Write the formulas for the Mean (Expected Value $\mu_X$) and Standard Deviation ($\sigma_X$) of a Discrete Random Variable $X$, and state how you must interpret $\mu_X$ in context.

• **Mean (Expected Value)**: $\mu_X = E(X) = \sum x_i P(x_i)$ (Note: $\mu_X$ does NOT have to be a possible integer value of $X$!). • **Standard Deviation**: $\sigma_X = \sqrt{\sum (x_i - \mu_X)^2 P(x_i)}$ • **Required AP Interpretation of $\mu_X$**: *'If we repeated [random process] **many, many times** (in the long run), the **average** [variable $X$, in context] would be about $\mu_X$ [units].'*

State the general structure of a Confidence Interval and the exact AP Statistics templates for interpreting (a) a Confidence Interval and (b) a Confidence Level ($C\%$).

• **General Formula**: $\text{Point Estimate} \pm \text{Margin of Error} = \text{Statistic} \pm (\text{Critical Value}) \times (\text{Standard Error})$. • **(a) Interpreting a Confidence Interval $(L, U)$**: *'We are [$C\%$] confident that the interval from $L$ to $U$ captures the **true population [parameter $\mu$ or $p$, in context]**.'* (**Never** mention the sample statistic $\bar{x}$ or $\hat{p}$, and **never** say 'there is a $95\%$ probability the parameter is in this interval'!). • **(b) Interpreting the Confidence Level ($C\%$)**: *'If we took many, many random samples of the same size $n$ from this population and constructed a [$C\%$] confidence interval from each sample, **about [$C\%$] of the resulting intervals would capture the true population [parameter, in context]**.'*

Define Type I Error, Type II Error, and the Power of a Test ($1 - \beta$), and state what increases the Power of a Hypothesis Test.

• **Type I Error (False Positive, probability $= \alpha$)**: **Rejecting $H_0$ when $H_0$ is actually TRUE**. • **Type II Error (False Negative, probability $= \beta$)**: **Failing to reject $H_0$ when $H_0$ is actually FALSE ($H_a$ is true)**. • **Power of a Test ($\text{Power} = 1 - \beta$)**: The probability that the test correctly **rejects a false $H_0$** when a specific alternative parameter value is true. • **Four Ways to Increase Power (and Decrease $\beta$)**: 1. **Increase sample size $n$** (reduces standard error; reduces BOTH $\beta$ and variability!). 2. **Increase significance level $\alpha$** (e.g., from $0.01$ to $0.05$; makes it easier to reject $H_0$, so $\alpha \uparrow \implies \beta \downarrow, \text{Power} \uparrow$). 3. **Larger effect size** (true parameter is farther from the null value). 4. **Decrease standard deviation $\sigma$** (reduce measurement error).

When choosing a significance level $\alpha$ ($0.01$ vs. $0.10$) before a test, how do you decide based on the real-world consequences of a Type I Error vs. a Type II Error?

• Choose a **SMALLER $\alpha$ (e.g., $\alpha = 0.01$)** when the consequence of a **Type I Error** (falsely rejecting $H_0$) is more severe (e.g., approving an expensive new drug with toxic side effects when it actually doesn't work). • Choose a **LARGER $\alpha$ (e.g., $\alpha = 0.10$)** when the consequence of a **Type II Error** (failing to detect a real problem) is more dangerous (e.g., testing drinking water for lead contamination where $H_0: \text{water is safe}$ and $H_a: \text{water is unsafe}$).

How do you solve a Tree Diagram / Reverse Conditional Probability (Bayes' Theorem) problem (such as a medical diagnostic test with false positives)?

Use **Bayes' Theorem** (or a 2x2 table with a hypothetical population of $10,000$): $P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{P(A)P(B \mid A)}{P(A)P(B \mid A) + P(A^c)P(B \mid A^c)}$ For example, if $1\%$ of a population has a disease ($P(D)=0.01$), a test is $95\%$ sensitive ($P(+ \mid D)=0.95$), and has a $4\%$ false-positive rate ($P(+ \mid D^c)=0.04$), then: $P(D \mid +) = \frac{(0.01)(0.95)}{(0.01)(0.95) + (0.99)(0.04)} = \frac{0.0095}{0.0491} \approx 0.1935$

In Combining Random Variables, what is the critical difference between $3X$ (multiplying a single observation by $3$, e.g., converting yards to feet) and $X_1 + X_2 + X_3$ (summing $3$ independent observations)?

Both have the exact same mean ($3\mu_X$), but their standard deviations differ! • **Linear Rescaling ($3X$)**: $\sigma_{3X} = \sqrt{3^2 \sigma_X^2} = \mathbf{3\sigma_X}$. • **Sum of $3$ Independent Observations ($X_1 + X_2 + X_3$)**: $\sigma_{X_1 + X_2 + X_3} = \sqrt{\sigma_X^2 + \sigma_X^2 + \sigma_X^2} = \mathbf{\sqrt{3}\,\sigma_X}$

When you Combine two Independent Random Variables ($W = aX \pm bY$), what are the Mean $\mu_W$, Variance $\sigma_W^2$, and Standard Deviation $\sigma_W$? ('Variances ALWAYS Add!')

For any two random variables $X$ and $Y$: • **Mean**: $\mu_{aX + bY} = a\mu_X + b\mu_Y$ and $\mu_{aX - bY} = a\mu_X - b\mu_Y$. • **Variance & Standard Deviation** (ONLY if $X$ and $Y$ are **INDEPENDENT**—**VARIANCES ALWAYS ADD, EVEN WHEN SUBTRACTING $X - Y$!**): $\sigma_{X \pm Y}^2 = \sigma_X^2 + \sigma_Y^2 \implies \sigma_{X \pm Y} = \sqrt{\sigma_X^2 + \sigma_Y^2}$ $\sigma_{aX \pm bY} = \sqrt{a^2\sigma_X^2 + b^2\sigma_Y^2}$

How do you conduct a $t$-Test or $t$-Interval for the Slope ($\beta$) of a Population Regression Line ($\mu_y = \alpha + \beta x$), and what are the LINE conditions?

