Inference for Two Populations: Independent Means, Matched Pairs, and Two Proportions

Course Structure, Exam Policies, and Learning Resources

• Conceptual Progression from Chapter 8 to Chapter 9:

  • Chapter 9 extends the single-population inferential methods established in Chapter 8 by introducing a second population and a second sample.

  • The core procedural steps for hypothesis testing remain identical across both chapters: formulating hypotheses, determining critical values or $p\text{-values}$, computing the test statistic, and making a formal statistical decision.

  • Chapter 10 continues this sequence, culminating in basic linear correlation and regression, leaving advanced regression techniques for graduate-level study.

• Examination Scheduling and Coverage:

  • Unit Exam: Covers Chapters 8, 9, and 10. Scheduling options include the Wednesday immediately preceding Thanksgiving break or the first Wednesday following Thanksgiving break. The Monday following break serves as a dedicated in-class review period.

  • Cumulative Final Examination: Administered after the Unit Exam, covering all course material comprehensively from Chapter 1 through Chapter 10.

• Assignment and Quiz Extension Policies:

  • Quizzes are automated such that modifications cannot be executed once an attempt is in progress; altering quiz timers or parameters during active administration triggers security assumptions regarding academic integrity.

  • Chapter 8 deadlines were extended to ensure fair opportunity without penalizing students.

  • While all students are permitted two attempts by default, an individual third attempt is granted exclusively upon direct email request for students seeking to improve their performance.

• Classroom Norms and Academic Support:

  • Punctual attendance is required; empty seating at the front rows must be occupied rather than congregating in the back.

  • Students experiencing scheduling conflicts or needing individualized assistance can access office hours, online support via Zoom, or campus tutoring at the Student Academic Support suites (SUSE).

Fundamental Distinctions: Independent vs. Dependent Samples

• Independent Samples:

  • Sample selection from Population 1 has no relationship, pairing, or statistical connection to sample selection from Population 2.

  • Values occurring in sample 1 are completely independent of the values appearing in sample 2.

  • Example: Comparing the academic test scores of a sample of male students to the test scores of an entirely separate sample of female students.

• Dependent Samples (Matched Pairs):

  • The data values in sample 1 are naturally paired or intrinsically related to corresponding values in sample 2.

  • Observations are collected as matched pairs representing the same subject or closely linked entities.

  • Example: Pre-test and post-test measurements collected on the same individuals ("before-and-after" designs), husband-and-wife pairs, or competing opponents matched within the same contest.

Hypothesis Testing for Two Independent Population Means ($\sigma_1, \sigma_2$ Unknown and Unequal)

• Underlying Assumptions and Distribution Requirements:

  • Population standard deviations $\sigma_1$ and $\sigma_2$ are unknown and are not assumed to be equal ($\sigma_1 \neq \sigma_2$); hence, pooling of variances is avoided.

  • The two samples are independent simple random samples.

  • Sample sizes must satisfy $n_1 > 30$ and $n_2 > 30$, unless the underlying populations are verified to be normally distributed (in which case $n < 30$ is acceptable).

  • Because population parameters are unknown, inferential calculations rely on the Student's $t$-distribution.

• Mathematical Notation:

  • Subscript 1 denotes parameters and statistics for Population 1: $\mu_1$ (population mean), $\bar{x}_1$ (sample mean), $s_1$ (sample standard deviation), and $n_1$ (sample size).

  • Subscript 2 denotes parameters and statistics for Population 2: $\mu_2$ (population mean), $\bar{x}_2$ (sample mean), $s_2$ (sample standard deviation), and $n_2$ (sample size).

