Comprehensive AP Statistics Notes: Inference for Means and Proportions

AP Statistics Exam Structure and Course Logistics

• AP Exam Administration Format (May 17 Paper-Pencil Administration):

  • Follows the standardized pre-COVID examination format.
  • Section I: Multiple Choice: Consists of $40$ multiple-choice questions with a time limit of $1.5$ hours ($90$ minutes).
  • Section II: Free Response: Consists of $6$ free-response questions with a time limit of $1.5$ hours ($90$ minutes).
  • Weighting: Both sections are equally weighted, each accounting for $50\%$ of the total AP composite score.
  • Score Reporting: AP exam scores are released in early July (typically around the July 4th holiday) and have no mathematical bearing on internal high school course grades.

• Academic Schedule and Review Logistics:

  • Quarter Grading Cutoff: Quarter grades posted in PowerSchool represent finalized marks; subsequent assignments (such as the Chapter 8 WebAssign) transition into the following quarter's grade book.
  • WebAssign Submission: Homework modules close at $11:59\,\text{PM}$ on scheduled review days.
  • In-Person Instruction ("Plan A"): Commences after spring break on Tuesday, April 13 (following a designated teacher workday on Monday, April 12).
  • Formula Sheets: Physical reference sheets are provided in class, though students may supply printed copies of the official AP reference packet.
  • Review Sessions: College Board broadcasts live daily review sessions (recorded for asynchronous viewing) during the $5:00\,\text{PM}$ to $6:00\,\text{PM}$ time block.

Combining Distributions and Sampling Distributions for Means

• Linear Combinations of Independent Normal Variables (Chapter 6 Review):

  • Contextual Example (Cholesterol Levels):
  • Men: Normal distribution with mean $\mu_M = 188$ and standard deviation $\sigma_M = 41$.
  • Boys: Normal distribution with mean $\mu_B = 170$ and standard deviation $\sigma_B = 30$.
  • Variable definition: $M$ is the cholesterol of a randomly selected man; $B$ is the cholesterol of a randomly selected boy.
  • Properties of the Difference Distribution ($M - B$):
  • Shape: Because both underlying parent distributions are normally distributed, their linear combination $M - B$ is symmetric, bell-shaped, and strictly normal.
  • Center: Means combine directly through subtraction:       $\mu_{M - B} = \mu_M - \mu_B = 188 - 170 = 18$
  • Spread (Variances Always Add): Standard deviations cannot be directly subtracted ($\sigma_M - \sigma_B = 41 - 30 = 11$ is incorrect). Variances ($\sigma^2$) must always be added regardless of whether the distributions are added or subtracted:       $\sigma_{M - B}^2 = \sigma_M^2 + \sigma_B^2 = (41)^2 + (30)^2 = 1681 + 900 = 2581$$\sigma_{M - B} = \sqrt{(41)^2 + (30)^2} = \sqrt{2581} \approx 50.8$
  • Probability Calculations on Combined Distributions:
  • Finding the probability that a randomly chosen boy has a higher cholesterol level than a randomly chosen man corresponds to $P(B > M)$, which is equivalent to $P(M - B < 0)$.
  • Calculating the standardized test statistic ($z$-score) relative to a boundary difference of $0$:       $z = \frac{x - \mu}{\sigma} = \frac{0 - 18}{50.8} \approx -0.35$
  • Consulting Table A for a $z$-score of $-0.35$ yields an area of $0.3632$.

• Sampling Distributions of the Difference Between Two Sample Means ($\bar{x}_1 - \bar{x}_2$):

