Let *X* and *Y* have independent normal distributions *X* and *Y*
are the same. So if the assumption of normality is valid, we would be interested in testing whether the two variances are equal and whether the two mean are equal.

Let first consider a test of the equality of the two means. When *X* and *Y* are independent and normally distributed, we can test hypotheses about their means using the same *t*-statistic that was used previously. Recall that the *t*-statistic used for constructing the confidence interval assumed that the variances of *X* and *Y* are equal. That is why we shall later consider a test for the equality of two variances.

Let start with an example and then let give a table that lists some hypotheses and critical regions.

#### Example 1

A botanist is interested in comparing the growth response of dwarf pea stems to two different levels of the hormone indoeacetic acid (IAA). Using 16-day-old pea plants, the botanist obtains 5-millimeter sections and floats these sections with different hormone concentrations to observe the effect of the hormone on the growth of the pea stem.

Let *X* and *Y* denote, respectively, the independent growths that can be attributed to the hormone during the first 26 hours after sectioning for *X* and *Y* are independent and normally distributed with common variance, respective random samples of size *n* and *m* give a test based on the statistic

where

*T* has a *t* distribution with *T* is less than

#### Example 2

In the example 1, the botanist measured the growths of pea stem segments, in millimeters, for *n*=11 observations of *X* given in the Table 1:

0.8 | 1.8 | 1.0 | 0.1 | 0.9 | 1.7 | 1.0 | 1.4 | 0.9 | 1.2 | 0.5 |

and *m*=13 observations of *Y* given in the Table 2:

1.0 | 0.8 | 1.6 | 2.6 | 1.3 | 1.1 | 2.4 | 1.8 | 2.5 | 1.4 | 1.9 | 2.0 | 1.2 |

For these data,

#### Notice that:

*p*-value of this test is 0.005 because