I'm quite new to statistics and I'm having some trouble solving this problem:
An unbiased dice is thrown $70$ times. Find the probability that the mean score is less than $3.3$.
Here's what I did:
Let $X$ be the score from a single throw then $E(X)=3.5$ and $Var(X)=35/12$. Let $\overline X$ be the mean value for 70 throws then $E(\overline X)=3.5$ and $Var(\overline X)=\frac{35}{12 * 70}=\frac{1}{24}$. We can use the Central Limit Theorem to approximate $\overline X$ so $\overline X$ is approximately $N(3.5,\frac{1}{24}$). Now it's quite easy to find the probability that the mean is less that $3.3$ just standardize and plug into a calculator, but somehow this gives the wrong answer. Anyone got any idea where I went wrong?
$$P(X<70*3.3) = P(z<\left(\dfrac{70*3.3-0.5-70*3.5}{\sqrt{\frac{70*35}{12}}}\right)$$ $$=0.155103$$