# Paired t test

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- Assumptions

- Dependent variable is normally distributed and is continuous (measured on ratio/interval scale of measurement).
- Independent variable is categorical (grouped into 2 dependent samples). Example: before-after

- 5-Step Hypothesis

## Null and Alternative Hypothesis

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

- Ha1: there is significant difference in the population means of two 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,n-1),t(a/2,n-1) (Two tailed test)

- -t(a,n-1) (Left tailed test)

- t(a,n-1) (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.

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