• Hypothesis Formulation:

  • Null Hypothesis: Assumes no difference between the two population means:         $H_0: \mu_1 = \mu_2 \quad \text{or} \quad H_0: \mu_1 - \mu_2 = 0$

  • Alternative Hypothesis: Reflects the substantive directional or non-directional claim:

    • Left-tailed: $H_1: \mu_1 < \mu_2 \quad (\mu_1 - \mu_2 < 0)$

    • Right-tailed: $H_1: \mu_1 > \mu_2 \quad (\mu_1 - \mu_2 > 0)$

    • Two-tailed: $H_1: \mu_1 \neq \mu_2 \quad (\mu_1 - \mu_2 \neq 0)$

• Test Statistic Formula:

  • Expanding the single-sample formula from Chapter 8 to account for the second population:         $t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

  • Under the assumption that $H_0$ is true, the hypothesized difference is set to zero: $(\mu_1 - \mu_2) = 0$.

• Exemplary Case Study: Background Paper Color and Creativity Scores:

  • Claim: Subjects using blue background paper achieve higher creativity scores than subjects using red background paper ($\mu_1 < \mu_2$, where Population 1 is Red and Population 2 is Blue).

  • Significance Level: $\alpha = 0.01$.

  • Sample Data:

    • Group 1 (Red background): $n_1 = 35$, $\bar{x}_1 = 3.39$, $s_1 = 0.97$

    • Group 2 (Blue background): $n_2 = 36$, $\bar{x}_2 = 3.97$, $s_2 = 0.63$

  • Hypotheses:         $H_0: \mu_1 - \mu_2 = 0$         $H_1: \mu_1 - \mu_2 < 0 \quad (\text{Left-tailed test})$

  • Critical Value: For a left-tailed test at $\alpha = 0.01$, the critical value is $t = -2.392$.

  • StatCrunch Navigation: Select Stat > T Stats > Two-Sample > With Summary. Input sample means, standard deviations, and sizes. Set the alternative hypothesis to < 0.

  • Decision Rule and Conclusion: The computed test statistic falls to the left of the critical value within the critical rejection region ($t < -2.392$), and the $p\text{-value} \le 0.01$. Therefore, reject $H_0$. There is sufficient sample evidence to support the claim that the red background group exhibits a lower mean creativity score than the blue background group.

Confidence Intervals for Two Independent Population Means

• Determining the Confidence Level ($1 - \alpha_{\text{CI}}$):

  • Two-Tailed Hypothesis Tests: Maintain the same significance level:         $\text{Confidence Level} = 1 - \alpha$

  • One-Tailed (Left- or Right-Tailed) Tests: Because a one-tailed test concentrates all of $\alpha$ in a single tail, constructing an equivalent two-sided confidence interval requires doubling the significance level:         $\alpha_{\text{CI}} = 2 \times \alpha$         $\text{Confidence Level} = 1 - 2\alpha$

  • Application to Example: For a one-tailed test with $\alpha = 0.01$:         $\alpha_{\text{CI}} = 2 \times 0.01 = 0.02$         $\text{Confidence Level} = 1 - 0.02 = 0.98 \quad (98\% \text{ confidence})$

• Statistical Decision-Making via Confidence Intervals:

  • The hypothesized difference under the null hypothesis is zero ($\mu_1 - \mu_2 = 0$).

  • If zero ($0$) is not contained within the confidence interval (both limits are strictly negative or strictly positive), reject $H_0$.

  • If zero ($0$) is contained within the confidence interval (spanning from a negative lower limit to a positive upper limit), fail to reject $H_0$.

Hypothesis Testing for Two Dependent Samples (Matched Pairs)

• Notation and Core Definitions:

  • $d = x_1 - x_2$: The difference between paired individual values in sample 1 and sample 2.

  • $\mu_d$: The hypothesized mean difference for the entire population of matched pairs.

  • $\bar{d}$: The mean of the differences calculated from the sample matched pairs:         $\bar{d} = \frac{\sum d}{n}$

  • $s_d$: The standard deviation of the differences calculated from the sample pairs.

  • $n$: The total number of matched pairs (the effective sample size equals the number of pairs, not the count of individual measurements).

  • $df = n - 1$: Degrees of freedom for the paired $t$-test.

• Test Statistic Formula:     $t = \frac{\bar{d} - \mu_d}{\frac{s_d}{\sqrt{n}}}$

  • Under the standard null hypothesis assumption that no net difference exists, $\mu_d = 0$.

