## 2.3 Carrying out the test: Paired $$t$$-test

We are now ready to carry out the paired $$t$$-test to test whether the true mean difference between before and after weights is different from 0. The results of the test are as follows:

Paired t-test

data:  anorexia$Prewt and anorexia$Postwt
t = -2.9376, df = 71, p-value = 0.004458
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-4.6399424 -0.8878354
sample estimates:
mean of the differences
-2.763889

We note the following:

• The $$p$$-value is 0.00446, which is less than 0.05, so we reject $$H_0$$. That is, we have enough evidence to conclude that the difference between before and after weights is statistically significant.
• The 95% confidence interval for $$\mu_D$$ is (-4.64, -0.89), meaning we are 95% confident that the average weight gain was between 4.64 and 0.89 lbs. Since this interval does not include 0, we can reject $$H_0$$.
• The test statistic is $$t = -2.9376$$
• The degrees of freedom is $$71$$
• The sample mean of the paired differences is $$-2.76$$.