7.3 Inference Of The Difference Of Two Means

The ability to compare the means of two different populations is a cornerstone of statistical analysis, allowing us to draw meaningful conclusions about whether observed differences are truly significant or simply due to random chance. In the realm of inferential statistics, Section 7.3 focuses specifically on inference of the difference of two means, providing a rigorous framework for testing hypotheses and constructing confidence intervals. This framework is essential in a wide array of fields, from medicine and engineering to social sciences and business, where comparing outcomes across different groups is a common objective.

Understanding the Core Concepts

At its heart, inference of the difference of two means seeks to determine if there's a statistically significant difference between the average values of a particular variable in two distinct populations. For example, we might want to compare the average test scores of students taught using two different methods, or the average sales figures for two different marketing campaigns.

Before diving into the specifics, let's clarify some key concepts:

• Population Mean (µ): The true average value of a variable within an entire population. Since it's often impractical to measure this directly, we rely on samples.
• Sample Mean (x̄): The average value of a variable calculated from a sample drawn from a population. This serves as an estimate of the population mean.
• Independent Samples: Samples drawn from two populations where the observations in one sample are not related to the observations in the other sample. This is a crucial assumption for many of the techniques we'll discuss.
• Hypothesis Testing: A procedure for determining whether there is enough evidence to reject a null hypothesis.
• Null Hypothesis (H₀): A statement of no effect or no difference. In the context of two means, it often takes the form H₀: µ₁ = µ₂ (the population means are equal).
• Alternative Hypothesis (H₁): A statement that contradicts the null hypothesis. It could be that µ₁ ≠ µ₂, µ₁ > µ₂, or µ₁ < µ₂.
• Significance Level (α): The probability of rejecting the null hypothesis when it is actually true. Common values are 0.05 (5%) and 0.01 (1%).
• P-value: The probability of observing a sample statistic as extreme as, or more extreme than, the one observed, assuming the null hypothesis is true. A small p-value suggests evidence against the null hypothesis.
• Confidence Interval: A range of values within which we are reasonably confident that the true difference between the population means lies.

The Different Scenarios

The specific method used for inference of the difference of two means depends on several factors, primarily whether the population variances are known or unknown, and whether the samples are independent or dependent (paired). We'll focus on the most common scenario: independent samples.

Within independent samples, we further distinguish between two cases:

1. Population Variances Known: This is a less common scenario in practice, but it provides a good starting point for understanding the underlying principles. When the population variances (σ₁² and σ₂²) are known, we can use the standard normal distribution (Z-distribution) for hypothesis testing and confidence interval construction.
2. Population Variances Unknown: This is the more realistic and frequently encountered scenario. When the population variances are unknown, we estimate them using the sample variances (s₁² and s₂²) and use the t-distribution for inference. This introduces a degree of uncertainty due to the estimation process.

Case 1: Population Variances Known

When the population variances are known, the test statistic for testing the difference between two means is given by:

z = (x̄₁ - x̄₂) / √(σ₁²/n₁ + σ₂²/n₂)

where:

• x̄₁ and x̄₂ are the sample means
• σ₁² and σ₂² are the population variances
• n₁ and n₂ are the sample sizes

Hypothesis Testing:

• Null Hypothesis (H₀): µ₁ = µ₂
• Alternative Hypothesis (H₁):
  • µ₁ ≠ µ₂ (two-tailed test)
  • µ₁ > µ₂ (right-tailed test)
  • µ₁ < µ₂ (left-tailed test)

Based on the chosen significance level (α) and the type of alternative hypothesis, we determine the critical value(s) from the Z-distribution. If the calculated test statistic (z) falls in the rejection region (beyond the critical value(s)), we reject the null hypothesis. Alternatively, we can calculate the p-value and compare it to α. If the p-value is less than α, we reject the null hypothesis.

Confidence Interval:

The confidence interval for the difference between two population means is:

(x̄₁ - x̄₂) ± zα/₂ * √(σ₁²/n₁ + σ₂²/n₂)

where zα/₂ is the z-score corresponding to the desired confidence level (e.g., for a 95% confidence interval, α = 0.05, and zα/₂ = 1.96).

Example:

Suppose we want to compare the average height of men and women. We know that the population standard deviation for men is 3 inches and for women is 2.5 inches. We take a sample of 50 men and find their average height to be 70 inches, and a sample of 60 women and find their average height to be 65 inches.

• x̄₁ (men) = 70 inches, n₁ = 50, σ₁ = 3 inches
• x̄₂ (women) = 65 inches, n₂ = 60, σ₂ = 2.5 inches

Let's test the hypothesis that men are taller than women at a significance level of 0.05.