• **Hypotheses**: $H_0: \beta = 0$ (no linear relationship between $x$ and $y$ in the population) vs. $H_a: \beta \ne 0$ (or $>0, <0$). • **Degrees of Freedom**: **$df = n - 2$**! • **Test Statistic & Interval** (reading $b$ and $SE_b$ from the explanatory variable row of computer output): $t = \frac{b - \beta_0}{SE_b} \quad \text{and} \quad b \pm t^*_{n-2}(SE_b)$ • **LINE Conditions**: **L**inear (scatterplot is linear, residual plot has no curved pattern), **I**ndependent ($10\%$ condition), **N**ormal (residuals are approximately Normal), **E**qual Variance (residual spread is constant across $x$), **R**andom.

What happens to the width of a Confidence Interval when (a) the Confidence Level increases from $90\%$ to $99\%$, or (b) the Sample Size $n$ is multiplied by $4$?

• **(a) Increasing Confidence Level ($90\% \to 99\%$)**: Increases the critical value $z^*$ (or $t^*$), which makes the Confidence Interval **WIDER** (to be more confident of capturing the parameter, you need a wider net). • **(b) Multiplying Sample Size $n$ by $4$**: Since $\sqrt{n}$ is in the denominator of the Standard Error, multiplying $n$ by $4$ cuts the Margin of Error (and interval width) in **HALF**.

When performing a One-Sample $z$-Test for $H_0: p = p_0$ versus a One-Sample $z$-Interval for $p$, which value of $p$ ($p_0$ vs. $\hat{p}$) do you use in the Large Counts Condition and the Standard Deviation/Error formula?

• **Hypothesis Test ($H_0: p = p_0$)**: Because we assume $H_0$ is true throughout the test, we use the **null hypothesized value $p_0$** for BOTH the Large Counts check ($np_0 \ge 10$ and $n(1 - p_0) \ge 10$) AND the test statistic denominator ($\sqrt{\frac{p_0(1 - p_0)}{n}}$)! • **Confidence Interval for $p$**: Because there is no $H_0$, we must plug in the **sample proportion $\hat{p}$** for BOTH Large Counts ($n\hat{p} \ge 10, n(1 - \hat{p}) \ge 10$) AND the Standard Error ($\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$).

What is the Sampling Distribution of the Difference Between Two Independent Sample Proportions ($\hat{p}_1 - \hat{p}_2$) and Two Independent Sample Means ($\bar{x}_1 - \bar{x}_2$)?

• **Difference in Sample Proportions ($\hat{p}_1 - \hat{p}_2$)**: - Mean: $\mu_{\hat{p}_1 - \hat{p}_2} = p_1 - p_2$. - Standard Deviation: $\sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{\frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2}}$. • **Difference in Sample Means ($\bar{x}_1 - \bar{x}_2$)**: - Mean: $\mu_{\bar{x}_1 - \bar{x}_2} = \mu_1 - \mu_2$. - Standard Deviation: $\sigma_{\bar{x}_1 - \bar{x}_2} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$.

Describe the Sampling Distribution of a Sample Mean ($\bar{x}$), including its Mean ($\mu_{\bar{x}}$), Standard Deviation ($\sigma_{\bar{x}}$), and the Central Limit Theorem (CLT).

When taking an SRS of size $n$ from a population with mean $\mu$ and standard deviation $\sigma$: • **Center**: $\mu_{\bar{x}} = \mu$. • **Spread ($10\%$ Condition: $n < 0.10N$)**: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ (Quadrupling sample size $n$ cuts sampling variability $\sigma_{\bar{x}}$ in **half**!). • **Shape & Central Limit Theorem (CLT)**: 1. If the **population distribution is Normal**, the sampling distribution of $\bar{x}$ is **exactly Normal for ANY sample size $n$**. 2. If the population distribution is **NOT Normal (skewed)**, the **Central Limit Theorem** guarantees that the sampling distribution of $\bar{x}$ is **approximately Normal if $n \ge 30$**.

If orange weights follow $X \sim N(\mu = 8\text{ oz}, \sigma = 1.5\text{ oz})$ and grapefruit weights follow $Y \sim N(\mu = 14\text{ oz}, \sigma = 2.0\text{ oz})$ independently, what is the probability that a randomly selected orange weighs more than half of a randomly selected grapefruit ($X > 0.5Y$)?

1. Rewrite the inequality as a linear combination compared to $0$: **$D = X - 0.5Y > 0$**. 2.Since $X$ and $Y$ are independent Normal random variables, $D$ is Normal with: - **Mean**: $\mu_D = \mu_X - 0.5\mu_Y = 8 - 0.5(14) = 8 - 7 = \mathbf{1.0\text{ oz}}$. - **Standard Deviation**: $\sigma_D = \sqrt{1^2\sigma_X^2 + (-0.5)^2\sigma_Y^2} = \sqrt{(1.5)^2 + (0.25)(2.0)^2} = \sqrt{2.25 + 1.00} = \sqrt{3.25} \approx \mathbf{1.803\text{ oz}}$. 3. Find $P(D > 0)$: $z = \frac{0 - 1.0}{1.803} = -0.555 \implies P(Z > -0.555) \approx \mathbf{0.710}$.

What are the BINS conditions for a Binomial Random Variable $X \sim B(n, p)$, and what are its Probability Formula, Mean $\mu_X$, and Standard Deviation $\sigma_X$?

• **BINS Conditions**: - **B — Binary**: Each trial results in 'Success' or 'Failure.' - **I — Independent**: Trials are independent (or satisfy the $10\%$ condition $n < 0.10N$ if sampling without replacement). - **N — Number**: Fixed number of trials $n$ is set in advance. - **S — Same Probability**: Constant probability of success $p$ on every trial. • **Probability of exactly $k$ successes in $n$ trials**: $P(X = k) = \binom{n}{k}p^k(1 - p)^{n - k}$ • **Mean & SD**: $\mu_X = np$ and $\sigma_X = \sqrt{np(1 - p)}$.

How does a Two-Sided Hypothesis Test ($H_a: \mu \ne \mu_0$) at significance level $\alpha$ correspond directly to a $(1 - \alpha)100\%$ Confidence Interval?

A two-sided hypothesis test at level $\alpha$ (e.g., $\alpha = 0.05$) will **reject $H_0: \mu = \mu_0$ if and only if the hypothesized null value $\mu_0$ falls OUTSIDE the corresponding $(1 - \alpha)100\%$ (e.g., $95\%$) confidence interval**! If $\mu_0$ is inside the $95\%$ confidence interval, then the two-sided $p\text{-value} > 0.05$ and we fail to reject $H_0$.

State the exact AP Statistics template for interpreting a $p$-Value in context.