  • Given independent simple random samples (SRS) of size $n_M = 25$ men and $n_B = 36$ boys:
  • Shape: Remains normally distributed.
  • Center: Remains unchanged; the expected value of the sampling distribution of differences in sample means is $\mu_{\bar{x}_M - \bar{x}_B} = 18$.
  • Spread: Standard deviation decreases as sample sizes increase according to $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$.
    • For men: $\sigma_{\bar{x}_M} = \frac{41}{\sqrt{25}} = \frac{41}{5} = 8.2$
    • For boys: $\sigma_{\bar{x}_B} = \frac{30}{\sqrt{36}} = \frac{30}{6} = 5.0$
    • Combined standard deviation of the sampling distribution:         $\sigma_{\bar{x}_M - \bar{x}_B} = \sqrt{\frac{\sigma_M^2}{n_M} + \frac{\sigma_B^2}{n_B}} = \sqrt{(8.2)^2 + (5.0)^2} = \sqrt{67.24 + 25} = \sqrt{92.24} \approx 9.6$
  • Standardized $z$-score for observing a difference in sample means less than $0$:       $z = \frac{0 - 18}{9.6} \approx -1.88$
  • Consulting Table A for $z = -1.88$ gives an area of $0.03$ ($3\%$).
  • Statistical Surmise: An event is formally classified as surprising or statistically unusual when its probability of occurrence under the assumed conditions is less than $5\%$ ($P < 0.05$).

Significance Tests and Confidence Intervals for Means

• Standard Error of the Sample Mean ($SE$ or $SEM$):

  • Standard error provides an estimate of the standard deviation of the sampling distribution when the population parameter $\sigma$ is unknown:     $SE_{\bar{x}} = \frac{s_x}{\sqrt{n}}$
  • Verbatim Contextual Interpretation: In repeated sampling, the average distance between the sample means ($\bar{x}$) and the true population mean ($\mu$) will be approximately equal to the calculated standard error value.
  • Numerical Applications:
  • Systolic blood pressure ($n = 27$, $s_x = 9.3$):       $SE_{\bar{x}} = \frac{9.3}{\sqrt{27}} \approx 1.79$
  • Yellowstone Willow Trees ($n = 23$, reported margin format $\bar{x} \pm 19.03$):       $19.03 = \frac{s_x}{\sqrt{23}} \implies s_x = 19.03 \times \sqrt{23} \approx 91.26$
  • The expression $\bar{x} \pm SE_{\bar{x}}$ does not constitute a valid confidence interval because it omits multiplication by a critical value ($t^*$ or $z^*$).

• Criteria for Selecting Between $z$ and $t$ Procedures:

  • Inferences regarding population means ($\mu$) require evaluating whether the population standard deviation ($\sigma$) is definitively known.
  • If $\sigma$ is known: Use the standard normal ($z$) distribution.
  • If $\sigma$ is unknown: Substitute the sample standard deviation ($s_x$) and use the Student's $t$-distribution with degrees of freedom $df = n - 1$.

• Critical Values ($t^*$) via Table B:

  • Determined by aligning the designated confidence level at the bottom baseline with the appropriate degrees of freedom on the left axis:
  • $95\%$ confidence interval with $n = 10$ ($df = 9$): $t^* = 2.262$.
  • $99\%$ confidence interval with $n = 20$ ($df = 19$): $t^* = 2.861$.
  • $95\%$ confidence interval with $n = 8$ ($df = 7$): $t^* = 2.365$.

• Four-Step Inference Framework for Means:

  • State: Identify population of interest and state parameters in context (e.g., $\mu$ = true mean vitamin C content in corn-soy blend [CSB] bags).
  • Plan: Verify the three core inference conditions:
  1. Random: Quoted directly from problem text (e.g., "random sample").
  2. Normal / Large Sample: Verified through parent population normality, the Central Limit Theorem ($n \ge 30$), or by evaluating sample graphs (dotplots, histograms, or normal probability plots). If $n < 30$, raw data must show an approximately linear normal probability plot or a dotplot displaying no extreme skewness and no outliers.
  3. 10% Condition (Independence): Ensure the population size is at least $10$ times the sample size ($N \ge 10n$).
  • Do: Compute interval using the formula:     $\bar{x} \pm t^* \left(\frac{s_x}{\sqrt{n}}\right)$
  • Conclude: Interpret the interval in context, utilizing exact phrasing: "We are [C]% confident that the interval from [lower] to [upper] captures the true mean [parameter in context]."

