Question1. Suppose $X_{1}$ is normally distributed with a mean of 4 and a standard deviation of 1 and suppose $X_{2}$ is normally distributed with a mean of 5 and a standard deviation of 2 . (a) Explain why $\bar{X}_{1}-\bar{X}_{2}$ is normally distributed, even though both $n_{1}$ and $n_{2}$ are small. (2 marks)

Studdy Solution
Explain the distribution of the difference of two independent normal variables:
- Since $X_1$ and $X_2$ are independent and normally distributed, the sample means $\bar{X}_1$ and $\bar{X}_2$ are also normally distributed. - The difference $\bar{X}_1 - \bar{X}_2$ is a linear combination of independent normal variables. - Therefore, $\bar{X}_1 - \bar{X}_2$ is normally distributed regardless of the sample sizes $n_1$ and $n_2$.
The explanation shows that $\bar{X}_1 - \bar{X}_2$ is normally distributed due to the properties of normal distributions and the independence of $X_1$ and $X_2$.