*'Assuming the null hypothesis is true (i.e., assuming [state $H_0$ in context]), there is a [$p\text{-value}$] probability of getting a sample [statistic $= \text{observed value}$] or more extreme (in the direction of $H_a$) **by random chance alone**.'*

Describe the Sampling Distribution of a Sample Proportion ($\hat{p}$), including its Mean ($\mu_{\hat{p}}$), Standard Deviation ($\sigma_{\hat{p}}$), the $10\%$ Condition, and the Large Counts Condition.

When taking an SRS of size $n$ from a population with true proportion $p$: • **Center (Unbiased Estimator)**: $\mu_{\hat{p}} = p$. • **Spread ($10\%$ Condition: $n < 0.10N$ for sampling without replacement)**: $\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}}$ • **Shape (Large Counts Condition for Normal Approximation)**: Approximately Normal if **BOTH $np \ge 10$ AND $n(1 - p) \ge 10$** (checking **expected** successes and failures).

What are the 4 required components ('STATE – PLAN – DO – CONCLUDE') of every full Hypothesis Test or Confidence Interval FRQ on the AP Statistics exam?

1. **STATE**: State $H_0$ and $H_a$ using parameter symbols ($p$ or $\mu$), **define the parameter in context**, and state the significance level $\alpha$ (or confidence level $C\%$). *(Never put sample symbols like $\hat{p}$ or $\bar{x}$ in $H_0$ or $H_a$!)* 2. **PLAN**: Name the specific inference procedure (e.g., *'One-sample $t$-test for a population mean'*) and check all **3 conditions** (Random, $10\%$, and Large Counts / Normal) **in context with numbers**. 3. **DO**: Compute the test statistic ($z$, $t$, or $\chi^2$), degrees of freedom (if applicable), and **$p$-value** (or confidence interval endpoints). 4. **CONCLUDE**: Explicitly compare the $p$-value to $\alpha$: *'Because the $p\text{-value} = \dots \alpha$: *'**fail to reject $H_0$**; we do **not** have convincing evidence that [$H_a$ in context]'*—never say 'accept $H_0!).

In a Two-Sample $z$-Test for the Difference Between Two Proportions ($H_0: p_1 - p_2 = 0$), why do we use the Pooled Sample Proportion ($\hat{p}_c$) in the test statistic, whereas a Two-Sample $z$-Interval does NOT pool?

• **Two-Sample $z$-Interval for $p_1 - p_2$ (Unpooled)**: $(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}$ • **Two-Sample $z$-Test for $H_0: p_1 = p_2$ (Pooled)**: Under $H_0$, the two populations share a **single common proportion $p_1 = p_2 = p$**, so we combine the two samples into the **Pooled Proportion**: $\hat{p}_c = \frac{X_1 + X_2}{n_1 + n_2}$ and compute: $z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}_c(1 - \hat{p}_c)\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$

In a Binomial distribution $X \sim B(n = 20, p = 0.30)$, how do you write the calculator command and formula for $P(X \ge 5)$ vs. $P(X < 5)$?

Because cumulative binomial distribution functions (`binomcdf(n, p, k)`) compute **$P(X \le k)$** for discrete integers: • **$P(X < 5)$**: Since $X$ takes integer values, $X < 5$ is strictly equivalent to **$P(X \le 4)$** (`binomcdf(20, 0.30, 4)`). • **$P(X \ge 5)$**: Use the complement rule: $P(X \ge 5) = 1 - P(X \le 4) = 1 - \sum_{k=0}^4 \binom{20}{k}(0.30)^k(0.70)^{20-k}$ *(Exam Tip: Always write out the binomial formula with at least two terms and an ellipsis $+ \dots +$ alongside any calculator result to guarantee full credit!)*

Compare the three Chi-Square ($\chi^2$) Tests on AP Statistics (Goodness-of-Fit, Homogeneity, and Independence), including their Null Hypotheses, Expected Counts, and Degrees of Freedom.

All three use the test statistic **$\chi^2 = \sum \frac{(\text{Observed} - \text{Expected})^2}{\text{Expected}}$** and require the **Large Counts Condition: ALL Expected Counts $\ge 5$** (never Observed counts!): 1. **$\chi^2$ Goodness-of-Fit (GOF) Test**: **1 sample, 1 categorical variable** with $k$ categories compared against a hypothesized distribution ($H_0$: proportions match the claimed values). $\text{Expected}_i = n \cdot p_i$; **$df = k - 1$**. 2. **$\chi^2$ Test for Homogeneity**: **2+ independent samples/treatment groups**, **1 categorical variable** ($H_0$: the true distribution of [variable] is the same across all populations/treatments). 3. **$\chi^2$ Test for Independence**: **1 single sample**, **2 categorical variables** in a two-way table ($H_0$: there is **no association** [they are independent] between Variable A and Variable B in the population). • For both Homogeneity and Independence: **$\text{Expected} = \frac{\text{Row Total} \times \text{Column Total}}{\text{Grand Total}}$** and **$df = (\text{Rows} - 1)(\text{Columns} - 1)$**.

How does a Geometric Random Variable $Y \sim \text{Geom}(p)$ differ from a Binomial Random Variable, and what are its Probability Formula, Mean, and Standard Deviation?

• **Difference**: A Binomial setting has a **fixed number of trials $n$**; a **Geometric setting has NO fixed $n$**—you repeat independent Bernoulli trials until you observe the **FIRST success** on trial $k$. • **Probability that the FIRST success occurs on the $k$-th trial**: $P(Y = k) = (1 - p)^{k - 1}p$ • **Mean (Expected trials to first success)**: $\mu_Y = \frac{1}{p}$. • **Standard Deviation**: $\sigma_Y = \frac{\sqrt{1 - p}}{p}$.

State the General Addition Rule, Conditional Probability Rule, and General Multiplication Rule, and distinguish Mutually Exclusive (Disjoint) Events from Independent Events.

• **General Addition Rule**: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. • **Conditional Probability**: $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$. • **General Multiplication Rule**: $P(A \cap B) = P(B) \cdot P(A \mid B)$. • **Mutually Exclusive (Disjoint) Events**: Cannot occur together, so **$P(A \cap B) = 0$**. • **Independent Events**: Knowing $B$ occurred does not change the probability of $A$, so **$P(A \mid B) = P(A)$** and **$P(A \cap B) = P(A) \cdot P(B)$**. • **Exam Trap**: Two events with positive probabilities ($P(A)>0, P(B)>0$) can **NEVER be both mutually exclusive and independent** (because if $A$ and $B$ are disjoint and $B$ occurs, $P(A \mid B) = 0 \ne P(A)$!).

