Comparing Two Population Parameters: Exhaustive Guide to Proportions and Means

Distinguishing between Independent and Dependent Sampling

• Objective 1: Distinguish between independent and dependent sampling.

• Definitions:   - Independent Sampling: A sampling method is considered independent when an individual selected for one sample does not dictate which individual is to be in a second sample.   - Dependent Sampling: A sampling method is dependent when an individual selected to be in one sample is used to determine the individual in the second sample.   - Matched-Pairs Samples: This is another term for dependent samples. In some cases, an individual might be matched against him- or herself (e.g., before-and-after measurements).

• Examples of Distinguishing Sampling Methods:   - Example (a): Hotel Prices: A researcher compares the price of a one-night stay at a Holiday Inn Express versus a Red Roof Inn. She randomly selects 8 towns where the hotel locations are close to each other.     - Classification: Dependent.     - Reasoning: Once a location is picked for the Holiday Inn Express, the Red Roof Inn must be chosen from the same or a nearby location.   - Example (b): Comic Preferences: A polling agency obtained a random sample of 500 young adults (18-39) and 550 senior adults (60-89) and asked if they prefer Marvel or DC Comics.     - Classification: Independent.     - Reasoning: The selection of young adults in the first sample is not related to the selection of seniors in the second sample.

Hypothesis Testing for Two Population Proportions (Independent Samples)

• Objective 2: Test hypotheses regarding two proportions from independent samples.

• Sampling Distribution Principles:   - To conduct inference, the sampling distribution of the difference of two proportions must be determined. Random assignment often suggests an approximately normal distribution.   - Parameters and Statistics:     - Suppose a simple random sample of size $n_1$ is taken from population 1 where $x_1$ individuals have a characteristic.     - A simple random sample of size $n_2$ is independently taken from population 2 where $x_2$ individuals have the same characteristic.     - The sample proportions are defined as $\hat{p}_1 = \frac{x_1}{n_1}$ and $\hat{p}_2 = \frac{x_2}{n_2}$.

• Distribution Properties:   - The sampling distribution of $\hat{p}_1 - \hat{p}_2$ is approximately normal if requirements are met.   - Mean: $\mu_{\hat{p}_1 - \hat{p}_2} = p_1 - p_2$.   - Standard Deviation: $\sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$.

• Standardized Test Statistic (Z-score):   - $Z = \frac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}}$   - This follows an approximate standard normal distribution.

• Requirements for the Hypothesis Test:   - Samples must be independently obtained via simple random sampling or a completely randomized experiment with two levels of treatment.   - Success-failure counts: $n_1 \hat{p}_1 (1 - \hat{p}_1) \ge 10$ and $n_2 \hat{p}_2 (1 - \hat{p}_2) \ge 10$.   - Independence of observations: Sample size must be no more than 5% of the population size ($n_1 \le 0.05N_1$ and $n_2 \le 0.05N_2$).

• Case Study 1: Diabetes Treatments (The ADOPT Study):   - Context: Does Avandia (a diabetes drug) increase heart attacks compared to other treatments?   - Data points:     - Group 1 (Avandia): $n_1 = 1456$, $x_1 = 27$, $\hat{p}_1 = \frac{27}{1456} \approx 0.0185$.     - Group 2 (Other): $n_2 = 2895$, $x_2 = 41$, $\hat{p}_2 = \frac{41}{2895} \approx 0.0142$.   - Requirement Verification:     - $1456(0.0185)(1-0.0185) = 26.44 \ge 10$ (Check).     - $2895(0.0142)(1-0.0142) = 40.53 \ge 10$ (Check).   - Hypotheses:     - $H_0: p_1 = p_2$ (or $H_0: p_1 - p_2 = 0$).     - H_1: p_1 > p_2 (or H_1: p_1 - p_2 > 0).   - Results: $Z_{stat} = 1.20$; $P\text{-value} = 0.1358$.   - Conclusion: Since 0.1358 > 0.05, we do not reject the null hypothesis. There is no significant evidence of an increased heart attack proportion.

• Case Study 2: Unplugging from Devices (Harris Interactive Survey):   - Context: Is there a difference in feelings about returning to a time before being "plugged in" between adults (35-54) and baby-boomers (55+)?   - Data points:     - Group 1 (Adults): $n_1 = 500$, $x_1 = 385$, $\hat{p}_1 = \frac{385}{500} = 0.77$.     - Group 2 (Boomers): $n_2 = 500$, $x_2 = 300$, $\hat{p}_2 = \frac{300}{500} = 0.60$.   - Requirement Verification:     - $500(0.77)(1-0.77) = 88.55 \ge 10$ (Check).     - $500(0.60)(1-0.60) = 120 \ge 10$ (Check).   - Hypotheses:     - $H_0: p_1 - p_2 = 0$.     - $H_1: p_1 - p_2 \neq 0$.   - Results: $Z_{stat} = 5.787$; P\text{-value} < 0.0001.   - Conclusion: Since the P-value is less than $0.05$, we reject the null hypothesis.

Confidence Intervals for Two Population Proportions

• Objective 3: Construct and interpret confidence intervals for the difference between two population proportions.

• Formula for a $(1 - \alpha) \times 100\%$ Confidence Interval:   - Lower Bound: $(\hat{p}_1 - \hat{p}_2) - Z_{\alpha/2} \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$   - Upper Bound: $(\hat{p}_1 - \hat{p}_2) + Z_{\alpha/2} \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

• Requirement Check: Same as the hypothesis test requirements ($SRS$, counts $\ge 10$, $n \le 0.05N$).

