Null hypothesis

Question

I have a random sample ${X_1,...,X_N}$, and we know that X bar $=3$, and the standard error $SE(X bar)=2 $. Suppose a new variable $Y_i$ is defined as $Y_i=0.17+0.1X_i$. The null hypothesis $H_0: E(Y_i)=0.25$ against a two-sided alternative at a 5% significance level.

My questions is: I know that the formula is $(Y-\mu)/SE$, and then see if it falls into the interval $[0.2, 0.3]$. I've found Y bar, wondering how to find $\mu$ and SE for Y. thanks!

Answer 1

(this should maybe be moved on xvalidated)

Using the usual properties for the mean and variance you have

• $\overline Y=0.17+0.1\overline X = 0.47$
• $SE(Y)=(0.1) SE(X) = 0.2 $

You want a two-sided 5% confidence interval so you're looking at the interval

$$0.47 \pm 1.96\times 0.2 = [0.078, 0.862]$$

and you cannot reject the null.