t-tests are used to determine whether the differences between the means of two normally distributed groups are statistically significant.

Independent t-tests are used to assess the differences between the means of two separate, unrelated groups. Also known as **2-sample t-tests**, **independent sample t-tests**, and **studentâ€™s t-tests**.

Consider the following 2 distributions.

**Group A**(left) has a normal distribution, with a mean of -1**Group B**(center) has a normal distribution, with a mean of +1When we plot them together (right), the difference is visible

```
##
## Welch Two Sample t-test
##
## data: listA and listB
## t = -53.878, df = 2992.2, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -2.078726 -1.932740
## sample estimates:
## mean of x mean of y
## -1.014098 0.991635
```

Large

**t-value**and a Small**p-value <.01**= we accept the hypothesis that the means of list A and list B show statistically significant differences at a 99% confidence interval.**listA mean**is estimated at approxiately -1 and**listB mean**is estimated at approximately 1.The mean differences are listed as well showing the lowest & highest possible differences at 95% confidence.

```
listA <- rnorm(1500, mean = -1) #Produce list of random values w/ mean of -1 & normal distribtion
listB <- rnorm(1500, mean = 1) #Produce list of random values w/ mean of 1 & normal distribtion
```

listA | listB |
---|---|

0.3331812 | 0.5523527 |

-1.6772201 | 2.5600491 |

0.9139560 | 1.2009382 |

-1.9383215 | 1.1982463 |

-0.0096800 | 1.0137419 |

-0.3920836 | 1.6985306 |

**t.test syntax:** t.test(variableA, variableB)

`t.test(listA,listB) #If data are contained in two separate lists`

```
##
## Welch Two Sample t-test
##
## data: listA and listB
## t = -55.359, df = 2997.5, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -2.083782 -1.941222
## sample estimates:
## mean of x mean of y
## -1.0154567 0.9970449
```

```
groupAB <- data.frame(listA, listB)%>%
gather(groupAB)
kable(head(groupAB))
```

groupAB | value |
---|---|

listA | 0.3331812 |

listA | -1.6772201 |

listA | 0.9139560 |

listA | -1.9383215 |

listA | -0.0096800 |

listA | -0.3920836 |

**t.test syntax:** t.test(continuous_var~binary_var, data=data)

`t.test(value~groupAB, data=groupAB) `

```
##
## Welch Two Sample t-test
##
## data: value by groupAB
## t = -55.359, df = 2997.5, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -2.083782 -1.941222
## sample estimates:
## mean in group listA mean in group listB
## -1.0154567 0.9970449
```

If two sets of observations are made on the same subjets, in a before-and-after or other similar scenario, you can used a paired t-test

**t.test syntax:** t.test(continuous_var~binary_var, paried=TRUE, data=data)

`t.test(value~groupAB, paired=TRUE, data=groupAB) `

```
##
## Paired t-test
##
## data: value by groupAB
## t = -56.03, df = 1499, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -2.082957 -1.942046
## sample estimates:
## mean of the differences
## -2.012502
```

Large

**t-value**and a Small**p-value <.01**= we accept the hypothesis that the means of listA and listB show statistically significant differences at a 99% confidence interval.The

**mean of the differences:**As the name implies, this shows the mean of the differences between groups.*When comparing experimental results on the same group of subjects, this might be considered something of an effect size*.