Why do we use the Student's $t$-Distribution ($df = n - 1$) instead of the Normal $z$-Distribution for Population Means ($\mu$), and what are the formulas for a One-Sample $t$-Interval and $t$-Test?

Because the true population standard deviation $\sigma$ is almost never known in practice, we estimate $\sigma$ using the sample standard deviation $s_x$. This extra estimation uncertainty gives the **$t$-distribution heavier tails (more spread)** than $N(0, 1)$, with **degrees of freedom $df = n - 1$** (as $df \to \infty$, the $t$-distribution approaches the standard Normal $z$-distribution). • **One-Sample $t$-Interval for $\mu$**: $\bar{x} \pm t^*_{n-1}\left(\frac{s_x}{\sqrt{n}}\right)$ • **One-Sample $t$-Test Statistic** (testing $H_0: \mu = \mu_0$): $t = \frac{\bar{x} - \mu_0}{s_x / \sqrt{n}} \quad (df = n - 1)$ • **Normal/Large Sample Condition for Means**: Population is Normal, OR $n \ge 30$ (CLT), OR if $n < 30$, **graph the sample data** (dotplot/boxplot) and verify there is **no strong skewness and no outliers**.

Suppose a $95\%$ Confidence Interval for the difference in two population means ($\mu_A - \mu_B$) is $(-1.2, +4.8)$. What can you conclude about whether there is a statistically significant difference between $\mu_A$ and $\mu_B$ at $\alpha = 0.05$?

Because **$0$ is INSIDE the $95\%$ confidence interval $(-1.2, +4.8)$**, $0$ is a plausible value for $\mu_A - \mu_B$ (meaning $\mu_A$ could equal $\mu_B$). Therefore, we **fail to reject $H_0: \mu_A - \mu_B = 0$** at the $\alpha = 0.05$ level; we do **not** have convincing statistical evidence of a difference between the two population means.

Write the formulas and conditions for a One-Sample $z$-Interval for a Population Proportion ($p$) and how to solve for the Minimum Sample Size $n$ needed for a desired Margin of Error ($ME$).

• **Conditions**: Random sample, $10\%$ condition ($n < 0.10N$), and Large Counts ($n\hat{p} \ge 10$ and $n(1 - \hat{p}) \ge 10$, using **observed** counts!). • **One-Sample $z$-Interval for $p$**: $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$ (Common critical values: $90\% \Rightarrow z^* = 1.645$; $95\% \Rightarrow z^* = 1.960$; $99\% \Rightarrow z^* = 2.576$). • **Sample Size Formula for Desired Margin of Error $ME$**: $z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \le ME$ (If no prior guess $\hat{p}$ is given, use the conservative worst-case **$\hat{p} = 0.50$**, and **always round $n$ UP to the next whole integer**!).

What is the Law of Large Numbers (LLN), and what is the 'Gambler's Fallacy'?

• **Law of Large Numbers**: As the number of independent trials $n$ increases toward infinity ($n \to \infty$), the **long-run cumulative sample proportion (or sample mean $\bar{x}$)** converges toward the true theoretical probability $p$ (or population mean $\mu$). • **Gambler's Fallacy ('Law of Small Numbers')**: The false belief that after a short streak of heads on a fair coin, tails is 'due' on the next flip to balance things out. Independent trials have **no memory**—long-run proportions converge by **swamping** early streaks with millions of future trials, not by compensating on the next flip.

If a computer regression output table reports a two-sided $p$-value of $0.036$ for the slope $b = +2.45$, what is the $p$-value for the one-sided test $H_a: \beta > 0$?

Computer regression output tables ALWAYS report the **two-sided $p$-value** ($2 \times P(T > |t|)$). Because the observed sample slope $b = +2.45$ is in the direction of the one-sided alternative $H_a: \beta > 0$, the **one-sided $p$-value is exactly half of the two-sided $p$-value**: $p\text{-value}_{\text{one-sided}} = \frac{0.036}{2} = \mathbf{0.018}$

Why does the Margin of Error ($ME = z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$) of a Confidence Interval NOT account for Nonresponse Bias, Voluntary Response Bias, or Undercoverage?

The **Margin of Error measures ONLY random sampling variability**—the chance variation from taking different random samples of size $n$. It does **NOT** include systematic errors from biased sampling methods, nonresponse, or poorly worded survey questions.

How do you decide whether to use a Paired $t$-Test (One-Sample $t$-Test on Differences $\mu_d$) versus a Two-Sample $t$-Test for the Difference Between Two Independent Means ($\mu_1 - \mu_2$)?

• **Paired $t$-Procedure ($H_0: \mu_d = 0$, $df = n_{\text{pairs}} - 1$)**: Used when data come from **Matched Pairs**—either two measurements on the **same subject** (e.g., Pre-test vs. Post-test, Left hand vs. Right hand) or pairs of subjects matched before random assignment. Compute $d_i = x_{1i} - x_{2i}$ for each pair and run a **one-sample $t$-test on the differences $\bar{d}$**: $t = \frac{\bar{d} - 0}{s_d / \sqrt{n_d}}$ • **Two-Sample $t$-Procedure ($H_0: \mu_1 - \mu_2 = 0$)**: Used when you have **two independent groups** (e.g., Treatment A vs. Treatment B in a completely randomized experiment, or SRS of men vs. SRS of women): $t = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

Why do we NEVER check the $10\%$ Condition when subjects in a randomized experiment are randomly assigned to treatment groups (without random sampling from a population)?

The **$10\%$ Condition ($n < 0.10N$)** is needed ONLY when **sampling without replacement from a finite population** to ensure observations are approximately independent. In a randomized experiment where volunteers are **randomly assigned to treatment groups** (not sampled from a population), there is no finite population sampling without replacement—instead, independence comes from random assignment and keeping subjects separate!

In a Chi-Square Test, after you reject $H_0$, how do you determine which specific category or cell contributed the most to the statistically significant result?

Inspect the individual **Chi-Square Components** (each cell's contribution to the $\chi^2$ sum): $\text{Component} = \frac{(\text{Observed} - \text{Expected})^2}{\text{Expected}}$ Identify the cell with the **largest $\frac{(O - E)^2}{E}$ component**, and state whether its observed count was **higher or lower** than expected under $H_0$.