• H₀: µ₁ = µ₂
• H₁: µ₁ > µ₂ (right-tailed test)

The test statistic is:

z = (70 - 65) / √(3²/50 + 2.5²/60) = 5 / √(0.18 + 0.104) = 5 / √0.284 ≈ 9.37

The critical value for a right-tailed test at α = 0.05 is 1.645. Since our test statistic (9.37) is much larger than 1.645, we reject the null hypothesis and conclude that there is significant evidence that men are taller than women.

A 95% confidence interval for the difference in average height is:

(70 - 65) ± 1.96 * √(3²/50 + 2.5²/60) = 5 ± 1.96 * √0.284 ≈ 5 ± 1.05

The 95% confidence interval is (3.95, 6.05) inches. This means we are 95% confident that the true difference in average height between men and women is between 3.95 and 6.05 inches.

Case 2: Population Variances Unknown

When the population variances are unknown, we must estimate them from the sample data. This introduces additional uncertainty, which is accounted for by using the t-distribution instead of the Z-distribution. There are two primary approaches within this case:

1. Equal Variances Assumed (Pooled t-test): If we have reason to believe that the population variances are equal (σ₁² = σ₂²), we can pool the sample variances to obtain a better estimate of the common variance.
2. Unequal Variances Assumed (Welch's t-test): If we don't have reason to believe that the population variances are equal, we should use a modified t-test that does not assume equal variances. This is often referred to as Welch's t-test.

a) Equal Variances Assumed (Pooled t-test):

The pooled variance (sₚ²) is calculated as:

sₚ² = [(n₁ - 1)s₁² + (n₂ - 1)s₂²] / (n₁ + n₂ - 2)

where:

• s₁² and s₂² are the sample variances

The test statistic is:

t = (x̄₁ - x̄₂) / √(sₚ²/n₁ + sₚ²/n₂)

The degrees of freedom for the t-distribution are:

df = n₁ + n₂ - 2

Hypothesis Testing:

The hypothesis testing procedure is similar to the case with known variances, but we use the t-distribution with the appropriate degrees of freedom to determine the critical value(s) or p-value.

Confidence Interval:

The confidence interval for the difference between two population means is:

(x̄₁ - x̄₂) ± tα/₂,df * √(sₚ²/n₁ + sₚ²/n₂)

b) Unequal Variances Assumed (Welch's t-test):

Welch's t-test is used when we cannot assume equal population variances. The test statistic is:

t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂)

The degrees of freedom for the t-distribution are approximated using the Welch-Satterthwaite equation:

df ≈ (s₁²/n₁ + s₂²/n₂)² / [(s₁²/n₁)² / (n₁ - 1) + (s₂²/n₂)² / (n₂ - 1)]

This formula generally results in non-integer degrees of freedom, which can be rounded down to the nearest integer.

Hypothesis Testing:

The hypothesis testing procedure is the same as before, but we use the t-distribution with the approximated degrees of freedom to determine the critical value(s) or p-value.

Confidence Interval:

The confidence interval for the difference between two population means is:

(x̄₁ - x̄₂) ± tα/₂,df * √(s₁²/n₁ + s₂²/n₂)

Example:

Suppose we want to compare the effectiveness of two different fertilizers on crop yield. We randomly assign 15 plots of land to fertilizer A and 12 plots to fertilizer B. The yields (in bushels per acre) are:

• Fertilizer A: x̄₁ = 85, s₁ = 10, n₁ = 15
• Fertilizer B: x̄₂ = 80, s₂ = 8, n₂ = 12

Let's test the hypothesis that fertilizer A results in a higher yield than fertilizer B at a significance level of 0.05. Since we don't know the population variances, we'll use the t-test. We need to decide whether to assume equal variances or not. A common rule of thumb is to check if the ratio of the larger sample variance to the smaller sample variance is less than 4. In this case, 10²/8² = 1.5625 < 4, so we can proceed with the pooled t-test (assuming equal variances). However, it's always a good practice to perform a formal test for equality of variances, such as Levene's test, to confirm this assumption. For demonstration, we will proceed with the pooled t-test.

• H₀: µ₁ = µ₂
• H₁: µ₁ > µ₂ (right-tailed test)

First, calculate the pooled variance:

sₚ² = [(15 - 1) * 10² + (12 - 1) * 8²] / (15 + 12 - 2) = (14 * 100 + 11 * 64) / 25 = (1400 + 704) / 25 = 2104 / 25 = 84.16

The test statistic is:

t = (85 - 80) / √(84.16/15 + 84.16/12) = 5 / √(5.61 + 7.01) = 5 / √12.62 ≈ 5 / 3.55 ≈ 1.41

The degrees of freedom are:

df = 15 + 12 - 2 = 25

The critical value for a right-tailed t-test with df = 25 and α = 0.05 is approximately 1.708. Since our test statistic (1.41) is less than 1.708, we fail to reject the null hypothesis. We don't have sufficient evidence to conclude that fertilizer A results in a higher yield than fertilizer B.