• Matched Pairs Versus Two-Sample Procedures:

  • Matched Pairs Designs: Each individual subject receives both competing treatments (e.g., measuring factory worker productivity under a music condition and a silent condition within alternate months assigned via coin toss), or measurements are taken in natural pairs (e.g., measuring weight loss and groove depth simultaneously across $16$ tires).
  • Analyzed by calculating the differences within pairs ($L_1 - L_2 \rightarrow L_3$) and running a one-sample $t$-test or one-sample $t$-interval on the column of differences ($\bar{x}_d$, $s_d$).
  • Tire Wear Study Results: The $95\%$ confidence interval on the mean difference ($\mu_{\text{weight}} - \mu_{\text{groove}}$) was calculated as $(2.8, 6.3)$. Because $0$ is not captured inside the interval and all values are positive, there is convincing evidence that the weight method yields larger estimates than the groove method. Reversing the subtraction order ($\mu_{\text{groove}} - \mu_{\text{weight}}$) yields identical magnitude with opposite signs: $(-6.275, -2.837)$.
  • Two-Sample Designs: Treatments are applied to separate, independently selected experimental units where each subject receives exactly one treatment level (e.g., assigning $50$ cars to tire brand A and $50$ separate cars to tire brand B; assigning $50$ women to diet A and $50$ separate women to diet B).

• Two-Sample $t$-Intervals for Differences in Means ($\mu_1 - \mu_2$):

  • Contextual Example (Flower Lengths and Hummingbird Beaks):
  • Red and yellow flower length data (in millimeters).
  • Visual inspection: Red flowers exhibit a higher center (estimated around $40\,\text{mm}$ to $41\,\text{mm}$) and greater variability/spread; yellow flowers are more compact with a center around $36\,\text{mm}$.
  • Model: Two-sample $t$-interval on raw data (STAT TESTS, option 0: 2-SampTInt).
  • Output ($95\%$ confidence): $(2.554, 4.482)$.
  • Concluding Interpretation: "We are 95% confident that the interval from 2.554 to 4.482 captures the true difference in the average lengths of the two different colors of flowers."
  • Inference from Interval: Since $0$ is not contained in the interval, there is statistically significant evidence of a difference in mean lengths between varieties. Furthermore, because the interval is entirely positive and constructed as $\mu_{\text{red}} - \mu_{\text{yellow}}$, red flowers are established as longer on average.

• Degrees of Freedom in Two-Sample $t$-Procedures:

  • Exact Degrees of Freedom: Graphing calculators calculate exact degrees of freedom via the complex Welch-Satterthwaite approximation formula, which produces non-integer degrees of freedom.
  • Conservative Degrees of Freedom: Applied by selecting degrees of freedom equal to one less than the smaller sample size:     $df = \min(n_1 - 1, n_2 - 1)$
  • When conservative degrees of freedom are prescribed on homework or exams, the exact calculator test output must be overridden. Find the conservative $p$-value using the $t$-cumulative distribution function (TCDF, accessed via 2nd VARS, option 6) with syntax:     $\text{TCDF}(\text{lower bound}, \text{upper bound}, df)$
  • For an upper-tail alternative test statistic ($t = 2.56$), input lower bound $= 2.56$, upper bound $= 9\text{E}99$ (entered using 2nd , above the 7 key to represent $9 \times 10^{99}$), and the conservative $df$.

Error Types, Statistical Power, and Practical Significance

• Type I and Type II Errors:

  • Type I Error: Rejecting the null hypothesis ($H_0$) when the null hypothesis is actually true.
  • Type II Error: Failing to reject the null hypothesis ($H_0$) when the null hypothesis is actually false.
  • Contextual Blood Pressure Example ($H_0: \mu = 130$, $H_a: \mu > 130$):
  • Type I Error Description: The computerized device concludes that an individual's mean systolic blood pressure exceeds $130$ when their blood pressure is actually normal ($130$ or lower).
  • Type I Consequence: The patient is directed to seek medical attention and visits a doctor unnecessarily.
  • Type II Error Description: The device determines that the individual's blood pressure is normal when their true mean blood pressure is significantly elevated above $130$.
  • Type II Consequence: The patient fails to seek medical intervention when they are experiencing medical distress, which represents the more severe error in this clinical context.
  • Scoring Rubric Requirement: Explanations of statistical error states that rely exclusively on textbook definitions without complete situational context receive zero credit on the AP examination.

• Statistical Power:

  • Definition: The probability of correctly rejecting a false null hypothesis ($1 - \beta$, where $\beta$ is the probability of committing a Type II error).
  • Methods to Increase Power:
  1. Increase Sample Size ($n$): Increases precision, narrows the sampling distribution's standard error, and expands the rejection region.
  2. Increase Significance Level ($\alpha$): Increases the probability of rejecting $H_0$ by shifting the critical threshold inwards.

