Last blog, we discussed about the Two sample z test and found that how we can compare the population mean of two machines or product.

Similar to two sample z test , we also compare the population mean of two machines using two sample t test. Now question arises that what may be the difference of two sample z and t test

When samples size are high then we should go with two sample z test and when sample size are small, choose two sample t test.

Two sample t test can be calculated in two ways, one when both the samples have equal variance and another one with unequal variance.

**Two sample t test with equal variance: **

Now Sp is the pooled variance that can be calculted as :

**Degree of freedom** can be calculated-> n1 + n2 – 2

Let us solve one problem to get clear understanding.

**Problem: **In a factory, samples are extracted from the Machine A and Machine B and we have to figure it out whether Machine A and B are producing of equal mean or mean of both the machine have been changed? (95% confidence) . Sample are given below.

Machine A | Machine B |

140 | 146 |

142 | 145 |

144 | 148 |

142 | 145 |

141 | 144 |

**Solution: ** **Null Hypothesis** -> Mean of Machine A =Mean of Machine B

**Alternate Hypothesis:** Mean of Machine A not equal to Mean of Machine B

Here we will see variance of two samples are approximately same so apply the formula of two sample t test wth equal variance. We also require mean of samples, variance of the columns to calculate the two sample t test.

Metrics | Machine A | Machine B |

Observations | 5 | 5 |

Mean | 141.8 | 145.6 |

Variance | 2.2 | 2.3 |

Pooled variance | 2.25 | |

degree of freedom | 8 | |

t critical for two tail test(95% confidence) | -2.3 and +2.3 |

Polled variance can be calulated to put the values of variance and observation in the formula and when we keep pooled variance, mean and number of observation in two sample t test formula, get :

t=-4 and t critical is given 2.3 , meaning we can **reject the null hypothesis. **

*Mean of Machine A and B has been changed.*

**Two Sample t test with unequal variance:**

And degree of freedom can be calculated as:

Let us solve one problem to get clear understanding.

**Problem: **In a factory, samples are extracted from the Machine A and Machine B and we have to figure it out whether Machine A and B are producing of equal mean or mean of both the machine have been changed? (95% confidence) . Sample are given below.

Machine A | Machine B |

140 | 134 |

142 | 152 |

144 | 167 |

142 | 140 |

141 | 130 |

**Solution: **

**Null Hypothesis** -> Mean of Machine A =Mean of Machine B **Alternate Hypothesis:** Mean of Machine A not equal to Mean of Machine B

Here we will see variance of two samples are not same so apply the formula of two sample t test wth unequal variance. We also require mean of samples, variance of the columns to calculate the two sample t test.

Metrics | Machine A | Machine B |

Observations | 5 | 5 |

Mean | 141.8 | 144.6 |

Variance | 2.2 | 225.8 |

t critical for two tail test | -2.77 and +2.77 |

When we put the mean and variance in the formula defined above :

t calculated=-.41 and it is less than t critical , meaning failed to reject the null hypothesis.

*Mean of Machine A and B are not changed.*

## 0 Comments