A 95% confidence interval for the difference in yield is:

(85 - 80) ± 2.060 * √(84.16/15 + 84.16/12) = 5 ± 2.060 * √12.62 ≈ 5 ± 2.060 * 3.55 ≈ 5 ± 7.31

The 95% confidence interval is (-2.31, 12.31) bushels per acre. Notice that this interval includes 0, which is consistent with our failure to reject the null hypothesis.

Paired Samples (Dependent Samples)

So far, we've focused on independent samples. However, in some situations, the samples are dependent or paired. This occurs when each observation in one sample is naturally paired with a corresponding observation in the other sample. Examples include:

• Measuring a patient's blood pressure before and after taking a medication.
• Comparing the fuel efficiency of two different car models using the same drivers and routes.
• Testing the performance of employees before and after a training program.

In the case of paired samples, we analyze the differences between the paired observations. Let dᵢ be the difference between the i-th pair of observations. We then calculate the mean difference (d̄) and the standard deviation of the differences (s_d).

The test statistic for testing the difference between two means with paired samples is:

t = d̄ / (sd / √n)

where n is the number of pairs.

The degrees of freedom for the t-distribution are:

df = n - 1

Hypothesis Testing:

The hypothesis testing procedure is similar to the independent samples case, but we use the t-distribution with df = n - 1.

Confidence Interval:

The confidence interval for the difference between two population means is:

d̄ ± tα/₂,df * (sd / √n)

Example:

Suppose we want to evaluate the effectiveness of a weight loss program. We measure the weight of 10 participants before and after the program. The data is shown below:

Participant	Weight Before (lbs)	Weight After (lbs)	Difference (lbs)
1	200	190	10
2	180	175	5
3	220	210	10
4	240	225	15
5	190	185	5
6	170	165	5
7	210	200	10
8	230	220	10
9	185	180	5
10	205	195	10

The mean difference is:

d̄ = (10 + 5 + 10 + 15 + 5 + 5 + 10 + 10 + 5 + 10) / 10 = 85 / 10 = 8.5

The standard deviation of the differences is (calculated using the differences column):

sd ≈ 3.54

Let's test the hypothesis that the weight loss program is effective (i.e., participants lose weight) at a significance level of 0.05.

• H₀: µ_d = 0 (no difference in weight)
• H₁: µ_d > 0 (weight loss) (right-tailed test)

The test statistic is:

t = 8.5 / (3.54 / √10) ≈ 8.5 / 1.12 ≈ 7.59

The degrees of freedom are:

df = 10 - 1 = 9

The critical value for a right-tailed t-test with df = 9 and α = 0.05 is approximately 1.833. Since our test statistic (7.59) is much larger than 1.833, we reject the null hypothesis and conclude that there is significant evidence that the weight loss program is effective.

A 95% confidence interval for the mean difference in weight is:

8.  5 ± 1.833 * (3.54 / √10) ≈ 8.5 ± 1.833 * 1.12 ≈ 8.5 ± 2.05

The 95% confidence interval is (6.45, 10.55) lbs. This means we are 95% confident that the true average weight loss is between 6.45 and 10.55 lbs.

Assumptions and Considerations

It's crucial to be aware of the assumptions underlying these statistical tests:

• Normality: The data (or the differences, in the case of paired samples) should be approximately normally distributed. This assumption is less critical with larger sample sizes due to the central limit theorem. However, if the data are severely non-normal, consider using non-parametric tests.
• Independence: The samples should be independent (except for paired samples, where the dependence is accounted for).
• Equal Variances (for Pooled t-test): If using the pooled t-test, the population variances should be approximately equal. As mentioned earlier, perform a test of equality of variances.
• Random Sampling: The data should be obtained through random sampling to ensure representativeness.

When assumptions are violated:

• Non-parametric tests: If the normality assumption is violated, consider using non-parametric alternatives like the Mann-Whitney U test (for independent samples) or the Wilcoxon signed-rank test (for paired samples).
• Transformations: Sometimes, data transformations (e.g., logarithmic transformation) can help to normalize the data.

Conclusion

Inference of the difference of two means is a powerful tool for comparing populations and drawing meaningful conclusions. By understanding the different scenarios, assumptions, and appropriate statistical tests, you can effectively analyze data and make informed decisions in a wide range of applications. Whether you're comparing the effectiveness of new treatments, evaluating marketing campaigns, or analyzing social trends, the principles outlined in this guide provide a solid foundation for rigorous statistical inference. Remember to always consider the context of your data, check assumptions, and interpret your results carefully. Selecting the correct test (Z-test, pooled t-test, Welch's t-test, or paired t-test) is paramount for obtaining valid and reliable insights.