• Exemplary Case Study: Presidential and Opponent Heights:

  • Context: Matching the physical heights of U.S. presidential candidates and their main political opponents running for the same office.

  • Claim: For the population of heights of presidents and their main opponents, the mean difference is greater than zero ($\mu_d > 0$), indicating presidents are systematically taller than their opponents.

  • Significance Level: $\alpha = 0.05$.

  • Assumptions: The sample has $n < 30$ pairs; therefore, the distribution of differences in the population is assumed to be normally distributed.

  • Hypotheses:         $H_0: \mu_d = 0$         $H_1: \mu_d > 0 \quad (\text{Right-tailed test})$

  • StatCrunch Procedure: Access Stat > T Stats > Paired. Select the presidential height column as Sample 1 and the opponent height column as Sample 2. Specify the hypothesis test as > 0.

  • Decision Rule and Graphical Area Interpretation:

    • The computed test statistic falls within the clear non-critical region (region of non-rejection) to the left of the critical cutoff.

    • The area representing the $p\text{-value}$ (tail area beyond the test statistic) is visibly larger than the significance level area $\alpha$ ($p\text{-value} > 0.05$).

    • Conclusion: Fail to reject $H_0$. There is insufficient sample evidence to conclude that presidents are significantly taller than their opponents.

Confidence Intervals for Dependent Samples (Matched Pairs)

• Confidence Level Adjustment for One-Tailed Paired Tests:

  • Given a one-tailed test with $\alpha = 0.05$, the confidence level must account for both tails:         $\alpha_{\text{CI}} = 2 \times 0.05 = 0.10$         $\text{Confidence Level} = 1 - 0.10 = 0.90 \quad (90\% \text{ confidence})$

• Application to the Presidential Height Dataset:

  • The resulting $90\%$ confidence interval spans from a lower limit of $-6.66$ to an upper limit of $14.6$:         $\text{CI: } (-6.66, 14.6)$

  • Because the interval spans across zero ($0$ is included within the bounds), the null value of zero difference remains plausible.

  • The inclusion of zero directly corroborates the decision from the hypothesis test: fail to reject $H_0$.

Hypothesis Testing for Two Independent Population Proportions

• Notation and Sample Parameters:

  • Population 1: $p_1$ (population proportion), $n_1$ (sample size), $x_1$ (number of successes).         $\hat{p}_1 = \frac{x_1}{n_1} \quad \text{and} \quad \hat{q}_1 = 1 - \hat{p}_1$

  • Population 2: $p_2$ (population proportion), $n_2$ (sample size), $x_2$ (number of successes).         $\hat{p}_2 = \frac{x_2}{n_2} \quad \text{and} \quad \hat{q}_2 = 1 - \hat{p}_2$

• Requirements for Testing Proportions:

  • Samples are simple random samples drawn independently from two populations.

  • Each of the two samples must contain at least 5 successes and at least 5 failures:         $x_1 \ge 5, \quad (n_1 - x_1) \ge 5, \quad x_2 \ge 5, \quad (n_2 - x_2) \ge 5$

• Pooled Sample Proportion Calculations:

  • Under the null hypothesis $H_0: p_1 = p_2$, the population proportions are presumed equal. Data from both samples are therefore combined to compute the pooled sample proportion (denoted $\bar{p}$):         $\bar{p} = \frac{x_1 + x_2}{n_1 + n_2}$         $\bar{q} = 1 - \bar{p}$

• Test Statistic Formula ($z$-distribution):     $z = \frac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{\sqrt{\bar{p}\bar{q}\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$

  • Where $(p_1 - p_2) = 0$ under the null hypothesis.

• Exemplary Case Study: Currency Denominations and Spending Habits:

  • Context: A behavioral study evaluating whether individuals exhibit different spending behaviors based on whether money is held in a single large bill versus smaller denominations.

  • Subjects: $89$ undergraduate students randomly assigned to one of two conditions, given the choice to retain the cash or purchase gum/mints.

  • Claim: Individuals holding money in large denominations are less likely to spend it relative to those possessing an equivalent amount in smaller denominations ($p_1 < p_2$).

