# Worksheet 17: The Normal Approximation

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1. For the distribution shown on the slides,
    a. What is the confidence level for the interval $[\hat{\mu} - 1, \hat{\mu} + 1]$?  That is, what is the probability that $|\hat{\mu} - \mu| \le 1$?


    b. What is the size $a$ of the confidence interval with confidence $0.8$? That is, how large do we need to take $a$ so that $\Pr[|\hat{\mu}-\mu| \le a] \ge 0.8$?




**The 68-95-99 rule:**  So as long as $n$ is large enough: $\Pr[|\hat{\mu_n}-\mu|\le C \frac{\sigma}{\sqrt{n}}] \ge \begin{cases}.68 & C = 1\\ .95 & C = 2\\ .997 & C = 3\end{cases}$

2. In the poll scenario, how large should I take $n$ to be 95\% sure that my estimate is within $\frac{1}{10}$ of the truth?



3. I was only able to poll $n = 49$ people. How confident am I that I am within $\frac{1}{10}$ of the truth?

    



**The 68-95-99 rule:**  So as long as $n$ is large enough: $\Pr[|\hat{\mu_n}-\mu|\le C \frac{\sigma}{\sqrt{n}}] \ge \begin{cases}.68 & C = 1\\ .95 & C = 2\\ .997 & C = 3\end{cases}$

4. I only polled $n = 49$ people. What is the error $a$ for my $95\%$ confidence interval?



5. In the microplastics scenario, suppose I know $\sigma_x = 10$. 

    a. How large should I take $n$ to be 99\% sure that my sample mean is within $\frac{1}{10}$ of the true microplastics concentration?


    b. Suppose I only took $n = 25$ samples. What is the size $a$ of the error for my $68\%$ confidence interval?