AP Statistics Flashcards: Setting Up Tests For Population Proportion

What this deck covers

This deck focuses on Setting Up Tests For 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 does the symbol $\alpha$ represent in hypothesis testing?

Answer: Level of significance. The probability threshold for rejecting the null hypothesis.

Flashcard 2: What is the critical value for a 95% confidence level in a two-tailed test?

Answer: $z^* = 1.96$. Bounds the middle 95% of the standard normal distribution.

Flashcard 3: What is the meaning of a $p$-value?

Answer: Probability of observing the data if $H_0$ is true. Measures how extreme the observed data is under the null assumption.

Flashcard 4: What symbol represents the sample proportion?

Answer: $\hat{p}$. The observed proportion calculated from sample data.

Flashcard 5: What is the standard error formula for a sample proportion?

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

Flashcard 6: Which distribution is used for a population proportion hypothesis test?

Answer: Standard normal distribution. Uses $z$-distribution when normal conditions are met.

Flashcard 7: Calculate the $p$-value from a $z$-score of 2.33 in a right-tailed test.

Answer: $p$-value ≈ 0.0099. Right-tail probability from $z = 2.33$ using standard normal table.

Flashcard 8: Identify the level of significance commonly used in hypothesis testing.

Answer: $\alpha = 0.05$. Common threshold for Type I error probability.

Flashcard 9: State the null hypothesis for a test if the claim is $p > 0.3$.

Answer: $H_0: p = 0.3$. Null always states equality, regardless of the alternative claim direction.

Flashcard 10: Find the test statistic for $\hat{p} = 0.6$, $p_0 = 0.5$, $n = 100$.

Answer: $z = 2.0$. $z = \frac{0.6-0.5}{\sqrt{\frac{0.5(0.5)}{100}}} = \frac{0.1}{0.05} = 2.0$

Flashcard 11: What is the alternative hypothesis for a left-tailed test of population proportion?

Answer: $H_a: p < p_0$. Tests if the proportion is less than the null value.

Flashcard 12: What is the purpose of setting up a test for a population proportion?

Answer: To determine if there is statistical evidence against $H_0$. Evaluates whether sample data supports or contradicts a population claim.

Flashcard 13: Provide an example of a two-tailed alternative hypothesis.

Answer: $H_a: p \neq 0.5$. Tests whether proportion differs from 0.5 in either direction.

Flashcard 14: What is the sample statistic in a proportion hypothesis test?

Answer: Sample proportion $\hat{p}$. The proportion calculated from the sample data collected.

Flashcard 15: What is the formula for the test statistic in a population proportion test?

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

Flashcard 16: What does $p_0$ represent in hypothesis testing?

Answer: The hypothesized population proportion. The claimed value of the population proportion being tested.

Flashcard 17: What is the formula for the confidence interval of a population proportion?

Answer: $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$. Provides a range of plausible values for the population proportion.

Flashcard 18: Calculate the $p$-value from a $z$-score of 2.33 in a right-tailed test.

Answer: $p$-value ≈ 0.0099. Right-tail probability from $z = 2.33$ using standard normal table.

Flashcard 19: What is the formula to calculate $z$-score in hypothesis testing?

Answer: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$. Standardizes the sample proportion for hypothesis testing.

Flashcard 20: State the alternative hypothesis for a claim $p < 0.4$.

Answer: $H_a: p < 0.4$. Alternative hypothesis matches the direction of the claim being tested.

Flashcard 21: What is the population parameter of interest in a proportion test?

Answer: Population proportion $p$. The true proportion in the entire population being studied.

Flashcard 22: What does $p_0$ represent in hypothesis testing?

Answer: The hypothesized population proportion. The claimed value of the population proportion being tested.

Flashcard 23: Calculate the sample proportion if 45 out of 100 show a characteristic.

Answer: $\hat{p} = 0.45$. Sample proportion equals number with characteristic divided by total.

Flashcard 24: State the null hypothesis for a test if the claim is $p > 0.3$.

Answer: $H_0: p = 0.3$. Null always states equality, regardless of the alternative claim direction.

Flashcard 25: Determine if $np_0 = 5$ and $n(1-p_0) = 5$ meet normal conditions.

Answer: No, conditions not satisfied. Both values must be at least 10 for normal approximation validity.

Flashcard 26: What is the formula for the test statistic in a population proportion test?

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

Flashcard 27: What is the alternative hypothesis for a two-sided test of population proportion?

Answer: $H_a: p \neq p_0$. Tests if the proportion differs from the null value in either direction.

Flashcard 28: What is the formula to calculate $z$-score in hypothesis testing?

Answer: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$. Standardizes the sample proportion for hypothesis testing.

Flashcard 29: What is Type I error in hypothesis testing?

Answer: Rejecting $H_0$ when it is true. Incorrectly concluding there is an effect when none exists.

Flashcard 30: Determine if $np_0 = 5$ and $n(1-p_0) = 5$ meet normal conditions.