• Case Study: Position on Divorce (Harris Interactive Survey):   - Groups: Religious ($n_1 = 970$, $x_1 = 834$, $\hat{p}_1 \approx 0.860$) and Not Religious ($n_2 = 1285$, $x_2 = 1157$, $\hat{p}_2 \approx 0.900$).   - Interval Calculation (95% Confidence):     - Lower Bound: $-0.0679$     - Upper Bound: $-0.0133$   - Interpretation: We are 95% confident that the proportion of religious people who find divorce acceptable is between $1.33\%$ and $6.79\%$ less than that of non-religious individuals.   - Note: Since 0 is not included in the interval and both bounds are negative (p_1 < p_2), we conclude there is a significant difference.

Sample Size Estimation for Proportions

• Objective 4: Determine the sample size necessary for estimating the difference between two population proportions.

• Calculation Rule: Always round up to the next integer for sample size calculations.

• Formula Case 1: Prior Estimates of $\hat{p}_1$ and $\hat{p}_2$ are available:   - $n = n_1 = n_2 = [\hat{p}_1(1-\hat{p}_1) + \hat{p}_2(1-\hat{p}_2)] \left(\frac{Z_{\alpha/2}}{E}\right)^2$

• Formula Case 2: Prior Estimates are unavailable:   - $n = n_1 = n_2 = 0.5 \left(\frac{Z_{\alpha/2}}{E}\right)^2$

• Case Study: Prenatal Care Mothers (15-19 years vs. 30-34 years):   - Goal: Estimate difference within 2 percentage points ($E = 0.02$) with 95% confidence.   - (a) With Prior Estimates ($\hat{p}_1 = 0.98$, $\hat{p}_2 = 0.992$):     - Result: Sample size required is $n = 265$.   - (b) Without Prior Estimates (Using 0.5):     - Result: Sample size required is $n = 4812$.

Inference about Two Means: Dependent Samples (Matched-Pairs)

• Objective 1: Test hypotheses for a population mean from matched-pairs data.

• Requirement Factors:   - Sample obtained by simple random sampling or matched-pair design.   - Samples are dependent.   - Differences are normally distributed with no outliers or $n \ge 30$.   - Sample size $n \le 0.05N$.

• Test Statistic for Matched Pairs:   - $t = \frac{\bar{d} - \mu_d}{s_d / \sqrt{n}}$   - Follows Student's t-distribution with $n - 1$ degrees of freedom ($d$ stands for "difference").

• Objective 2: Confidence Intervals for Matched-Pairs Data.   - Formula: $\bar{d} \pm t_{\alpha/2} \left(\frac{s_d}{\sqrt{n}}\right)$   - Point estimate $\pm$ margin of error.

• Case Study: Hotel Prices (Hampton Inn vs. La Quinta):   - Data collected from Dallas, Tampa Bay, St. Louis, Seattle, San Diego, Chicago, New Orleans, Phoenix, Atlanta, and Orlando ($n = 10$).   - Objective: Construct 95% CI for the mean difference.

• Case Study: Disney Wait Times:   - Comparing "Pirates of the Caribbean" Wait Times to "Tiana's Bayou Adventure" (formerly Splash Mountain).   - Data is paired by specific day and time of day (18 observations).   - Hypothesis: Wait times for Tiana's are longer than Pirates.

Inference about Two Means: Independent Samples

• Objective 1: Test hypotheses regarding the difference of two independent means.

• Behrens-Fisher Problem: The case where population variances are unequal and unknown. The solution used is Welch's approximate t.

• Welch's t-test Statistic:   - $t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$   - Degrees of Freedom ($df$): Use the smaller of $n_1 - 1$ or $n_2 - 1$.

• Requirements:   - Simple random samples or randomized experiment.   - Independent samples.   - Populations are normally distributed or $n_1 \ge 30$ and $n_2 \ge 30$.   - Sample size is no more than 5% of the population.

• Comparison Caution on Pooling:   - Pooled two-sample t-tests are for equal variances. Because equality of variance is hard to verify, Welch's t is always preferred.

• Case Study: Exam Paper Color:   - Question: Does the color of paper (White vs. Marine Blue) affect results?   - Hypotheses:     - $H_0: \mu_1 = \mu_2$ (or $\mu_1 - \mu_2 = 0$).     - H_1: \mu_1 > \mu_2 (or \mu_1 - \mu_2 > 0).   - Results: $t_0 \approx 2.216$; $P\text{-value} = 0.0175$.   - Conclusion: Since 0.0175 < 0.05, reject the null hypothesis. There is evidence that scores are higher on white paper.

• Objective 2: Confidence Intervals for Independent Means.   - Formula: $(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$   - Example: Salary by Degree:     - Engineer ($n_1=80$, $\bar{x}_1=60,300$, $s_1=10,822$) vs. Psychology ($n_2=110$, $\bar{x}_2=34,200$, $s_2=10,698$).     - 95% CI result: Between $22,972.98 and $29,224.03.

Summary: Which Method to Use?

• Step 1: Determine the Parameter:   - Proportion ($p$): Use Normal distribution ($Z$) provided $n\hat{p}(1-\hat{p}) \ge 10$ and Sample size $\le 5\%$ of population.   - Mean ($\mu$): Categorize by sampling method.

• Step 2: Determine Sampling Method:   - Independent Samples (Proportion): Use $Z = \frac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}}$.   - Independent Samples (Mean): Use Student's t with Welch's adjustment $t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$, $df = \min(n_1-1, n_2-1)$.   - Dependent Samples (Mean/Matched-Pairs): Use Student's t with $t = \frac{\bar{d} - \mu_d}{s_d / \sqrt{n}}$, $df = n - 1$.

• Final Examples for Identification:   - Economic System Fair Poll: Comparing 200 Democrats and 160 Republicans (Independent, Proportion).   - Pulse Rates: Testing pulse before and after a fright (Dependent, Mean).