  • Significance Level: $\alpha = 0.05$.

  • Sample Data:

    • Group 1 (Large denomination - single bill): $n_1 = 46$, $x_1 = 12$ spent the money             $\hat{p}_1 = \frac{12}{46} \approx 0.261$

    • Group 2 (Small denomination - four quarters): $n_2 = 43$, $x_2 = 27$ spent the money             $\hat{p}_2 = \frac{27}{43} \approx 0.628$

  • Hypotheses:         $H_0: p_1 = p_2 \quad (p_1 - p_2 = 0)$         $H_1: p_1 < p_2 \quad (p_1 - p_2 < 0) \quad [\text{Left-tailed}]$

  • StatCrunch Procedure: Navigate to Stat > Proportion Stats > Two Sample > With Summary. Enter successes and total observations for both groups. Set the alternative hypothesis to < 0.

  • Decision and Interpretation: The resulting $z$ test statistic falls far into the left-tail critical region, yielding a $p\text{-value} \le 0.05$. Reject $H_0$. There is sufficient statistical evidence to support the claim that individuals holding large denominations are significantly less likely to spend their funds than those holding small denominations.

Confidence Intervals for Two Independent Proportions and Discrepancy Analysis

• Confidence Level Determination for a One-Tailed Proportion Test:

  • Given $\alpha = 0.05$ in a one-tailed test, double the alpha for the confidence interval:         $\alpha_{\text{CI}} = 2 \times 0.05 = 0.10$         $\text{Confidence Level} = 1 - 0.10 = 0.90 \quad (90\% \text{ confidence})$

• Evaluating Null Values in the Interval:

  • If the resulting confidence interval does not contain zero, it indicates that the null hypothesis value of zero difference should be rejected.

• Theoretical Cause of Discrepancies Between Tests and Intervals:

  • Occasionally, a two-sample hypothesis test of proportions and a two-sample confidence interval for proportions lead to conflicting conclusions regarding $H_0$.

  • Mathematical Cause: The two inferential techniques employ different standard error calculations:

    • Hypothesis Testing: Assumes the null hypothesis is true ($p_1 = p_2$) and relies on the pooled standard error based on the combined sample proportion $\bar{p}$:             $\text{SE}_{\text{Test}} = \sqrt{\bar{p}\bar{q}\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}$

    • Confidence Intervals: Does not presume the null hypothesis is true; instead, it models variance using the separate, unpooled sample proportions:             $\text{SE}_{\text{CI}} = \sqrt{\frac{\hat{p}_1\hat{q}_1}{n_1} + \frac{\hat{p}_2\hat{q}_2}{n_2}}$

  • When an interval fails to match a hypothesis test decision, the divergence is attributable to this structural difference in standard error formulas rather than computational error.

General Decision Rules for Statistical Hypothesis Testing

• The Critical Value Method:

  • Identify the critical boundaries using distribution tables or software based on $\alpha$ and degrees of freedom (or the standard normal curve for proportions).

  • Reject $H_0$ if the test statistic falls into the extreme rejection tail(s) (the critical region).

  • Fail to reject $H_0$ if the test statistic falls in the central, non-critical region.

• The $P$-Value Method:

  • Compute the probability of obtaining a test statistic at least as extreme as the observed value, assuming the null hypothesis is true.

  • If $p\text{-value} \le \alpha$: Reject $H_0$ ("If the $p$ is low, the null must go").

  • If $p\text{-value} > \alpha$: Fail to reject $H_0$.

Questions & Discussion

• Scope of the Final Examination:

  • Question: Is there an additional exam after the upcoming Unit Exam, or does the Unit Exam conclude all material?

  • Response: The upcoming Unit Exam covers new material through Chapter 10. Following that, there is a comprehensive, cumulative Final Exam covering all material from Chapter 1 through Chapter 10.

• Post-Exam Class Attendance:

  • Question: Are students expected to attend classes after the upcoming exam has concluded?

  • Response: Attendance and scheduling for the remaining class periods after the Thanksgiving break will be formally evaluated and determined together upon returning to class.