The independent samples t-test is a powerful tool for comparing means between two unrelated groups. It's used to determine if there's a significant difference between populations, like test scores in public vs private schools or salaries in different departments.

This statistical method involves calculating a t-statistic, determining degrees of freedom, and comparing results to critical values. Understanding its assumptions and how to interpret confidence intervals is crucial for drawing accurate conclusions from your data analysis.

Independent Samples T-Test

Scenarios for independent samples t-test

• Compares means of two independent groups not related or paired in any way
  • Comparing test scores of students in two different schools (public vs private)
  • Comparing salaries of employees in two different departments (marketing vs sales)
• Dependent variable must be continuous measured on an interval or ratio scale (height, weight, temperature)
• Independent variable must be categorical with only two levels (male/female, treatment/control)

Conducting and interpreting t-tests

• State null and alternative hypotheses
  • Null hypothesis ($H_0$): Means of the two populations are equal ($\mu_1 = \mu_2$)
  • Alternative hypothesis ($H_a$): Means of the two populations are not equal ($\mu_1 \neq \mu_2$)
• Calculate t-statistic using formula:
  • $t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$
    • $\bar{x}_1$ and $\bar{x}_2$ represent sample means
    • $s_1^2$ and $s_2^2$ represent sample variances
    • $n_1$ and $n_2$ represent sample sizes
• Determine degrees of freedom (df) using formula:
  • $df = n_1 + n_2 - 2$
• Find critical t-value based on significance level (α) and degrees of freedom
• Compare calculated t-statistic to critical t-value
  1. If |t| > critical t-value, reject null hypothesis
  2. If |t| ≤ critical t-value, fail to reject null hypothesis
• Interpret results in context of problem (e.g., significant difference in test scores between public and private schools)

Confidence intervals for population means

• Provides range of plausible values for difference between two population means
• Formula for confidence interval:
  • $(\bar{x}_1 - \bar{x}2) \pm t{\alpha/2, df} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$
    • $t_{\alpha/2, df}$ represents critical t-value based on significance level (α) and degrees of freedom (df)
• Interpreting confidence interval:
  • If interval contains zero, insufficient evidence to conclude population means differ
  • If interval does not contain zero, evidence suggests population means differ (e.g., 95% confidence interval for difference in salaries between marketing and sales departments: $1000 to $5000)

Assumptions of independent samples t-test

• Independence: Observations within each sample must be independent of each other
  • Randomly selected samples from population
  • Samples not related or paired
• Normality: Populations from which samples are drawn must be normally distributed
  • If sample sizes are large (n > 30), t-test is robust to violations of normality
• Equal variances: Variances of the two populations must be equal
  • If sample sizes are equal, t-test is robust to violations of equal variances
  • If sample sizes are unequal and variances are unequal, use Welch's t-test