T test for independent samples with equal variances

• Assumptions

1. Dependent variable is continuous (measured on interval or ratio scale of measurement).
   
2. Independent variable is categorical (grouped into 2 independent samples)
   
3. There is no relationship/dependency between 2 samples
4. Method of simple random sampling is used
5. Data is normally distributed
6. Homogeneity of variance between 2 independent samples. This can be tested using Levene’s Test or F test
   
   

• 5-Step Hypothesis

Null and Alternative Hypothesis

• Ho: there is no significant difference in the population means of 2 groups. μ1-μ2=0

• Ha1: there is significant difference in the population means of 2 groups. μ1-μ2≠0 (Two tailed test)

Ha2: population mean of sample 1 is less than that of sample 2. μ1-μ2<0  (Left tailed test)

Ha3: population mean of sample 1 is greater than that of sample 2. μ1-μ2>0  (Right tailed test)

Test Statistic

Critical value

• -t(a/2,n1+n2-2),t(a/2,n1+n2-2) (Two tailed test)
• -t(a,n1+n2-2) (Left tailed test)
• t(a,n1+n2-2) (Right tailed test)

P-value

• 2*(1-P(T≤|t|)  (Two tailed test)
• P(T≤t)  (Left tailed test)
• P(T≥t)  (Right tailed test)

Decision rule

• Reject Ho if |t| > t(a/2, df) or p-value < alpha (two tailed test)
• Reject Ho if –t < -t(a, df) or p-value < alpha (left tailed test)
• Reject Ho if t > t(a, df) or p-value < alpha (right tailed test)

• Confidence Interval

Standard error (SE) and margin of error (ME) is given by:

100(1-alpha)% Confidence interval for the population mean difference is given by:

This implies I am 100(1-alpha)% confident that estimated population mean difference between two samples lies in the obtained interval. If confidence interval contains 0, I fail to reject null hypothesis ho and conclude that there is no significant difference in the means of two samples.