• Duality Between Confidence Intervals and Two-Sided Tests:

  • A confidence interval and a two-sided significance test yield equivalent conclusions provided the confidence level ($C$) and significance level ($\alpha$) satisfy:     $C + \alpha = 1.0 \quad (100\%)$
  • An $\alpha = 0.10$ significance test pairs with a $90\%$ confidence interval; an $\alpha = 0.05$ significance test pairs with a $95\%$ confidence interval.
  • Water Consumption Case ($H_0: \mu = 5$, $n = 24$): A $90\%$ confidence interval from MiniTab that fails to capture the claimed parameter value ($\mu = 5$) constitutes definitive grounds to reject $H_0$ at the $\alpha = 0.10$ significance level.
  • Significance Comparison Case ($p\text{-value} = 0.06$):
  • At $95\%$ confidence ($\alpha = 0.05$): Since $p\text{-value} = 0.06 > \alpha = 0.05$, the test fails to reject $H_0$, meaning the value $10$ is captured inside the $95\%$ confidence interval.
  • At $90\%$ confidence ($\alpha = 0.10$): Since $p\text{-value} = 0.06 < \alpha = 0.10$, the test rejects $H_0$, meaning the value $10$ is not captured inside the $90\%$ confidence interval.

• Statistical Significance Versus Practical Significance:

  • SAT Coaching Program Case Study ($H_0: \mu = 518$, $H_a: \mu > 518$, $n = 100$, $s = 114$):
  • Part A ($\bar{x} = 536.7$): Results in a $p$-value greater than $0.05$, failing to achieve statistical significance.
  • Part B ($\bar{x} = 537.0$): Results in a $p$-value less than $0.05$, achieving formal statistical significance.
  • Part C (Comparative Interpretation): An increase of only $0.3$ scale points on the SAT exam shifts the empirical result into statistical significance due to sample sizing, but an increase of a third of a point carries no practical utility or real-world benefit.

Confidence Intervals for a Single Population Proportion

• Standard Deviation Versus Standard Error of Proportions:

  • Theoretical standard deviation of the sample proportion $\hat{p}$:     $\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}}$
  • Because the population parameter $p$ is unknown, the sample statistic $\hat{p}$ must be substituted, creating the standard error ($SE$):     $SE_{\hat{p}} = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

• Inference Conditions for One-Sample Proportion Intervals:

  1. Random: Quoted verbatim from the prompt (e.g., "simple random sample").
  2. Normal (Large Counts Condition): The anticipated counts of successes and failures must both be at least $10$, evaluated using the sample proportion $\hat{p}$:     $n\hat{p} \ge 10 \quad \text{and} \quad n(1 - \hat{p}) \ge 10$
  3. 10% Condition: The population size must be at least $10$ times the sample size ($N \ge 10n$) to sample without replacement safely.

• Computational Construction:

  • Formula for a one-proportion $z$-interval:     $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$
  • Proportions strictly require the critical value $z^*$; the $t$-distribution is never utilized for proportions.
  • Standard critical values ($z^*$) obtained from the bottom row ($df = \infty$) of Table B:
  • $90\%$ confidence: $z^* = 1.645$
  • $95\%$ confidence: $z^* = 1.960$
  • $99\%$ confidence: $z^* = 2.576$

• Worked Example: High School Students Working Part-Time:

  • Data: SRS of $n = 2673$ high school students; $X = 1524$ maintain part-time employment.     $\hat{p} = \frac{1524}{2673} \approx 0.5701$
  • Condition Check:
  • Random: Stated as an SRS.
  • Large Counts: $n\hat{p} = 2673 \left(\frac{1524}{2673}\right) = 1524 \ge 10$; $n(1 - \hat{p}) = 2673 - 1524 = 1149 \ge 10$.
  • 10% Condition: Population of high school students must be at least $10 \times 2673 = 26,730$.
  • Interval Computation ($99\%$ Confidence, $z^* = 2.576$):$0.5701 \pm 2.576 \sqrt{\frac{(0.5701)(0.4299)}{2673}}$
  • Margin of Error: $2.576 \times 0.00958 \approx 0.0247$
  • Resulting Boundaries: $(0.545, 0.595)$
  • Concluding Interpretation: "I am 99% confident that the true proportion of high school students who work part-time falls between 0.545 and 0.595."
  • Inference: Because the entire interval lies strictly above $0.50$ ($50\%$), the data provides evidence that a majority of high school students work part-time.

