AP Statistics Flashcards: Confidence Interval For A Population Proportion

What this deck covers

This deck focuses on Confidence Interval For A 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: What is the effect of increasing sample size on CI width?

Answer: Decreases CI width. Larger $n$ reduces standard error, making the interval narrower.

Flashcard 2: Calculate the margin of error for $\bar{p} = 0.6$, $n = 100$, $z^* = 1.96$.

Answer: 0.096. Using formula: $1.96 \times \sqrt{\frac{0.6(0.4)}{100}} = 0.096$.

Flashcard 3: Calculate the 95% CI for $\bar{p} = 0.45$, $n = 200$.

Answer: $(0.385, 0.515)$. Using $0.45 \pm 1.96\sqrt{\frac{0.45(0.55)}{200}} = (0.385, 0.515)$.

Flashcard 4: What is the typical confidence level used in constructing confidence intervals?

Answer: 95%. Most commonly used standard in statistical practice.

Flashcard 5: What is the sample proportion for 80 successes in 200 trials?

Answer: $\bar{p} = 0.4$. Calculated as $\frac{80}{200} = 0.4$.

Flashcard 6: What happens to the CI if the confidence level is increased?

Answer: The CI becomes wider. Higher confidence requires larger critical value, increasing margin of error.

Flashcard 7: Calculate the 90% CI for $\bar{p} = 0.75$, $n = 150$.

Answer: $(0.693, 0.807)$. Using $0.75 \pm 1.645\sqrt{\frac{0.75(0.25)}{150}} = (0.693, 0.807)$.

Flashcard 8: What is the effect of a larger sample size on the standard error?

Answer: Decreases the standard error. Standard error is inversely proportional to $\sqrt{n}$.

Flashcard 9: Identify the margin of error when $z^* = 1.96$ and SE = 0.05.

Answer: 0.098. Margin of error equals $z^* \times SE = 1.96 \times 0.05 = 0.098$.

Flashcard 10: What is the critical value for an 80% confidence level?

Answer: $z^* = 1.282$. Captures the middle 80% of the standard normal distribution.

Flashcard 11: State the margin of error formula for a population proportion CI.

Answer: $z^* \, \sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$. Half-width of the confidence interval around $\bar{p}$.

Flashcard 12: Find the standard error for $\bar{p} = 0.5$, $n = 250$.

Answer: $0.0316$. Using $\sqrt{\frac{0.5(0.5)}{250}} = 0.0316$.

Flashcard 13: Find and correct the error: $CI = \bar{p} \, \pm \, z^* \, \sqrt{\frac{1-\bar{p}}{n}}$.

Answer: $CI = \bar{p} \, \pm \, z^* \, \sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$. Missing $\bar{p}$ term in the numerator of the standard error formula.

Flashcard 14: What is the impact of a small sample size on confidence intervals?

Answer: Results in wider confidence intervals. Small $n$ increases standard error, reducing precision.

Flashcard 15: What is the formula for the confidence interval of a population proportion?

Answer: $\bar{p} \, \pm \, z^* \, \sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$. Standard formula using sample proportion, critical value, and standard error.

Flashcard 16: What is the minimum sample size needed for a valid confidence interval?

Answer: Both $np \, \geq \, 10$ and $n(1-p) \, \geq \, 10$. Ensures normal approximation is valid for both successes and failures.

Flashcard 17: What is the standard error formula for a population proportion?

Answer: $\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$. Measures the sampling variability of the sample proportion.

Flashcard 18: Find the standard error for $\bar{p} = 0.5$, $n = 250$.

Answer: $0.0316$. Using $\sqrt{\frac{0.5(0.5)}{250}} = 0.0316$.

Flashcard 19: Identify the sample proportion if 30 out of 100 are successful.

Answer: $\bar{p} = 0.3$. Sample proportion equals number of successes divided by sample size.

Flashcard 20: Identify the effect of a narrower confidence interval on precision.

Answer: Increases precision. Narrower intervals provide more precise estimates of the parameter.

Flashcard 21: What is the sample proportion for 80 successes in 200 trials?

Answer: $\bar{p} = 0.4$. Calculated as $\frac{80}{200} = 0.4$.

Flashcard 22: State the z-score associated with a 90% confidence interval.

Answer: $z^* = 1.645$. Captures the middle 90% of the standard normal distribution.

Flashcard 23: State the z-score associated with a 90% confidence interval.

Answer: $z^* = 1.645$. Captures the middle 90% of the standard normal distribution.

Flashcard 24: What is the relationship between CI width and variability?

Answer: Higher variability increases CI width. More variability requires wider intervals to maintain confidence level.

Flashcard 25: What is the effect of a larger sample size on the standard error?

Answer: Decreases the standard error. Standard error is inversely proportional to $\sqrt{n}$.

Flashcard 26: What is the relationship between CI width and variability?

Answer: Higher variability increases CI width. More variability requires wider intervals to maintain confidence level.

Flashcard 27: Identify the effect of a narrower confidence interval on precision.

Answer: Increases precision. Narrower intervals provide more precise estimates of the parameter.

Flashcard 28: What is the interpretation of a 95% confidence interval?

Answer: 95% confident the true proportion is within the interval. Describes the long-run capture rate of the true parameter.

Flashcard 29: State the margin of error formula for a population proportion CI.

Answer: $z^* \, \sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$. Half-width of the confidence interval around $\bar{p}$.

Flashcard 30: What mathematical operation is used to adjust the sample proportion in CI?

Answer: Addition and subtraction. The $\pm$ creates the interval bounds around the point estimate.

Flashcard 31: What is the effect of increasing sample size on CI width?

Answer: Decreases CI width. Larger $n$ reduces standard error, making the interval narrower.

