State the approximate distribution of Y giving its Parameters and work out $ P(Y <22)$

Question

This is the question [ I tried solving it and the X-N (21,4.2^2/100) is correct but the probability is wrong its spouse to be 0.954. I have no teacher to ask as I am self-studying any help would be much appreciated. this is my work thank you!

This is similar example:

Answer 1

When you correct your standard deviation for $\bar Y$, you will get with $n=50$ observations

$$Z = \frac{\bar Y - \mu}{\frac{\sigma}{\sqrt n}}\stackrel{\approx}{\sim}N(0,1)$$

and $$P\left(\bar Y < 22\right)= P\left(Z < \frac{22-21}{\frac{4.2}{\sqrt{50}}}\right)=P\left(Z<\frac{\sqrt{50}}{4.2}\right)\approx 0.954$$