AP Statistics: Probability, Sampling Distributions, Confidence Intervals & Hypothesis Tests (z, t, Chi-Square)

High-yield AP Statistics Units 4–9 review covering probability rules, Bayes' Theorem, discrete/binomial/geometric random variables, CLT and sampling distributions, 4-step STATE-PLAN-DO-CONCLUDE inference, 1/2-sample z and t procedures, Type I/II errors, Power, Chi-Square tests, and regression slope t-tests.

Level: intermediate
Languages: en → en
Cards: 32
Markdown: AP Statistics: Probability, Sampling Distributions, Confidence Intervals & Hypothesis Tests (z, t, Chi-Square) (.md)

Flashcards (32)

#	Front (Term / Question)	Back (Definition / Answer)
1	Write the formulas for the Mean (Expected Value $\mu_X$) and Standard Deviation ($\sigma_X$) of a Discrete Random Variable $X$, and state how you must interpret $\mu_X$ in context.	• Mean (Expected Value): $\mu_X = E(X) = \sum x_i P(x_i)$ (Note: $\mu_X$ does NOT have to be a possible integer value of $X$!). • Standard Deviation: $\sigma_X = \sqrt{\sum (x_i - \mu_X)^2 P(x_i)}$ • Required AP Interpretation of $\mu_X$: 'If we repeated [random process] many, many times* (in the long run), the average [variable $X$, in context] would be about $\mu_X$ [units].'*
2	State the general structure of a Confidence Interval and the exact AP Statistics templates for interpreting (a) a Confidence Interval and (b) a Confidence Level ($C\%$).	• General Formula: $\text{Point Estimate} \pm \text{Margin of Error} = \text{Statistic} \pm (\text{Critical Value}) \times (\text{Standard Error})$. • (a) Interpreting a Confidence Interval $(L, U)$: 'We are [$C\%$] confident that the interval from $L$ to $U$ captures the true population [parameter $\mu$ or $p$, in context].' (Never mention the sample statistic $\bar{x}$ or $\hat{p}$, and never say 'there is a $95\%$ probability the parameter is in this interval'!). • (b) Interpreting the Confidence Level ($C\%$): 'If we took many, many random samples of the same size $n$ from this population and constructed a [$C\%$] confidence interval from each sample, about [$C\%$] of the resulting intervals would capture the true population [parameter, in context].'
3	Define Type I Error, Type II Error, and the Power of a Test ($1 - \beta$), and state what increases the Power of a Hypothesis Test.	• Type I Error (False Positive, probability $= \alpha$): Rejecting $H_0$ when $H_0$ is actually TRUE. • Type II Error (False Negative, probability $= \beta$): Failing to reject $H_0$ when $H_0$ is actually FALSE ($H_a$ is true). • Power of a Test ($\text{Power} = 1 - \beta$): The probability that the test correctly rejects a false $H_0$ when a specific alternative parameter value is true. • Four Ways to Increase Power (and Decrease $\beta$): 1. Increase sample size $n$ (reduces standard error; reduces BOTH $\beta$ and variability!). 2. Increase significance level $\alpha$ (e.g., from $0.01$ to $0.05$; makes it easier to reject $H_0$, so $\alpha \uparrow \implies \beta \downarrow, \text{Power} \uparrow$). 3. Larger effect size (true parameter is farther from the null value). 4. Decrease standard deviation $\sigma$ (reduce measurement error).
4	When choosing a significance level $\alpha$ ($0.01$ vs. $0.10$) before a test, how do you decide based on the real-world consequences of a Type I Error vs. a Type II Error?	• Choose a SMALLER $\alpha$ (e.g., $\alpha = 0.01$) when the consequence of a Type I Error (falsely rejecting $H_0$) is more severe (e.g., approving an expensive new drug with toxic side effects when it actually doesn't work). • Choose a LARGER $\alpha$ (e.g., $\alpha = 0.10$) when the consequence of a Type II Error (failing to detect a real problem) is more dangerous (e.g., testing drinking water for lead contamination where $H_0: \text{water is safe}$ and $H_a: \text{water is unsafe}$).
5	How do you solve a Tree Diagram / Reverse Conditional Probability (Bayes' Theorem) problem (such as a medical diagnostic test with false positives)?	Use Bayes' Theorem (or a 2x2 table with a hypothetical population of $10,000$): $P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{P(A)P(B \mid A)}{P(A)P(B \mid A) + P(A^c)P(B \mid A^c)}$ For example, if $1\%$ of a population has a disease ($P(D)=0.01$), a test is $95\%$ sensitive ($P(+ \mid D)=0.95$), and has a $4\%$ false-positive rate ($P(+ \mid D^c)=0.04$), then: $P(D \mid +) = \frac{(0.01)(0.95)}{(0.01)(0.95) + (0.99)(0.04)} = \frac{0.0095}{0.0491} \approx 0.1935$
6	In Combining Random Variables, what is the critical difference between $3X$ (multiplying a single observation by $3$, e.g., converting yards to feet) and $X_1 + X_2 + X_3$ (summing $3$ independent observations)?	Both have the exact same mean ($3\mu_X$), but their standard deviations differ! • Linear Rescaling ($3X$): $\sigma_{3X} = \sqrt{3^2 \sigma_X^2} = \mathbf{3\sigma_X}$. • Sum of $3$ Independent Observations ($X_1 + X_2 + X_3$): $\sigma_{X_1 + X_2 + X_3} = \sqrt{\sigma_X^2 + \sigma_X^2 + \sigma_X^2} = \mathbf{\sqrt{3}\,\sigma_X}$
7	When you Combine two Independent Random Variables ($W = aX \pm bY$), what are the Mean $\mu_W$, Variance $\sigma_W^2$, and Standard Deviation $\sigma_W$? ('Variances ALWAYS Add!')	For any two random variables $X$ and $Y$: • Mean: $\mu_{aX + bY} = a\mu_X + b\mu_Y$ and $\mu_{aX - bY} = a\mu_X - b\mu_Y$. • Variance & Standard Deviation (ONLY if $X$ and $Y$ are INDEPENDENT—VARIANCES ALWAYS ADD, EVEN WHEN SUBTRACTING $X - Y$!): $\sigma_{X \pm Y}^2 = \sigma_X^2 + \sigma_Y^2 \implies \sigma_{X \pm Y} = \sqrt{\sigma_X^2 + \sigma_Y^2}$ $\sigma_{aX \pm bY} = \sqrt{a^2\sigma_X^2 + b^2\sigma_Y^2}$
