AP Statistics Reference Guide: Confidence Intervals and Significance Tests

One-Sample and Two-Sample Confidence Intervals for Proportions

• One-Sample z-Interval for a Proportion     * Statistic: Represents the sample proportion, denoted as $\hat{p}$.     * Parameter: Represents the population proportion, denoted as $p$.     * Conditions for Inference:         * Randomness: The data must come from a random sample.         * Independence (10% Rule): The sample size $n$ must be less than or equal to $10\%$ of the population size ($n \le 10\%N$).         * Large Counts: The number of successes and failures must both be at least 10, specifically $n\hat{p} \ge 10$ and $n(1 - \hat{p}) \ge 10$.     * Formula: $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$     * Calculator Command: 1-PropZInt

• Two-Sample z-Interval for a Difference in Proportions     * Statistic: The difference between two sample proportions, denoted as $\hat{p}_1 - \hat{p}_2$.     * Parameter: The difference between two population proportions, denoted as $p_1 - p_2$.     * Conditions for Inference:         * Randomness/Independence: Requires independent random samples or a randomized experiment.         * Independence (10% Rule): For both samples, the size must be less than or equal to $10\%$ of their respective populations ($n_1 \le 10\%N_1$ and $n_2 \le 10\%N_2$).         * Large Counts: Successes and failures for both groups must be at least 10: $n_1\hat{p}_1 \ge 10$, $n_1(1 - \hat{p}_1) \ge 10$, $n_2\hat{p}_2 \ge 10$, and $n_2(1 - \hat{p}_2) \ge 10$.     * Formula: $(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}$     * Calculator Command: 2-PropZInt

Confidence Intervals for Means and Slope

• One-Sample t-Interval for a Mean (Including Paired t-Interval)     * Statistic: The sample mean, denoted as $\bar{x}$.     * Parameter: The population mean, denoted as $\mu$.     * Conditions for Inference:         * Randomness: Data must come from a random sample or a randomized experiment.         * Independence (10% Rule): Sample size must satisfy $n \le 10\%N$.         * Normality/Large Sample: The population distribution must be approximately normal (either given by the problem or sample data must show no strong skew or outliers) OR the sample size must be at least 30 ($n \ge 30$).     * Formula: $\bar{x} \pm t^* \frac{s}{\sqrt{n}}$     * Degrees of Freedom ($df$): $df = n - 1$     * Calculator Command: TInterval

• Two-Sample t-Interval for a Difference in Means     * Statistic: The difference between two sample means, denoted as $\bar{x}_1 - \bar{x}_2$.     * Parameter: The difference between two population means, denoted as $\mu_1 - \mu_2$.     * Conditions for Inference:         * Randomness/Independence: Independent random samples or a randomized experiment.         * Independence (10% Rule): For both groups, $n_1 \le 10\%N_1$ and $n_2 \le 10\%N_2$.         * Normality/Large Sample: For each group, the population distribution must be approximately normal (given or sample data shows no strong skew/outliers) OR each group's sample size must be at least 30 ($n \ge 30$).     * Formula: $(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$     * Degrees of Freedom ($df$): Conservatively calculated as the smaller of $n_1 - 1$ and $n_2 - 1$, or determined more accurately via technology.     * Calculator Command: 2-SampTInt

• t-Interval for a Slope     * Statistic: The sample slope, denoted as $b$.     * Parameter: The population slope, denoted as $\beta$.     * Conditions for Inference:         * Linearity: The relationship between $x$ and $y$ must be fairly linear.         * Independence (10% Rule): Sample size must satisfy $n \le 10\%N$.         * Normality: For each value of $x$, the distribution of $y$ must be approximately normal.         * Equal Variance: For each value of $x$, the variable $y$ must have the same standard deviation.         * Randomness: Data must come from a random sample or randomized experiment.     * Formula: $b \pm t^* SE_b$     * Degrees of Freedom ($df$): $df = n - 2$     * Calculator Command: LinRegTInt

Significance Tests for Proportions

• One-Sample z-Test for a Proportion     * Null Hypothesis ($H_0$): $H_0: p = p_0$     * Conditions for Inference:         * Randomness: Random sample.         * Independence (10% Rule): $n \le 10\%N$.         * Large Counts: Based on the null value ($p_0$), $np_0 \ge 10$ and $n(1 - p_0) \ge 10$.     * Test Statistic Formula: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$     * Calculator Command: 1-PropZTest

