The 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 , determining , 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
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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
(H0): Means of the two populations are equal (μ1=μ2)
(Ha): Means of the two populations are not equal (μ1=μ2)
Calculate t-statistic using formula:
t=n1s12+n2s22xˉ1−xˉ2
xˉ1 and xˉ2 represent sample means
s12 and s22 represent sample variances
n1 and n2 represent sample sizes
Determine degrees of freedom (df) using formula:
df=n1+n2−2
Find critical t-value based on significance level (α) and degrees of freedom
Compare calculated t-statistic to critical t-value
If |t| > critical t-value, reject null hypothesis
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 :
(xˉ1−xˉ2)±tα/2,dfn1s12+n2s22
tα/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: 1000to5000)
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
: 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