• Sample Size Determination Based on Margin of Error:

  • The margin of error ($ME$) equation is isolated and solved for $n$:     $ME = z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$
  • Conservative Estimation Principle: When no prior estimate for $\hat{p}$ is provided, use $\hat{p} = 0.5$. The product $(0.5)(0.5) = 0.25$ maximizes the variance, ensuring the calculated sample size is large enough to guarantee the target margin of error.
  • Customer Service Satisfaction Problem:
  • Desired margin of error $ME = 0.03$, confidence level $= 95\%$ ($z^* = 1.960$), unknown $\hat{p}$ (use $0.5$):       $0.03 = 1.960 \sqrt{\frac{(0.5)(0.5)}{n}}$$\frac{0.03}{1.960} \approx 0.015306$$(0.015306)^2 \approx 0.00023428 = \frac{0.25}{n}$$n = \frac{0.25}{0.00023428} \approx 1067.111$
  • Decimal results are valid on AP scoring rubrics, but practical sampling demands rounding up to the next integer ($n = 1068$).

Hypothesis Tests for a Single Population Proportion

• Hypothesis Formulation and Parameters:

  • Parameter definition: $p$ = true population proportion.
  • Null hypothesis: $H_0: p = p_0$.
  • Alternative hypothesis: $H_a: p > p_0$, $H_a: p < p_0$, or $H_a: p \ne p_0$.

• Inference Conditions for One-Sample Tests:

  1. Random: Explicitly verified via prompt text.
  2. Normal (Large Counts Condition): Must be evaluated using the null value ($p_0$), because significance tests assume $H_0$ is true:     $n p_0 \ge 10 \quad \text{and} \quad n(1 - p_0) \ge 10$
  • Under no circumstances can the Central Limit Theorem be cited to establish normality for categorical proportions.
  1. 10% Condition: $N \ge 10n$.

• Test Statistic and Standard Error Computation:

  • General formula for a standardized test statistic:     $\text{Test Statistic} = \frac{\text{Statistic} - \text{Parameter}}{\text{Standard Error}}$
  • Formula for a one-proportion $z$-test:     $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$
  • The standard error in the denominator must incorporate the assumed null parameter ($p_0$), rather than the sample statistic ($\hat{p}$).
  • Calculator Execution: STAT TESTS, option 5 (1-PropZTest). Option 1 (Z-Test) is reserved strictly for population means.

• Worked Example: Testing High School Employment Rates:

  • Problem Claim: An article asserts that $55\%$ of all high school students work part-time. An SRS of $n = 2673$ shows $X = 1524$ working.
  • Hypotheses:$H_0: p = 0.55$$H_a: p > 0.55$
  • Where $p$ represents the true proportion of all high school students who work part-time.
  • Normality Check:$n p_0 = 2673(0.55) = 1470.15 \ge 10$$n(1 - p_0) = 2673(0.45) = 1202.85 \ge 10$
  • Test Statistic Computation:$\hat{p} = \frac{1524}{2673} \approx 0.5701$$z = \frac{0.5701 - 0.55}{\sqrt{\frac{(0.55)(0.45)}{2673}}} = \frac{0.0201}{\sqrt{0.00009259}} = \frac{0.0201}{0.009622} \approx 2.094$
  • $P$-Value Derivation:
  • Via Table A: Looking up $z = 2.09$ yields a cumulative left-tail area of $0.9817$. Because the test is right-tailed ($H_a: p > 0.55$):       $p\text{-value} = 1 - 0.9817 = 0.0183$
  • Via Calculator (1-PropZTest): Produces $z = 2.094$ and $p\text{-value} = 0.0181$.

