Question 1

Distribution of sample mean

Asuming \(x_1\), \(x_2\),…,\(x_n\) is a random sample of size n from a distribution with mean \(\mu\) and standard deviation \(\sigma\).

Then the sample mean is \[\bar{x} = \frac{1}{n}\sum_{i=1}^{n}{x_i}\]
\(\bar{x}\) follows a distribution with mean \(\mu\) and standard deviation \(\sigma/\sqrt{n}\)
Sample Mean \(\bar{x}\) approximately is N(\(\mu,\sigma^2/n\))
\(\sigma\) the population standard deviation, which is unknown
\(\bar{z}= \frac{\bar{x}-\mu}{\sigma/\sqrt{n}}\) is approximately N(0,1)

Question 2

The \(100(1-\alpha)%\) confidence interval (CI) for the difference \(\mu_1 -\mu_2\)
is constrcuted as follows:

Difference between sample means \(d = \bar{y_1} - \bar{y_2}\)
Set \(s_p = \sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}}\)
CI: \((\bar{y_1}-\bar{y_2}) \pm t_\alpha/2S_p \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}\)

where \(t_\alpha/2S\) is the \((1-\alpha/2)%\) quantile of a Student’s t distribution with \(df= n_1+n_2-2\)