. Captures the middle 80% of the standard normal distribution.
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Identify the effect of a narrower confidence interval on precision.
Identify the effect of a narrower confidence interval on precision.
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Increases precision. Narrower intervals provide more precise estimates of the parameter.
Increases precision. Narrower intervals provide more precise estimates of the parameter.
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How does variability in data affect the CI width?
How does variability in data affect the CI width?
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Greater variability increases CI width. Higher variability increases uncertainty, requiring wider intervals.
Greater variability increases CI width. Higher variability increases uncertainty, requiring wider intervals.
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What is the standard error formula for a population proportion?
What is the standard error formula for a population proportion?
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$\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$. Measures the sampling variability of the sample proportion.
$\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$. Measures the sampling variability of the sample proportion.
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What is the relationship between CI width and variability?
What is the relationship between CI width and variability?
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Higher variability increases CI width. More variability requires wider intervals to maintain confidence level.
Higher variability increases CI width. More variability requires wider intervals to maintain confidence level.
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What mathematical operation is used to adjust the sample proportion in CI?
What mathematical operation is used to adjust the sample proportion in CI?
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Addition and subtraction. The $\pm$ creates the interval bounds around the point estimate.
Addition and subtraction. The $\pm$ creates the interval bounds around the point estimate.
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What is the effect of a larger sample size on the standard error?
What is the effect of a larger sample size on the standard error?
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Decreases the standard error. Standard error is inversely proportional to $\sqrt{n}$.
Decreases the standard error. Standard error is inversely proportional to $\sqrt{n}$.
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How does the confidence level affect the critical value?
How does the confidence level affect the critical value?
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Higher confidence level increases critical value. More confidence requires larger $z^*$ to capture more area.
Higher confidence level increases critical value. More confidence requires larger $z^*$ to capture more area.
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What is the sample proportion for 80 successes in 200 trials?
What is the sample proportion for 80 successes in 200 trials?
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$\bar{p} = 0.4$. Calculated as $\frac{80}{200} = 0.4$.
$\bar{p} = 0.4$. Calculated as $\frac{80}{200} = 0.4$.
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Calculate the 90% CI for $\bar{p} = 0.75$, $n = 150$.
Calculate the 90% CI for $\bar{p} = 0.75$, $n = 150$.
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$(0.693, 0.807)$. Using $0.75 \pm 1.645\sqrt{\frac{0.75(0.25)}{150}} = (0.693, 0.807)$.
$(0.693, 0.807)$. Using $0.75 \pm 1.645\sqrt{\frac{0.75(0.25)}{150}} = (0.693, 0.807)$.
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State the z-score associated with a 90% confidence interval.
State the z-score associated with a 90% confidence interval.
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$z^* = 1.645$. Captures the middle 90% of the standard normal distribution.
$z^* = 1.645$. Captures the middle 90% of the standard normal distribution.
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What is the critical value for a 99% confidence level?
What is the critical value for a 99% confidence level?
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$z^* = 2.576$. Captures the middle 99% of the standard normal distribution.
$z^* = 2.576$. Captures the middle 99% of the standard normal distribution.
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Calculate the 95% CI for $\bar{p} = 0.45$, $n = 200$.
Calculate the 95% CI for $\bar{p} = 0.45$, $n = 200$.
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$(0.385, 0.515)$. Using $0.45 \pm 1.96\sqrt{\frac{0.45(0.55)}{200}} = (0.385, 0.515)$.
$(0.385, 0.515)$. Using $0.45 \pm 1.96\sqrt{\frac{0.45(0.55)}{200}} = (0.385, 0.515)$.
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What happens to the CI if the confidence level is increased?
What happens to the CI if the confidence level is increased?
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The CI becomes wider. Higher confidence requires larger critical value, increasing margin of error.
The CI becomes wider. Higher confidence requires larger critical value, increasing margin of error.
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What is the effect of increasing sample size on CI width?
What is the effect of increasing sample size on CI width?
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Decreases CI width. Larger $n$ reduces standard error, making the interval narrower.
Decreases CI width. Larger $n$ reduces standard error, making the interval narrower.
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Identify the sample proportion if 30 out of 100 are successful.
Identify the sample proportion if 30 out of 100 are successful.
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$\bar{p} = 0.3$. Sample proportion equals number of successes divided by sample size.
$\bar{p} = 0.3$. Sample proportion equals number of successes divided by sample size.
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State the z-score associated with a 90% confidence interval.
State the z-score associated with a 90% confidence interval.
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$z^* = 1.645$. Captures the middle 90% of the standard normal distribution.
$z^* = 1.645$. Captures the middle 90% of the standard normal distribution.
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