• Standardized Concluding Decision Structure:

  • When no significance level is stated, the default threshold is $\alpha = 0.05$.
  • Formal Synthesis: "Because the $p$-value ($0.018$) is less than the significance level ($\alpha = 0.05$), we reject the null hypothesis. There is convincing evidence that the true proportion of high school students who work part-time is greater than 0.55."
  • Scoring Rubric Penalty Alert: Stating "I accept the null hypothesis" results in an automatic deduction of points on AP exam scoring rubrics. The acceptable terminology is strictly restricted to "reject the null hypothesis" or "fail to reject the null hypothesis."

Two-Sample Inference for Differences Between Population Proportions

• Two-Sample Proportion Confidence Intervals:

  • Used to estimate the true difference between two independent population proportions ($p_1 - p_2$).
  • Model Formula:$(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}$
  • Large Counts Condition: Requires verifying four distinct inequalities using sample proportions:     $n_1\hat{p}_1 \ge 10, \quad n_1(1 - \hat{p}_1) \ge 10, \quad n_2\hat{p}_2 \ge 10, \quad n_2(1 - \hat{p}_2) \ge 10$
  • Calculator Execution: STAT TESTS, option B (2-PropZInt).

• Two-Sample Proportion Significance Tests:

  • Null and Alternative Hypotheses:$H_0: p_1 = p_2 \quad (p_1 - p_2 = 0)$$H_a: p_1 < p_2, \quad p_1 > p_2, \quad \text{or} \quad p_1 \ne p_2$
  • Pooled (Combined) Sample Proportion ($\hat{p}_c$):
  • Because the null hypothesis assumes $p_1 = p_2$, the two samples are pooled to form a single combined parameter estimate:       $\hat{p}_c = \frac{X_1 + X_2}{n_1 + n_2}$
  • The pooled proportion $\hat{p}_c$ must be used when checking the Large Counts condition for both groups in a significance test:       $n_1\hat{p}_c \ge 10, \quad n_1(1 - \hat{p}_c) \ge 10, \quad n_2\hat{p}_c \ge 10, \quad n_2(1 - \hat{p}_c) \ge 10$
  • Standardized Test Statistic ($z$):$z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}_c(1 - \hat{p}_c)\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\frac{\hat{p}_c(1 - \hat{p}_c)}{n_1} + \frac{\hat{p}_c(1 - \hat{p}_c)}{n_2}}}$
  • Calculator Execution: STAT TESTS, option 6 (2-PropZTest).

• Comprehensive Case Study: High Scope Educational Preschool Research:

  • Study Design: Evaluates the long-term effects of preschool on low-income children in Michigan. Group 1: $n_1 = 62$ children who attended preschool; Group 2 (control): $n_2 = 61$ children from identical backgrounds who did not attend preschool. The response variable is the need for social services as adults over a $10$-year period.
  • Observed Counts:
  • Preschool group ($1$): $X_1 = 38$, $n_1 = 62$ $\implies \hat{p}_1 = \frac{38}{62} \approx 0.6129$
  • Control group ($2$): $X_2 = 49$, $n_2 = 61$ $\implies \hat{p}_2 = \frac{49}{61} \approx 0.8033$
  • Confidence Interval Construction ($95\%$ Confidence):
  • Large counts check:
    • Group 1: $62\left(\frac{38}{62}\right) = 38 \ge 10$; $62\left(\frac{24}{62}\right) = 24 \ge 10$.
    • Group 2: $61\left(\frac{49}{61}\right) = 49 \ge 10$; $61\left(\frac{12}{61}\right) = 12 \ge 10$.
  • 10% Condition: Because both samples are drawn from the same general subpopulation (three-to-four-year-old low-income children in Michigan), evaluate the larger threshold: there are safely at least $10 \times 62 = 620$ such children in Michigan.
  • Computation: Critical value $z^* = 1.960$.       $\left(\frac{38}{62} - \frac{49}{61}\right) \pm 1.960 \sqrt{\frac{\left(\frac{38}{62}\right)\left(\frac{24}{62}\right)}{62} + \frac{\left(\frac{49}{61}\right)\left(\frac{12}{61}\right)}{61}}$
    • Calculator Output (2-PropZInt): Interval is $(-0.3474, -0.0334)$.
  • Interpretation Requirement: The word difference is mandatory in concluding interpretations. "I am 95% confident that the true difference in the proportion of kids from preschool and non-preschool groups needing social services as adults falls between -0.3474 and -0.0334."
  • Significance Test Execution:
  • Hypotheses: $H_0: p_1 = p_2$ versus $H_a: p_1 < p_2$.
  • Pooled Proportion:       $\hat{p}_c = \frac{38 + 49}{62 + 61} = \frac{87}{123} \approx 0.7073$
  • Normality via Pooling:       $n_1\hat{p}_c = 62\left(\frac{87}{123}\right) \approx 43.85 \ge 10, \quad n_1(1 - \hat{p}_c) = 62\left(\frac{36}{123}\right) \approx 18.15 \ge 10$$n_2\hat{p}_c = 61\left(\frac{87}{123}\right) \approx 43.15 \ge 10, \quad n_2(1 - \hat{p}_c) = 61\left(\frac{36}{123}\right) \approx 17.85 \ge 10$
  • Standardized Test Statistic:       $z = \frac{\left(\frac{38}{62} - \frac{49}{61}\right) - 0}{\sqrt{\left(\frac{87}{123}\right)\left(\frac{36}{123}\right)\left(\frac{1}{62} + \frac{1}{61}\right)}} \approx -2.320$
  • Test Output: Standardized test statistic $z = -2.320$; $p\text{-value} = 0.0101$.
  • Concluding Synthesis ($\alpha = 0.05$): Because the $p$-value ($0.0101$) is less than $\alpha = 0.05$, we reject $H_0$. There is convincing evidence that the proportion of children needing adult social services is significantly lower among those who attended preschool than those who did not.

