A nutritionist wants to compare the effects of two different diets on weight loss. After $8$ weeks, the following statistics are recorded:

Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Use degrees of freedom equal to the smaller of $n_1-1$ and $n_2-1$. Construct a $95\%$ confidence interval for the difference in mean weight loss, where the difference is defined as the mean for the Diet X group minus the mean for the Diet Y group.