Inference for Two Independent Means: Comprehensive Study Notes

Overview of Two-Population Inference for Means

The study of two-population inference (Section 12.1) focuses on comparing a single quantitative variable across two distinct categories.
This transition moves away from comparing a sample mean ( $\bar{x}$ ) to a specific numerical parameter (e.g., comparing a mean to exactly $30$ minutes) to comparing the means of two independent groups against each other.
Variable Structure: Each analysis involves one categorical variable (defining the groups) and one quantitative variable (the measurement being compared).
- Example: Comparing the Essex campus to the Dundalk campus. The categorical variable is the "Campus Type" and the quantitative variable is the "Commute Time."
- Example: Comparing GPA by gender. The categorical variable is "Gender" and the quantitative variable is "GPA."
Application Examples:
    - Effectiveness of a placebo versus a medical treatment.
    - Popularity levels of two different political candidates.
    - Weight times in teller services comparing a single line versus individual lines.
    - Longevity of different battery brands.
    - Cholesterol levels in patients on medication versus traditional treatment.
Note on Proportions: While a difference in proportions is a standard statistical topic, it was removed from this semester's curriculum ("on the chopping block") to prioritize means.

Hypotheses and Notation for Two Means

Notation:
    - Population Means: $\mu_1$ and $\mu_2$ .
    - Sample Means: $\bar{x}_1$ and $\bar{x}_2$ .
    - Sample Standard Deviations: $s_1$ and $s_2$ .
    - Sample Sizes: $n_1$ and $n_2$ .
    - Subscripts are essential to differentiate between the two distinct populations.
Null Hypothesis ( $H_0$ ):
- The null hypothesis always assumes no difference between the two populations: $H_0: \mu_1 = \mu_2$ .
- It can also be expressed as $H_0: \mu_1 - \mu_2 = 0$ . This version is helpful for understanding software inputs in tools like R-Guru, where the value compared is zero.
Alternative Hypothesis ( $H_a$ ):
    - This identifies the nature of the suspected difference:
        - Two-tailed (Difference): $H_a: \mu_1 \neq \mu_2$ .
        - Right-tailed (Greater than): H_a: \mu_1 > \mu_2.
        - Left-tailed (Less than): H_a: \mu_1 < \mu_2.
Importance of Order: The direction of the inequality depends on which group is assigned as population 1 and population 2. If groups are switched, the inequality must be reversed to maintain the logical claim.

Visualization and Box Plots

Box plots are an effective visual tool for comparing a quantitative variable across categories.
They allow for a quick assessment of the center (median) and the spread (interquartile range) of data.
Interpretation:
    - In a teller service wait-time example, a single-line median of $4.5$ minutes is compared to an individual-line median of $6$ minutes.
    - If box plots show significant overlap, a formal hypothesis test is required to determine if the $1.5$ -minute difference is statistically significant.
    - If the boxes were entirely separated along the y-axis, the difference would be visually obvious and likely significant without further testing.

Conditions for Hypothesis Testing

Before conducting a t-test for two means, the following conditions must be met for both samples:
- Sample Size: Both sample sizes ( $n_1$ and $n_2$ ) must be greater than or equal to $30$ ( $n \ge 30$ ).
- Normality: If the sample size is less than $30$ , the population must be approximately normal (bell-shaped curve) with no heavy skews or extreme outliers.
In many textbook scenarios where sample sizes are small, practitioners must explicitly state the assumption that the data is normally distributed.

The Test Statistic and R-Guru Procedure

Test Statistic Formula:
- The difference in sample means is divided by the standard error of the difference:
- $t = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$
R-Guru Workflow:
    1. Navigate to Analysis -> Mean Inference -> Two Population.
    2. Under the Summary tab, enter labels and data for both factors:
        - Factor 1: $\bar{x}_1$ , $s_1$ , and $n_1$ .
        - Factor 2: $\bar{x}_2$ , $s_2$ , and $n_2$ .
    3. Under the Population 1 & 2 tab:
        - Select Test for Hypothesis.
        - Set the difference value to $0$ .
        - Select the appropriate inequality ( $<$ , $>$ , or $\neq$ ).
        - Select the t-statistic.
        - Set the significance level ( $\alpha$ ). Default is $0.05$ (5%) if not specified.
Decision Rules (Same as Chapter 11):
- If p-value $\le \alpha$ : Reject the null hypothesis ( $H_0$ ).
- If p-value > \alpha: Fail to reject the null hypothesis ( $H_0$ ).