Answer: No, conditions not satisfied. Both values must be at least 10 for normal approximation validity.

Flashcard 31: What is the alternative hypothesis for a right-tailed test of population proportion?

Answer: $H_a: p > p_0$. Tests if the proportion is greater than the null value.

Flashcard 32: Identify the level of significance commonly used in hypothesis testing.

Answer: $\alpha = 0.05$. Common threshold for Type I error probability.

Flashcard 33: Calculate the sample proportion if 45 out of 100 show a characteristic.

Answer: $\hat{p} = 0.45$. Sample proportion equals number with characteristic divided by total.

Flashcard 34: State the alternative hypothesis for a claim $p < 0.4$.

Answer: $H_a: p < 0.4$. Alternative hypothesis matches the direction of the claim being tested.

Flashcard 35: What is the sample statistic in a proportion hypothesis test?

Answer: Sample proportion $\hat{p}$. The proportion calculated from the sample data collected.

Flashcard 36: What symbol represents the hypothesized population proportion?

Answer: $p_0$. The specific value claimed for the population proportion in $H_0$.

Flashcard 37: What is the population parameter of interest in a proportion test?

Answer: Population proportion $p$. The true proportion in the entire population being studied.

Flashcard 38: What symbol represents the sample proportion?

Answer: $\hat{p}$. The observed proportion calculated from sample data.

Flashcard 39: What decision should be made if $p$-value > $\alpha$?

Answer: Fail to reject the null hypothesis. Insufficient evidence to conclude the null hypothesis is false.

Flashcard 40: What is the decision rule for a hypothesis test?

Answer: Reject $H_0$ if $p$-value < $\alpha$. Compare probability to significance level to make conclusion.

Flashcard 41: What is the null hypothesis for a test of population proportion?

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

Flashcard 42: What is the condition for a normal approximation to be valid in a proportion test?

Answer: $np_0 \geq 10$ and $n(1-p_0) \geq 10$. Ensures at least 10 successes and 10 failures for normal approximation.

Flashcard 43: What is Type II error in hypothesis testing?

Answer: Failing to reject $H_0$ when it is false. Missing a real effect that actually exists in the population.

Flashcard 44: What is the critical value for a 90% confidence level in a two-tailed test?

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

Flashcard 45: Choose the correct test if $H_a: p \neq 0.25$.

Answer: Two-tailed test. Not-equal alternative requires testing both tails of the distribution.

Flashcard 46: What is Type II error in hypothesis testing?

Answer: Failing to reject $H_0$ when it is false. Missing a real effect that actually exists in the population.

Flashcard 47: Find $z$-score for $\hat{p} = 0.55$, $p_0 = 0.5$, $n = 200$.

Answer: $z = 1.414$. $z = \frac{0.55-0.5}{\sqrt{\frac{0.5(0.5)}{200}}} = \frac{0.05}{\sqrt{0.00125}} = 1.414$

Flashcard 48: What does the symbol $\alpha$ represent in hypothesis testing?

Answer: Level of significance. The probability threshold for rejecting the null hypothesis.

Flashcard 49: Find the $z$-score if $\hat{p} = 0.62$, $p_0 = 0.5$, $n = 150$.

Answer: $z = 2.449$. $z = \frac{0.62-0.5}{\sqrt{\frac{0.5(0.5)}{150}}} = \frac{0.12}{0.0408} = 2.449$

Flashcard 50: Provide an example of a two-tailed alternative hypothesis.

Answer: $H_a: p \neq 0.5$. Tests whether proportion differs from 0.5 in either direction.

Flashcard 51: What is the critical value for a 95% confidence level in a two-tailed test?

Answer: $z^* = 1.96$. Bounds the middle 95% of the standard normal distribution.

Flashcard 52: What does a $p$-value less than $\alpha$ indicate?

Answer: Reject the null hypothesis. Strong evidence against $H_0$ when probability is below threshold.

Flashcard 53: Find and correct the error: Null hypothesis is $H_0: \mu = p_0$.

Answer: Correct: $H_0: p = p_0$. Null hypothesis should use $p$ for proportion, not $\mu$ for mean.

Flashcard 54: Find $z$-score for $\hat{p} = 0.55$, $p_0 = 0.5$, $n = 200$.

Answer: $z = 1.414$. $z = \frac{0.55-0.5}{\sqrt{\frac{0.5(0.5)}{200}}} = \frac{0.05}{\sqrt{0.00125}} = 1.414$

Flashcard 55: What is Type I error in hypothesis testing?

Answer: Rejecting $H_0$ when it is true. Incorrectly concluding there is an effect when none exists.

Flashcard 56: Calculate standard error for $\hat{p} = 0.7$, $n = 100$.

Answer: $0.0458$. $SE = \sqrt{\frac{0.7(1-0.7)}{100}} = \sqrt{\frac{0.21}{100}} = 0.0458$