• Two-Sample z-Test for a Difference in Proportions     * Null Hypothesis ($H_0$): $H_0: p_1 - p_2 = 0$     * Conditions for Inference:         * Randomness/Independence: Independent random samples or localized randomized experiment.         * Independence (10% Rule): $n_1 \le 10\%N_1$ and $n_2 \le 10\%N_2$.         * Large Counts: Based on the pooled proportion $\hat{p}_c$, where $\hat{p}_c = \frac{x_1 + x_2}{n_1 + n_2}$. The conditions are: $n_1\hat{p}_c \ge 10$, $n_1(1 - \hat{p}_c) \ge 10$, $n_2\hat{p}_c \ge 10$, and $n_2(1 - \hat{p}_c) \ge 10$.     * Test Statistic Formula: $z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\frac{\hat{p}_c(1 - \hat{p}_c)}{n_1} + \frac{\hat{p}_c(1 - \hat{p}_c)}{n_2}}}$     * Calculator Command: 2-PropZTest

Significance Tests for Means and Slope

• One-Sample t-Test for a Mean (Including Paired t-Test)     * Null Hypothesis ($H_0$): $H_0: \mu = \mu_0$     * Conditions for Inference:         * Randomness: Random sample or randomized experiment.         * Independence (10% Rule): $n \le 10\%N$.         * Normality/Large Sample: Population is approximately normal (given or no skew/outliers in sample data) OR $n \ge 30$.     * Test Statistic Formula: $t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$     * Degrees of Freedom ($df$): $df = n - 1$     * Calculator Command: T-Test

• Two-Sample t-Test for a Difference in Means     * Null Hypothesis ($H_0$): $H_0: \mu_1 - \mu_2 = 0$     * Conditions for Inference:         * Randomness/Independence: Independent random samples or randomized experiment.         * Independence (10% Rule): $n_1 \le 10\%N_1$ and $n_2 \le 10\%N_2$.         * Normality/Large Sample: For each group, the population is approximately normal (given or no skew/outliers) OR $n \ge 30$.     * Test Statistic Formula: $t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$     * Degrees of Freedom ($df$): Smaller of $n_1 - 1$ and $n_2 - 1$, or calculated by technology.     * Calculator Command: 2-SampTTest

• t-Test for a Slope     * Null Hypothesis ($H_0$): $H_0: \beta = \beta_0$     * Conditions for Inference:         * Linearity: Relationship between $x$ and $y$ is linear.         * Independence (10% Rule): $n \le 10\%N$.         * Normality: Y-distribution is approximately normal for each $x$.         * Equal Variance: Constant standard deviation of $y$ for all $x$.         * Randomness: Random sample or experiment.     * Test Statistic Formula: $t = \frac{b - \beta_0}{SE_b}$     * Degrees of Freedom ($df$): $df = n - 2$     * Calculator Command: LinRegTTest

Chi-Square tests

• Chi-Square ($\chi^2$) Test for Goodness-of-Fit     * Hypotheses:         * Null Hypothesis ($H_0$): The claimed distribution of the categorical variable is correct.         * Alternative Hypothesis ($H_a$): The claimed distribution of the categorical variable is incorrect.     * Conditions for Inference:         * Randomness: Data comes from a random sample or randomized experiment.         * Independence (10% Rule): $n \le 10\%N$.         * Expected Counts: All expected counts must be greater than 5.     * Formula: $\chi^2 = \sum \frac{(\text{observed} - \text{expected})^2}{\text{expected}}$     * Degrees of Freedom ($df$): $df = \text{number of categories} - 1$     * Calculator Command: χ²GOF-Test

• Chi-Square ($\chi^2$) Test for Homogeneity     * Hypotheses:         * Null Hypothesis ($H_0$): There is no difference in the distribution of the categorical variable across populations or treatments.         * Alternative Hypothesis ($H_a$): There is a difference in the distribution of the categorical variable across populations or treatments.     * Conditions for Inference:         * Randomness: Random samples from each population or a randomized experiment.         * Independence (10% Rule): $n \le 10\%N$.         * Expected Counts: All expected counts must be greater than 5.     * Formula: $\chi^2 = \sum \frac{(\text{observed} - \text{expected})^2}{\text{expected}}$     * Degrees of Freedom ($df$): $df = (\text{number of rows} - 1) \times (\text{number of columns} - 1)$     * Calculator Command: χ²-Test

• Chi-Square ($\chi^2$) Test for Independence     * Hypotheses:         * Null Hypothesis ($H_0$): There is no association between two categorical variables in a given population (i.e., the variables are independent).         * Alternative Hypothesis ($H_a$): Two categorical variables in a population are associated (i.e., the variables are dependent).     * Conditions for Inference:         * Randomness: Data comes from a random sample or randomized experiment.         * Independence (10% Rule): $n \le 10\%N$.         * Expected Counts: All expected counts must be greater than 5.     * Formula: $\chi^2 = \sum \frac{(\text{observed} - \text{expected})^2}{\text{expected}}$     * Degrees of Freedom ($df$): $df = (\text{number of rows} - 1) \times (\text{number of columns} - 1)$     * Calculator Command: χ²-Test matches the homogeneity command.