8	How do you conduct a $t$-Test or $t$-Interval for the Slope ($\beta$) of a Population Regression Line ($\mu_y = \alpha + \beta x$), and what are the LINE conditions?	• Hypotheses: $H_0: \beta = 0$ (no linear relationship between $x$ and $y$ in the population) vs. $H_a: \beta \ne 0$ (or gt;0, <0$). • Degrees of Freedom: $df = n - 2$! • Test Statistic & Interval (reading $b$ and $SE_b$ from the explanatory variable row of computer output): $t = \frac{b - \beta_0}{SE_b} \quad \text{and} \quad b \pm t^_{n-2}(SE_b)$ • LINE Conditions: Linear (scatterplot is linear, residual plot has no curved pattern), Independent ($10\%$ condition), Normal (residuals are approximately Normal), Equal Variance (residual spread is constant across $x$), R*andom.
9	What happens to the width of a Confidence Interval when (a) the Confidence Level increases from $90\%$ to $99\%$, or (b) the Sample Size $n$ is multiplied by $4$?	• (a) Increasing Confidence Level ($90\% \to 99\%$): Increases the critical value $z^$ (or $t^$), which makes the Confidence Interval WIDER (to be more confident of capturing the parameter, you need a wider net). • (b) Multiplying Sample Size $n$ by $4$: Since $\sqrt{n}$ is in the denominator of the Standard Error, multiplying $n$ by $4$ cuts the Margin of Error (and interval width) in HALF.
10	When performing a One-Sample $z$-Test for $H_0: p = p_0$ versus a One-Sample $z$-Interval for $p$, which value of $p$ ($p_0$ vs. $\hat{p}$) do you use in the Large Counts Condition and the Standard Deviation/Error formula?	• Hypothesis Test ($H_0: p = p_0$): Because we assume $H_0$ is true throughout the test, we use the null hypothesized value $p_0$ for BOTH the Large Counts check ($np_0 \ge 10$ and $n(1 - p_0) \ge 10$) AND the test statistic denominator ($\sqrt{\frac{p_0(1 - p_0)}{n}}$)! • Confidence Interval for $p$: Because there is no $H_0$, we must plug in the sample proportion $\hat{p}$ for BOTH Large Counts ($n\hat{p} \ge 10, n(1 - \hat{p}) \ge 10$) AND the Standard Error ($\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$).
11	What is the Sampling Distribution of the Difference Between Two Independent Sample Proportions ($\hat{p}_1 - \hat{p}_2$) and Two Independent Sample Means ($\bar{x}_1 - \bar{x}_2$)?	• Difference in Sample Proportions ($\hat{p}_1 - \hat{p}_2$): - Mean: $\mu_{\hat{p}_1 - \hat{p}_2} = p_1 - p_2$. - Standard Deviation: $\sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{\frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2}}$. • Difference in Sample Means ($\bar{x}_1 - \bar{x}_2$): - Mean: $\mu_{\bar{x}_1 - \bar{x}_2} = \mu_1 - \mu_2$. - Standard Deviation: $\sigma_{\bar{x}_1 - \bar{x}_2} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$.
12	Describe the Sampling Distribution of a Sample Mean ($\bar{x}$), including its Mean ($\mu_{\bar{x}}$), Standard Deviation ($\sigma_{\bar{x}}$), and the Central Limit Theorem (CLT).	When taking an SRS of size $n$ from a population with mean $\mu$ and standard deviation $\sigma$: • Center: $\mu_{\bar{x}} = \mu$. • Spread ($10\%$ Condition: $n < 0.10N$): $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ (Quadrupling sample size $n$ cuts sampling variability $\sigma_{\bar{x}}$ in half!). • Shape & Central Limit Theorem (CLT): 1. If the population distribution is Normal, the sampling distribution of $\bar{x}$ is exactly Normal for ANY sample size $n$. 2. If the population distribution is NOT Normal (skewed), the Central Limit Theorem guarantees that the sampling distribution of $\bar{x}$ is approximately Normal if $n \ge 30$.
13	If orange weights follow $X \sim N(\mu = 8\text{ oz}, \sigma = 1.5\text{ oz})$ and grapefruit weights follow $Y \sim N(\mu = 14\text{ oz}, \sigma = 2.0\text{ oz})$ independently, what is the probability that a randomly selected orange weighs more than half of a randomly selected grapefruit ($X > 0.5Y$)?	1. Rewrite the inequality as a linear combination compared to $0$: $D = X - 0.5Y > 0$. 2.Since $X$ and $Y$ are independent Normal random variables, $D$ is Normal with: - Mean: $\mu_D = \mu_X - 0.5\mu_Y = 8 - 0.5(14) = 8 - 7 = \mathbf{1.0\text{ oz}}$. - Standard Deviation: $\sigma_D = \sqrt{1^2\sigma_X^2 + (-0.5)^2\sigma_Y^2} = \sqrt{(1.5)^2 + (0.25)(2.0)^2} = \sqrt{2.25 + 1.00} = \sqrt{3.25} \approx \mathbf{1.803\text{ oz}}$. 3. Find $P(D > 0)$: $z = \frac{0 - 1.0}{1.803} = -0.555 \implies P(Z > -0.555) \approx \mathbf{0.710}$.
14	What are the BINS conditions for a Binomial Random Variable $X \sim B(n, p)$, and what are its Probability Formula, Mean $\mu_X$, and Standard Deviation $\sigma_X$?	• BINS Conditions: - B — Binary: Each trial results in 'Success' or 'Failure.' - I — Independent: Trials are independent (or satisfy the $10\%$ condition $n < 0.10N$ if sampling without replacement). - N — Number: Fixed number of trials $n$ is set in advance. - S — Same Probability: Constant probability of success $p$ on every trial. • Probability of exactly $k$ successes in $n$ trials: $P(X = k) = \binom{n}{k}p^k(1 - p)^{n - k}$ • Mean & SD: $\mu_X = np$ and $\sigma_X = \sqrt{np(1 - p)}$.
15	How does a Two-Sided Hypothesis Test ($H_a: \mu \ne \mu_0$) at significance level $\alpha$ correspond directly to a $(1 - \alpha)100\%$ Confidence Interval?	A two-sided hypothesis test at level $\alpha$ (e.g., $\alpha = 0.05$) will reject $H_0: \mu = \mu_0$ if and only if the hypothesized null value $\mu_0$ falls OUTSIDE the corresponding $(1 - \alpha)100\%$ (e.g., $95\%$) confidence interval! If $\mu_0$ is inside the $95\%$ confidence interval, then the two-sided $p\text{-value} > 0.05$ and we fail to reject $H_0$.
16	State the exact AP Statistics template for interpreting a $p$-Value in context.	'Assuming the null hypothesis is true (i.e., assuming [state $H_0$ in context]), there is a [$p\text{-value}$] probability of getting a sample [statistic $= \text{observed value}$] or more extreme (in the direction of $H_a$) by random chance alone.'
