Let $X$ be the number of times that a fair coin, flipped $40$ times, lands heads.
$\text{(a)}$ Find the probability that $X=20$.
$\text{(b)}$ Use the normal approximations and then compare them to the exact solution.
Suppose that when a signal having value $\mu$ is transmitted from location $A$ the value received at location $B$ is normally distributed with mean $\mu$ and variance $4$. That is, if $\mu$ is sent, then the value received is $\mu+N$ where $N$, representing noise, is normal with mean $0$ and variance $4$. To reduce error, suppose the same value is sent $9$ times. If the successive values received are: $5,8,5,12,15,7,,7,5,6,5,10,5$. Construct a $5$ percent confidence interval for $\mu$.
For question 1(a), is the answer $0.1254$?
For question 1(b), is the answer $0.08883$?
For question 2, is the answer $[7.693, 10.31]$ ?
Your 1(a) answer of about $0.1254$ is correct.
Your 1(b) answer is wrong and should be about $0.1262 \approx \frac{1}{\sqrt{20\pi} }$ or perhaps about $0.1256$. I am not sure what your error is, but you might be using a standard deviation of $\sqrt{np}$ rather than the better $\sqrt{np(1-p)}$, i.e. $\sqrt{20}$ rather than $\sqrt{10}$.
If $f(x)$ is the probability density function and $F(x)$ the cumulative distribution function of a normal distribution with mean $20$ and standard deviation $\sqrt{10}$ then you might want to consider something like $f(20)$ or $F(20.5)-F(19.5)$.