What proportion are above x of sample size n where X ~ N(0,1) Homework

Question

I have a homework question that I'm not quiet sure of. It follows as so:

Consider a random variable $X$ that has a standard normal distribution with mean $\mu=0$ and standard deviation $\sigma=1$. What proportion of the means of samples of size 10 are greater than 0.60?

This is what I did:

$$X \sim N(\mu=0, \sigma=1)$$

$$Z = \frac{X - \mu}{\sigma/\sqrt(n)}$$

$$Z = \frac{X - 0}{1/\sqrt(10)}$$

$$= 0.189$$

Thank you,

Answer 1

Let $\bar{X}$ be the sample mean. Then $\bar{X}$ has normal distribution, mean $0$, standard deviation $\frac{1}{\sqrt{10}}$.

We want to find $\Pr(\bar{X}\gt 0.6)$. This is equal to $$\Pr\left(Z\gt \frac{0.6-0}{1/\sqrt{10}}\right),$$ where $Z$ is standard normal.

Note that $\dfrac{0.6}{1/\sqrt{10}}\approx 1.897$. Now you can find the required information from tables of the standard normal, or from appropriate software.