Suppose we want to estimate the pH of a mysterious liquid. Let the true pH be 5. We take t readings and let $X_i$ be the value returned by the $i^{th}$ reading. You should assume that the readings are independent and that $E(X_i) = 5$ and $var(X_i) = 2$. Let $Y = (X_1 + X_2 + . . . + X_t)/t$ be the average of these readings.
1) Find the mean and variance of Y
2) How large does t need to be such that P(4.9 < Y < 5.1) ≥ 0.99? Show your work.
Any help with this question would be appreciated!
To get you started
See that in this case, $Y = \bar{x}$ for $t$ independent events, so we are searching for the sampling distribution of $\bar{x}$.
Assuming the sample size $t$ is sufficiently large, we can say that $Y$ follows a normal distribution with mean $5$ and variance $\frac{2}{t}$