# T-test for Paired Samples

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Instructions:
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This calculator conducts a t-test for two paired samples. This test applies when you have two samples that are dependent (paired or matched). Please select the null and alternative hypotheses, type the sample data and the significance level, and the results of the t-test for two dependent samples will be displayed for you:

## The T-Test For Paired Samples

More about the
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t-test for two dependent samples
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so you can understand in a better way the results delivered by the solver.

### How do you calculate a paired t-test?

A t-test for two paired samples is a hypothesis test that attempts to make a claim about the population means (\(\mu_1\) and \(\mu_2\)). More specifically, a t-test uses sample information to assess how plausible it is for difference \(\mu_1\) - \(\mu_2\) to be equal to zero. The test has two non-overlaping hypotheses, the null and the alternative hypothesis. The null hypothesis is a statement about the population parameter which indicates no effect, and the alternative hypothesis is the complementary hypothesis to the null hypothesis. The main properties of the t-test for two paired samples are:

- The test required two dependent samples, which are actually paired or matched or we are dealing with repeated measures (measures taken from the same subjects)
- As with all hypotheses tests, depending on our knowledge about the "no effect" situation, the t-test can be two-tailed, left-tailed or right-tailed
- The main principle of hypothesis testing is that the null hypothesis is rejected if the test statistic obtained is sufficiently unlikely under the assumption that the null hypothesis is true
- The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true
- In a hypothesis tests there are two types of errors. Type I error occurs when we reject a true null hypothesis, and the Type II error occurs when we fail to reject a false null hypothesis

### How do you manually calculate a paired t-test? What formula do you use?

The formula for a t-statistic for two dependent samples is:

\[t = \frac{\bar D}{s_D/\sqrt{n}}\]where \(\bar D = \bar X_1 - \bar X_2\) is the mean difference and \(s_D\) is the sample standard deviation of the differences \(\bar D = X_1^i - X_2^i\), for \(i=1, 2, ... , n\).

### Paired t test example

**Question**: Assume that you have the following sample of paired data.

Sample 1 | Sample 2 | Difference = Sample 1 - Sample 2 | |

4 | 2 | 2 | |

5 | 3 | 2 | |

6 | 4 | 2 | |

5 | 5 | 0 | |

4 | 6 | -2 | |

3 | 4 | -1 | |

5 | 3 | 2 | |

Average | 4.571 | 3.857 | 0.714 |

St. Dev. | 0.976 | 1.345 | 1.704 |

n | 7 | 7 | 7 |

Can the null hypothesis that the population mean difference is zero be rejected at the .05 significance level.

Solution:

From the sample data, it is found that the corresponding sample means are:

\[\bar X_1 = 4.571\]\[\bar X_2 = 3.857\]Also, the provided sample standard deviations are:

\[ s_1 = 0.976 \]\[ s_2 = 1.345 \]and the sample size is n = 7. For the score differences we have

\[ \bar D = 0.714 \]\[ s_D = 1.704 \]__(1) Null and Alternative Hypotheses__

The following null and alternative hypotheses need to be tested:

\[ \begin{array}{ccl} H_0: \mu_D & = & 0 \\\\ \\\\ H_a: \mu_D & \ne & 0 \end{array}\]This corresponds to a two-tailed test, for which a t-test for two paired samples be used.

__(2) Rejection Region__

Based on the information provided, the significance level is \(\alpha = 0.05\), and the critical value for a two-tailed test is \(t_c = 2.447\).

The rejection region for this two-tailed test is \(R = \{t: |t| > 2.447\}\)

__(3) Test Statistics__

The t-statistic is computed as follows:

\[ \begin{array}{ccl} t & = & \displaystyle \frac{\bar D}{s_D/ \sqrt n} \\\\ \\\\ & = & \displaystyle \frac{0.714}{1.704/ \sqrt{7}} \\\\ \\\\ & = & 1.109 \end{array}\]__(4) Decision about the null hypothesis__

Since it is observed that \(|t| = 1.109 \le t_c = 2.447\), it is then concluded that *the null hypothesis is not rejected.*

Using the P-value approach: The p-value is \(p = 0.31\), and since \(p = 0.31 \ge 0.05\), it is concluded that the null hypothesis is not rejected.

__(5) Conclusion__

It is concluded that the null hypothesis Ho *is not rejected.* Therefore, there is not enough evidence to claim that the population mean difference
\(\mu_D = \mu_1 - \mu_2\) is different than 0, at the \(\alpha = 0.05\) significance level.

*Confidence Interval*

The 95% confidence interval is \(-0.862 < \mu_D < 2.291\).

### What is the non-parametric alternative of the paired t-test?

This is a parametric test that should be used only if the normality assumption is met. If it fails, you should use instead this Wilcoxon Signed Ranks test . This paired t-test calculator deals with mean and standard deviation of pairs.

### Other t-test applications

Often times you have two samples that are not paired, in which case you would use a t-test for two independent samples calculator. Notice that in that case the samples don't have to necessarily have the same size.