17	Describe the Sampling Distribution of a Sample Proportion ($\hat{p}$), including its Mean ($\mu_{\hat{p}}$), Standard Deviation ($\sigma_{\hat{p}}$), the $10\%$ Condition, and the Large Counts Condition.	When taking an SRS of size $n$ from a population with true proportion $p$: • Center (Unbiased Estimator): $\mu_{\hat{p}} = p$. • Spread ($10\%$ Condition: $n < 0.10N$ for sampling without replacement): $\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}}$ • Shape (Large Counts Condition for Normal Approximation): Approximately Normal if BOTH $np \ge 10$ AND $n(1 - p) \ge 10$ (checking expected successes and failures).
18	What are the 4 required components ('STATE – PLAN – DO – CONCLUDE') of every full Hypothesis Test or Confidence Interval FRQ on the AP Statistics exam?	1. STATE: State $H_0$ and $H_a$ using parameter symbols ($p$ or $\mu$), define the parameter in context, and state the significance level $\alpha$ (or confidence level $C\%$). (Never put sample symbols like $\hat{p}$ or $\bar{x}$ in $H_0$ or $H_a$!) 2. PLAN: Name the specific inference procedure (e.g., 'One-sample $t$-test for a population mean') and check all 3 conditions (Random, $10\%$, and Large Counts / Normal) in context with numbers. 3. DO: Compute the test statistic ($z$, $t$, or $\chi^2$), degrees of freedom (if applicable), and $p$-value (or confidence interval endpoints). 4. CONCLUDE: Explicitly compare the $p$-value to $\alpha$: 'Because the $p\text{-value} = \dots < \alpha = 0.05$, we reject $H_0$. We have convincing statistical evidence that [restate $H_a$ in context].' (If $p\text{-value} > \alpha$: 'fail to reject $H_0$; we do not* have convincing evidence that [$H_a$ in context]'*—never say 'accept $H_0#39;!).
19	In a Two-Sample $z$-Test for the Difference Between Two Proportions ($H_0: p_1 - p_2 = 0$), why do we use the Pooled Sample Proportion ($\hat{p}_c$) in the test statistic, whereas a Two-Sample $z$-Interval does NOT pool?	• Two-Sample $z$-Interval for $p_1 - p_2$ (Unpooled): $(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}$ • Two-Sample $z$-Test for $H_0: p_1 = p_2$ (Pooled): Under $H_0$, the two populations share a single common proportion $p_1 = p_2 = p$, so we combine the two samples into the Pooled Proportion: $\hat{p}_c = \frac{X_1 + X_2}{n_1 + n_2}$ and compute: $z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}_c(1 - \hat{p}_c)\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$
20	In a Binomial distribution $X \sim B(n = 20, p = 0.30)$, how do you write the calculator command and formula for $P(X \ge 5)$ vs. $P(X < 5)$?	Because cumulative binomial distribution functions (`binomcdf(n, p, k)`) compute $P(X \le k)$ for discrete integers: • $P(X < 5)$: Since $X$ takes integer values, $X < 5$ is strictly equivalent to $P(X \le 4)$ (`binomcdf(20, 0.30, 4)`). • $P(X \ge 5)$: Use the complement rule: $P(X \ge 5) = 1 - P(X \le 4) = 1 - \sum_{k=0}^4 \binom{20}{k}(0.30)^k(0.70)^{20-k}$ (Exam Tip: Always write out the binomial formula with at least two terms and an ellipsis $+ \dots +$ alongside any calculator result to guarantee full credit!)
21	Compare the three Chi-Square ($\chi^2$) Tests on AP Statistics (Goodness-of-Fit, Homogeneity, and Independence), including their Null Hypotheses, Expected Counts, and Degrees of Freedom.	All three use the test statistic $\chi^2 = \sum \frac{(\text{Observed} - \text{Expected})^2}{\text{Expected}}$ and require the Large Counts Condition: ALL Expected Counts $\ge 5$ (never Observed counts!): 1. $\chi^2$ Goodness-of-Fit (GOF) Test: 1 sample, 1 categorical variable with $k$ categories compared against a hypothesized distribution ($H_0$: proportions match the claimed values). $\text{Expected}_i = n \cdot p_i$; $df = k - 1$. 2. $\chi^2$ Test for Homogeneity: 2+ independent samples/treatment groups, 1 categorical variable ($H_0$: the true distribution of [variable] is the same across all populations/treatments). 3. $\chi^2$ Test for Independence: 1 single sample, 2 categorical variables in a two-way table ($H_0$: there is no association [they are independent] between Variable A and Variable B in the population). • For both Homogeneity and Independence: $\text{Expected} = \frac{\text{Row Total} \times \text{Column Total}}{\text{Grand Total}}$ and $df = (\text{Rows} - 1)(\text{Columns} - 1)$.
22	How does a Geometric Random Variable $Y \sim \text{Geom}(p)$ differ from a Binomial Random Variable, and what are its Probability Formula, Mean, and Standard Deviation?	• Difference: A Binomial setting has a fixed number of trials $n$; a Geometric setting has NO fixed $n$—you repeat independent Bernoulli trials until you observe the FIRST success on trial $k$. • Probability that the FIRST success occurs on the $k$-th trial: $P(Y = k) = (1 - p)^{k - 1}p$ • Mean (Expected trials to first success): $\mu_Y = \frac{1}{p}$. • Standard Deviation: $\sigma_Y = \frac{\sqrt{1 - p}}{p}$.
23	State the General Addition Rule, Conditional Probability Rule, and General Multiplication Rule, and distinguish Mutually Exclusive (Disjoint) Events from Independent Events.	• General Addition Rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. • Conditional Probability: $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$. • General Multiplication Rule: $P(A \cap B) = P(B) \cdot P(A \mid B)$. • Mutually Exclusive (Disjoint) Events: Cannot occur together, so $P(A \cap B) = 0$. • Independent Events: Knowing $B$ occurred does not change the probability of $A$, so $P(A \mid B) = P(A)$ and $P(A \cap B) = P(A) \cdot P(B)$. • Exam Trap: Two events with positive probabilities ($P(A)>0, P(B)>0$) can NEVER be both mutually exclusive and independent (because if $A$ and $B$ are disjoint and $B$ occurs, $P(A \mid B) = 0 \ne P(A)$!).
