Chapter 6

Chapter 6: Comparing Two Means (Independent Samples)

Course: STAT 1 3: Introduction to Statistical Methods for Life and Health SciencesInstructor: J.H. Sparks Date: Winter 2025

Two-Sample Problems: Quantitative

• This chapter focuses on comparing two means, an important aspect of statistical inference.

• Review of previous chapters: confidence intervals, hypothesis tests, inferences about means and proportions.

• Key Property: Independence or Dependence of Samples

  • Chapter 6 assumes samples are independent.

  • Chapter 7 will explore dependent samples.

• Goals of Chapter 6:

  • Compare populations and assess normality (Section 6.1+)

  • Compare means via simulation (Section 6.2) and theoretical approaches (Section 6.3)

Section 6.1: Comparing Two Groups: Quantitative Response

• Importance of checking assumptions about population distributions.

• Use descriptive statistics and distributions from sample data.

• Descriptive Measures:

  • Five Number Summary for data visualization.

  • Boxplot construction to assess data symmetry.

  • Normal probability plot to evaluate distribution normality.

Re-Examining Variability & Measures of Relative Standing

• Definition of the p-th Percentile:

  • A value such that p% of observations fall below it.

• Review of Chapter 2 concepts:

  • Measures of Central Tendency: mean, median (50th percentile), mode.

  • Measures of Variability: range, variance, standard deviation.

  • Introduction of Quartiles:

    • 25th percentile: Q1 (Lower quartile)

    • 50th percentile: Median (M)

    • 75th percentile: Q3 (Upper quartile)

Finding the Quartiles

• For an odd number of observations, the median is the middle value in an ordered set.

• For an even number of observations, the median is the average of the two middle values.

• Methodology for locating quartiles:

  1. For odd count: omit the median when locating Q1 and Q3.

  2. For even count: include all observations when locating Q1 and Q3.

• Note: Definitions of quartiles may vary by statistician or software.

IQR & the Five-Number Summary

• Interquartile Range (IQR): Difference between Q3 and Q1, IQR = Q3 - Q1

• Preferred measure of variation with median as the center.

• The five-number summary includes:

  • Minimum

  • Q1

  • Median (M)

  • Q3

  • Maximum

• IQR provides insight into variation in data across the four quarters.

Detecting Outliers

• Use the five-number summary and IQR to identify outliers.

• Inner Fences:

  • Lower Inner Fence: Q1 - 1.5 * IQR

  • Upper Inner Fence: Q3 + 1.5 * IQR

• Observations outside these fences are potential outliers.

• Outer Fences:

  • Lower Outer Fence: Q1 - 3 * IQR

  • Upper Outer Fence: Q3 + 3 * IQR

• Adjacent values are the most extreme that are still within the inner fences.

Boxplots

• Boxplots visualize quantitative data.

• Created by constructing a box at Q1 and Q3, with median shown as a dividing line.

• Whiskers extend to adjacent values and dots indicate potential outliers.

• Developed by Professor John Tukey (also introduced the stem-and-leaf plot).

Comparisons and Caveats

• Boxplots allow for quick comparisons of five-number summaries across groups.

• Limitations:

  • May lose detailed shape of distribution.

  • Cannot identify multimodal distributions or clusters.

• Recommended to combine with dotplots or histograms for small data sets.

Examining Normality

• Importance of checking normality for valid statistical techniques.

• Procedures for checking:

  1. Histogram should be bell-shaped for large samples.

  2. Compute intervals around mean and evaluate percentage of data in each.

Normal Probability Plot

• A normal probability plot (Q-Q plot) evaluates normality by comparing observed and expected values.

• A roughly linear plot indicates normal distribution; deviations suggest otherwise.

Section 6.2: Comparing Two Groups: Quantitative Response

• This section investigates differences between two samples.

• Hypotheses:

  • Null Hypothesis (H0): No association between treatment and group.

  • Alternative Hypothesis (H1): There is an association between treatment and group.

Example: Alcohol's Effect on Reaction Times

• Investigated the effect of alcohol on driving reaction times with two groups (alcohol vs placebo).

• Aim: To determine if alcohol affects average reaction times.

• 95% confidence interval will be constructed for the difference in reaction times.

Constructing a Test Statistic

• Method involves shuffling data to simulate random reassignments of treatments.

• Aim: Determine if means differ significantly.

Confidence Interval

• The sampling distribution gives mean and standard deviation for the difference in means.

Section 6.3: Comparing Two Means: Theoretical Approach

• Transition from simulation to theoretical method for mean comparisons.

• Point estimate for difference between population means using sample means.

Validity Conditions for the Two-Independent Sample t-Procedure

• Conditions for conducting statistical inference include:

  1. Large samples (n1 ≥ 30, n2 ≥ 30) for normal approximation.

  2. About normal or symmetric distributions for smaller samples.

  3. Robust conditions for 20+ observations if distributions are not skewed.

Equal vs. Unequal Variances

• Discussion of procedures for small samples and equal/unequal variances.

• Special procedures apply based on variance equality:

  • Pooled estimates used for equal variances.

  • Satterthwaite approximation for unequal variances.

Confidence Interval & Relation to Hypothesis Testing

• Establish the relationship between confidence intervals and hypothesis tests.

R Output and Final Notes

• Use of R for hypothesis tests and confidence intervals.

• Implementation of non-parametric methods such as Wilcoxon Rank Sum Test and Kolmogorov-Smirnov test as alternatives.