Inferences Comparing Two Population Means: Independent and Dependent Samples

Inferences Comparing Two Population Means

Introduction

Statistical inference about two population means is a fundamental topic in inferential statistics. It involves comparing the means of two populations using sample data, which can be either independent or dependent. The choice of method depends on the relationship between the samples.

Types of Samples

Independent Samples

Two samples are independent if the selection of subjects from one population is unrelated to the selection from the other population.

• Example: Comparing test scores from students in two different schools.

Dependent Samples (Paired Samples)

Samples are dependent if each subject in one sample is paired with a subject in the other sample, often forming similar pairs of objects.

• Example: Measuring blood pressure before and after treatment in the same individuals.

Inferences About Two Population Means: Independent Samples

Notation and Parameters

Population I	Population II
$\mu_1$ = Population mean $\sigma_1$ = Population Std. Dev.	$\mu_2$ = Population mean $\sigma_2$ = Population Std. Dev.
Sample I	Sample II
$\bar{y}_1$ = Sample mean $s_1$ = Sample Std. Dev. $n_1$ = Sample size	$\bar{y}_2$ = Sample mean $s_2$ = Sample Std. Dev. $n_2$ = Sample size

Conditions for Using the t-Distribution

• Two random samples must be selected, one from each population of interest.

• The samples must be independent.

• Either both populations must be normally distributed, or both sample sizes ($n_1$, $n_2$) must be at least 30.

• The population standard deviations ($\sigma_1$, $\sigma_2$) are unknown.

Sampling Distribution of $\bar{y}_1 - \bar{y}_2$

• Mean: $\mu_1 - \mu_2$

• Standard Error: $\sigma_{\bar{y}_1 - \bar{y}_2} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$ Note: In practice, use sample standard deviations $s_1$ and $s_2$.

Test Statistic for Independent Samples

When population standard deviations are unknown:

• Test Statistic: $t = \frac{(\bar{y}_1 - \bar{y}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

Confidence Interval for $\mu_1 - \mu_2$

• Formula: $(\bar{y}_1 - \bar{y}_2) \pm t \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

• $t$ is the critical value from the t-distribution.

Elements of a t-Test of Hypothesis

• Null Hypothesis ($H_0$): $\mu_1 - \mu_2 = 0$

• Alternative Hypothesis ($H_a$): $\mu_1 - \mu_2 < 0$, $\mu_1 - \mu_2 > 0$, or $\mu_1 - \mu_2 \neq 0$

• Decision Rule: If p-value < $\alpha$, reject $H_0$.

Example Application: Braces Study

A laboratory compares the mean wearing times for standard braces and new bands in children. Forty children are randomly assigned to each group. The number of days required for correction is recorded. The goal is to determine if the new bands require less time on average, using a significance level of 0.01.

• Type I Error: Concluding the new bands are different when they are not.

• Type II Error: Failing to detect a difference when one exists.

• Confidence Interval: Estimate the difference in mean wearing time with a specified level of confidence.

Inferences About Two Population Means: Dependent Samples

Notation and Parameters

Population of Differences
$\mu_D$ = Population mean of differences $\sigma_D$ = Population Std. Dev. of differences
Sample of Differences
$\bar{y}_D$ = Sample mean of differences $s_D$ = Sample Std. Dev. of differences $n$ = Sample size

Conditions for Using the t-Statistic

• A random sample of paired observations must be selected.

• The population of differences must be normally distributed, or the sample size $n$ must be at least 30.

• The population standard deviation $\sigma_D$ is unknown.

Test Statistic for Dependent Samples

• Test Statistic: $t = \frac{\bar{y}_D - \mu_D}{s_D / \sqrt{n}}$

Confidence Interval for $\mu_D$

• Formula: $\bar{y}_D \pm t \frac{s_D}{\sqrt{n}}$

• $t$ is the critical value from the t-distribution.

Elements of a t-Test of Hypothesis

• Null Hypothesis ($H_0$): $\mu_D = 0$

• Alternative Hypothesis ($H_a$): $\mu_D < 0$, $\mu_D > 0$, or $\mu_D \neq 0$

• Decision Rule: If p-value < $\alpha$, reject $H_0$.

Example Application: Biofeedback Blood Pressure Study

Researchers measure blood pressure in subjects before and after biofeedback training. The difference in measurements is analyzed to determine if training reduces blood pressure, using a paired t-test and a 95% confidence interval to estimate the mean reduction.

Statistical Analysis Using Software

Interpreting Output Tables

Statistical software (e.g., StatCrunch) provides hypothesis test results and confidence intervals. The following table summarizes typical output:

Hypothesis Test Results	90% Confidence Interval Results
Difference Sample Diff. Std. Err. DF T-Stat P-value	Difference Sample Diff. Std. Err. DF L Limit U Limit
1.3 1.2931443 13.029 1.0050316 0.166	1.3 1.2931443 13.029 -0.97875273 3.5787527

Additional info: The table above is inferred from the context and typical StatCrunch output for two-sample t-tests.

Steps in Hypothesis Testing

1. State the parameters of interest.

2. Formulate null and alternative hypotheses.

3. State the decision rule (based on significance level and p-value).

4. Calculate the test statistic and p-value.

5. Make the decision (reject or fail to reject $H_0$).

6. Interpret the results in context.

Types of Errors

• Type I Error: Incorrectly rejecting a true null hypothesis.

• Type II Error: Failing to reject a false null hypothesis.

• Application: In the context of work time lost due to sickness, a Type I error would mean concluding a difference exists when it does not.

Summary Table: Independent vs. Dependent Samples

Feature	Independent Samples	Dependent Samples
Sample Relationship	Unrelated	Paired/Matched
Test Statistic	$t = \frac{(\bar{y}_1 - \bar{y}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$	$t = \frac{\bar{y}_D - \mu_D}{s_D / \sqrt{n}}$
Confidence Interval	$(\bar{y}_1 - \bar{y}_2) \pm t \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$	$\bar{y}_D \pm t \frac{s_D}{\sqrt{n}}$
Typical Application	Comparing two groups	Before-and-after studies

Additional info: Table structure and content inferred for clarity and completeness.