24	Why do we use the Student's $t$-Distribution ($df = n - 1$) instead of the Normal $z$-Distribution for Population Means ($\mu$), and what are the formulas for a One-Sample $t$-Interval and $t$-Test?	Because the true population standard deviation $\sigma$ is almost never known in practice, we estimate $\sigma$ using the sample standard deviation $s_x$. This extra estimation uncertainty gives the $t$-distribution heavier tails (more spread) than $N(0, 1)$, with degrees of freedom $df = n - 1$ (as $df \to \infty$, the $t$-distribution approaches the standard Normal $z$-distribution). • One-Sample $t$-Interval for $\mu$: $\bar{x} \pm t^_{n-1}\left(\frac{s_x}{\sqrt{n}}\right)$ • One-Sample $t$-Test Statistic* (testing $H_0: \mu = \mu_0$): $t = \frac{\bar{x} - \mu_0}{s_x / \sqrt{n}} \quad (df = n - 1)$ • Normal/Large Sample Condition for Means: Population is Normal, OR $n \ge 30$ (CLT), OR if $n < 30$, graph the sample data (dotplot/boxplot) and verify there is no strong skewness and no outliers.
25	Suppose a $95\%$ Confidence Interval for the difference in two population means ($\mu_A - \mu_B$) is $(-1.2, +4.8)$. What can you conclude about whether there is a statistically significant difference between $\mu_A$ and $\mu_B$ at $\alpha = 0.05$?	Because $0$ is INSIDE the $95\%$ confidence interval $(-1.2, +4.8)$, $0$ is a plausible value for $\mu_A - \mu_B$ (meaning $\mu_A$ could equal $\mu_B$). Therefore, we fail to reject $H_0: \mu_A - \mu_B = 0$ at the $\alpha = 0.05$ level; we do not have convincing statistical evidence of a difference between the two population means.
26	Write the formulas and conditions for a One-Sample $z$-Interval for a Population Proportion ($p$) and how to solve for the Minimum Sample Size $n$ needed for a desired Margin of Error ($ME$).	• Conditions: Random sample, $10\%$ condition ($n < 0.10N$), and Large Counts ($n\hat{p} \ge 10$ and $n(1 - \hat{p}) \ge 10$, using observed counts!). • One-Sample $z$-Interval for $p$: $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$ (Common critical values: $90\% \Rightarrow z^* = 1.645$; $95\% \Rightarrow z^* = 1.960$; $99\% \Rightarrow z^* = 2.576$). • Sample Size Formula for Desired Margin of Error $ME$: $z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \le ME$ (If no prior guess $\hat{p}$ is given, use the conservative worst-case $\hat{p} = 0.50$, and always round $n$ UP to the next whole integer!).
27	What is the Law of Large Numbers (LLN), and what is the 'Gambler's Fallacy'?	• Law of Large Numbers: As the number of independent trials $n$ increases toward infinity ($n \to \infty$), the long-run cumulative sample proportion (or sample mean $\bar{x}$) converges toward the true theoretical probability $p$ (or population mean $\mu$). • Gambler's Fallacy ('Law of Small Numbers'): The false belief that after a short streak of heads on a fair coin, tails is 'due' on the next flip to balance things out. Independent trials have no memory—long-run proportions converge by swamping early streaks with millions of future trials, not by compensating on the next flip.
28	If a computer regression output table reports a two-sided $p$-value of $0.036$ for the slope $b = +2.45$, what is the $p$-value for the one-sided test $H_a: \beta > 0$?	Computer regression output tables ALWAYS report the two-sided $p$-value ($2 \times P(T > \|t\|)$). Because the observed sample slope $b = +2.45$ is in the direction of the one-sided alternative $H_a: \beta > 0$, the one-sided $p$-value is exactly half of the two-sided $p$-value: $p\text{-value}_{\text{one-sided}} = \frac{0.036}{2} = \mathbf{0.018}$
29	Why does the Margin of Error ($ME = z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$) of a Confidence Interval NOT account for Nonresponse Bias, Voluntary Response Bias, or Undercoverage?	The Margin of Error measures ONLY random sampling variability—the chance variation from taking different random samples of size $n$. It does NOT include systematic errors from biased sampling methods, nonresponse, or poorly worded survey questions.
30	How do you decide whether to use a Paired $t$-Test (One-Sample $t$-Test on Differences $\mu_d$) versus a Two-Sample $t$-Test for the Difference Between Two Independent Means ($\mu_1 - \mu_2$)?	• Paired $t$-Procedure ($H_0: \mu_d = 0$, $df = n_{\text{pairs}} - 1$): Used when data come from Matched Pairs—either two measurements on the same subject (e.g., Pre-test vs. Post-test, Left hand vs. Right hand) or pairs of subjects matched before random assignment. Compute $d_i = x_{1i} - x_{2i}$ for each pair and run a one-sample $t$-test on the differences $\bar{d}$: $t = \frac{\bar{d} - 0}{s_d / \sqrt{n_d}}$ • Two-Sample $t$-Procedure ($H_0: \mu_1 - \mu_2 = 0$): Used when you have two independent groups (e.g., Treatment A vs. Treatment B in a completely randomized experiment, or SRS of men vs. SRS of women): $t = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$
31	Why do we NEVER check the $10\%$ Condition when subjects in a randomized experiment are randomly assigned to treatment groups (without random sampling from a population)?	The $10\%$ Condition ($n < 0.10N$) is needed ONLY when sampling without replacement from a finite population to ensure observations are approximately independent. In a randomized experiment where volunteers are randomly assigned to treatment groups (not sampled from a population), there is no finite population sampling without replacement—instead, independence comes from random assignment and keeping subjects separate!
32	In a Chi-Square Test, after you reject $H_0$, how do you determine which specific category or cell contributed the most to the statistically significant result?	Inspect the individual Chi-Square Components (each cell's contribution to the $\chi^2$ sum): $\text{Component} = \frac{(\text{Observed} - \text{Expected})^2}{\text{Expected}}$ Identify the cell with the largest $\frac{(O - E)^2}{E}$ component, and state whether its observed count was higher or lower than expected under $H_0$.