Case Study 1: Senior vs. Freshman Study Habits

Context: A study compares the average time spent studying per week by seniors versus freshmen.
Hypotheses:
- $H_0: \mu_1 = \mu_2$ (No difference).
- H_a: \mu_1 > \mu_2 (Seniors study more than freshmen).
Data Provided:
- Seniors (Group 1): $\bar{x}_1 = 15.6$ hours, $s_1 = 3.9$ , $n_1 = 60$ .
- Freshmen (Group 2): $\bar{x}_2 = 13.7$ hours, $s_2 = 4.8$ , $n_2 = 75$ .
Analysis:
- Level of significance ( $\alpha$ ): $0.05$ .
- p-value: $0.006$ .
Conclusion: Since 0.006 < 0.05, reject the null hypothesis. There is sufficient evidence to suggest that seniors, on average, study more than freshmen.

Case Study 2: Cordless Phone Range Comparison

Context: Comparing the long-distance range of two cordless phone brands.
Data Provided:
- Phone 1: $\bar{x}_1 = 1390$ units, $s_1 = 36$ , $n_1 = 5$ .
- Phone 2: $\bar{x}_2 = 1340$ units, $s_2 = 33$ , $n_2 = 11$ .
Conditions: Since sample sizes are small (n < 30), we must assume the data is normally distributed.
Analysis:
    - Null Hypothesis ( $H_0$ ): $\mu_1 = \mu_2$ .
    - Alternative Hypothesis ( $H_a$ ): \mu_1 > \mu_2 (Claim: Phone 1 is better than Phone 2).
    - Significance Level ( $\alpha$ ): $0.01$ (1%).
    - Resulting p-value was found to be greater than $0.01$ .
Conclusion: Fail to reject the null hypothesis. At the $1\%$ significance level, there is not enough evidence to suggest Phone 1 has a longer average range than Phone 2, even though a $5\%$ or $10\%$ level might have yielded a different result.

Case Study 3: Cholesterol Medication vs. Placebo

Context: Testing if a drug has a greater decrease in cholesterol compared to a placebo.
Data Provided:
- Drug (Group 1): $\bar{x}_1 = 22.9$ , $s_1 = 4.4$ , $n_1 = 49$ .
- Placebo (Group 2): $\bar{x}_2 = 20.9$ , $s_2 = 20.5$ , $n_2 = 35$ .
Analysis:
    - Hypotheses: $H_0: \mu_1 = \mu_2$ ; H_a: \mu_1 > \mu_2.
    - Significance Level ( $\alpha$ ): $0.05$ .
    - Test Statistic (t): $2.64$ .
    - p-value: $0.005$ .
Conclusion: Since 0.005 < 0.05, reject the null hypothesis. There is enough evidence to suggest the drug results in a greater average decrease in cholesterol compared to the placebo.

Questions & Discussion

Student Inquiry on R-Guru Settings: A student asked about the "greater than" inequality setting in R-Guru regarding comparing the means in the senior/freshman study.
Response: The instructor confirmed that the setting should show the difference ( $\mu_1 - \mu_2$ ) is greater than zero, which aligns with the hypothesis that the first group is larger than the second.
Student Inquiry on Variances: A student asked about assuming population variances are equal for the cordless phone problem.
Response: The instructor noted that while the book may suggest this, in this course, standard deviations are typically treated as unknown and variances are not assumed to be equal for these specific t-tests.
Student Inquiry on p-value meaning: A student asked for clarification on what the p-value means in context.
Response: The instructor explained that it determines whether the data provides enough evidence to reject the null hypothesis based on its size relative to $\alpha$ . Small p-values indicate strong evidence against the null.