Questions and Discussion

• Question Regarding Quarterly Grading:

  • Query: Is the grade currently displayed in PowerSchool the final, closed grade for the academic quarter?
  • Response: Yes, all assessments for the quarter are concluded and accounted for; the upcoming Chapter 8 WebAssign serves as the first entry for the next grading quarter.

• Question Regarding Final Exam AP Weighting:

  • Query: What percentage of the high school course grade is determined by the final score on the AP examination?
  • Response: The AP examination has zero impact on the course grade assigned at the school. Scores are released independently by the College Board in July.

• Question Regarding Exam Review Structure:

  • Query: Will the two designated review weeks prior to the AP exam consist primarily of working through practice problems and released multiple-choice questions?
  • Response: Yes, review time is dedicated to analyzing previously released authentic College Board examination questions to build familiarity with test patterns.

• Question Regarding Formula Sheets for Plan A Return:

  • Query: Should students returning to physical classrooms under "Plan A" bring printed formula sheets, or will materials be distributed?
  • Response: Extra hard copies will be distributed in the classroom, though students are welcome to supply their own personal printed copies.

• Question Regarding Wording for Confidence Levels with Unequal Samples:

  • Query: In free-response explanations of confidence level interpretations (e.g., Problem 46, Part C), should sample sizes be stated, especially when the two samples are of different sizes?
  • Response: Specific sample sizes do not need to be stated if they differ. The key is to state the repeated sampling concept: "If we took repeated samples of the same populations and calculated confidence intervals for each, approximately 95% of the resulting intervals would capture the true difference in proportions/means." Stating "repeated samples of the same size" is only appropriate when the two sample sizes are identical.

• Question Regarding Detail Level for Error Consequences:

  • Query: On free-response questions requiring descriptions of Type I and Type II errors and their consequences, is it necessary to elaborate on extreme health outcomes?
  • Response: Extensive elaboration is not needed, but descriptions must be contextualized. For a Type II error, stating that the patient was told their blood pressure was fine when it was elevated, resulting in them not seeing a doctor when they should have, provides sufficient context. Generic textbook definitions without context receive no credit.

• Question Regarding Discrepancies Between Calculator and Table $P$-Values:

  • Query: On free-response questions, the calculator gives a $p$-value of $0.0181$ while the normal probability table gives $0.0183$. Which should be written down, and does using the table require showing the subtraction step ($1 - 0.9817$)?
  • Response: Both calculations are recognized on scoring rubrics, and credit is awarded for either method. Showing the subtraction step ($1 - 0.9817$) indicates that Table A was used, whereas writing $0.0181$ indicates calculator usage. As long as the recorded value rounds reasonably close to $0.018$, students will not be penalized.