Flashcard 32: How does variability in data affect the CI width?

Answer: Greater variability increases CI width. Higher variability increases uncertainty, requiring wider intervals.

Flashcard 33: Which statistical distribution is used for the critical value in proportion CI?

Answer: Standard normal distribution. Used because sample proportions are approximately normally distributed.

Flashcard 34: What is the interpretation of a 95% confidence interval?

Answer: 95% confident the true proportion is within the interval. Describes the long-run capture rate of the true parameter.

Flashcard 35: State the effect of a higher confidence level on CI width.

Answer: Increases the width of the CI. Higher confidence requires larger critical value and margin of error.

Flashcard 36: What is the critical value for a 99% confidence level?

Answer: $z^* = 2.576$. Captures the middle 99% of the standard normal distribution.

Flashcard 37: Find and correct the error: $CI = \bar{p} \, \pm \, z^* \, \sqrt{\frac{1-\bar{p}}{n}}$.

Answer: $CI = \bar{p} \, \pm \, z^* \, \sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$. Missing $\bar{p}$ term in the numerator of the standard error formula.

Flashcard 38: Calculate the 95% CI for $\bar{p} = 0.45$, $n = 200$.

Answer: $(0.385, 0.515)$. Using $0.45 \pm 1.96\sqrt{\frac{0.45(0.55)}{200}} = (0.385, 0.515)$.

Flashcard 39: State the effect of a higher confidence level on CI width.

Answer: Increases the width of the CI. Higher confidence requires larger critical value and margin of error.

Flashcard 40: What is the impact of a small sample size on confidence intervals?

Answer: Results in wider confidence intervals. Small $n$ increases standard error, reducing precision.

Flashcard 41: What assumptions must be met to construct a CI for a proportion?

Answer: Random sample and $np, n(1-p) \, \geq \, 10$. Required for valid normal approximation to the sampling distribution.

Flashcard 42: How does variability in data affect the CI width?

Answer: Greater variability increases CI width. Higher variability increases uncertainty, requiring wider intervals.

Flashcard 43: What does $z^*$ represent in the confidence interval formula?

Answer: The critical value from the standard normal distribution. Corresponds to the desired confidence level (e.g., 1.96 for 95%).

Flashcard 44: What does $z^*$ represent in the confidence interval formula?

Answer: The critical value from the standard normal distribution. Corresponds to the desired confidence level (e.g., 1.96 for 95%).

Flashcard 45: What happens to the CI if the confidence level is increased?

Answer: The CI becomes wider. Higher confidence requires larger critical value, increasing margin of error.

Flashcard 46: What is the critical value for an 80% confidence level?

Answer: $z^* = 1.282$. Captures the middle 80% of the standard normal distribution.

Flashcard 47: Define the term 'population proportion'.

Answer: The fraction or percentage of the population with a particular characteristic. Represents the parameter $p$ we're trying to estimate with confidence intervals.

Flashcard 48: What is the standard error formula for a population proportion?

Answer: $\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$. Measures the sampling variability of the sample proportion.

Flashcard 49: Calculate the margin of error for $\bar{p} = 0.6$, $n = 100$, $z^* = 1.96$.

Answer: 0.096. Using formula: $1.96 \times \sqrt{\frac{0.6(0.4)}{100}} = 0.096$.

Flashcard 50: What mathematical operation is used to adjust the sample proportion in CI?

Answer: Addition and subtraction. The $\pm$ creates the interval bounds around the point estimate.

Flashcard 51: What does $\bar{p}$ represent in a confidence interval formula?

Answer: Sample proportion. The point estimator for the population proportion $p$.

Flashcard 52: Identify the sample proportion if 30 out of 100 are successful.

Answer: $\bar{p} = 0.3$. Sample proportion equals number of successes divided by sample size.

Flashcard 53: What is the formula for the confidence interval of a population proportion?

Answer: $\bar{p} \, \pm \, z^* \, \sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$. Standard formula using sample proportion, critical value, and standard error.

Flashcard 54: What is the typical confidence level used in constructing confidence intervals?

Answer: 95%. Most commonly used standard in statistical practice.

Flashcard 55: What assumptions must be met to construct a CI for a proportion?

Answer: Random sample and $np, n(1-p) \, \geq \, 10$. Required for valid normal approximation to the sampling distribution.

Flashcard 56: Calculate the 90% CI for $\bar{p} = 0.75$, $n = 150$.

Answer: $(0.693, 0.807)$. Using $0.75 \pm 1.645\sqrt{\frac{0.75(0.25)}{150}} = (0.693, 0.807)$.

Flashcard 57: Identify the margin of error when $z^* = 1.96$ and SE = 0.05.

Answer: 0.098. Margin of error equals $z^* \times SE = 1.96 \times 0.05 = 0.098$.

Flashcard 58: What does $\bar{p}$ represent in a confidence interval formula?

Answer: Sample proportion. The point estimator for the population proportion $p$.

Flashcard 59: Define the term 'population proportion'.

Answer: The fraction or percentage of the population with a particular characteristic. Represents the parameter $p$ we're trying to estimate with confidence intervals.

Flashcard 60: Which statistical distribution is used for the critical value in proportion CI?

Answer: Standard normal distribution. Used because sample proportions are approximately normally distributed.

Flashcard 61: How does the confidence level affect the critical value?

Answer: Higher confidence level increases critical value. More confidence requires larger $z^*$ to capture more area.

Flashcard 62: What is the critical value for a 99% confidence level?

Answer: $z^* = 2.576$. Captures the middle 99% of the standard normal distribution.

Flashcard 63: How does the confidence level affect the critical value?

Answer: Higher confidence level increases critical value. More confidence requires larger $z^*$ to capture more area.