Using the Central Limit Theorem to approximate a binomial probability

Question

Here's my problem:

In Ohio, 55% of the population support the republican candidate in an upcoming election. 200 people are polled at random. If we suppose that each person’s vote (for or against) is a Bernoulli random variable with probability p, and votes are independent,

(a) Show that the number of people polled that support the democratic candidate X has distribution Bin(200, .45) and calculate the mean and variance.

(b) Calculate directly the probability that more than half of the polled people will vote for the democratic candidate. Tell me the equation that you used to solve this.

(c) Use the CLT to approximate the Binomial probability and calculate the approximate probability that half of the polled people will vote for the democratic candidate

And here's what I got so far:

Part a: Let us suppose if X number of people are supporting the democratic candidate, then there can be $\binom {200} {X}$ possible ways to select the people $\binom {200} {X} (0.45)^X (0.55)^{1-X}$ Therefore the given distribution is binomial distribution with n=200, p =0.55 and 1-p = 0.45

According to the theorem, the mean of the probability distribution is given as $E(X) = n*p = 200 * 0.45 = 90$

The variance of probability distribution is given as $E(X^2) - (E(X))^2 = np(1-p)$

For this problem,

$200*(0.45)*(1-0.45) = 49.5$

Part b: More than half of the people voting for the democratic candidate would be equal to $\sum\limits_{i=101}^{i=200} \binom {200} {i} (0.45)^i (0.55)^{200-i}$

Part c I'm at a total loss.

I'm very new to these sorts of problems and suspect I might be way off the mark on every part. Any guidance would be appreciated. (Apologies if this is way too long a problem, I can split it up.)

Answer 1

Hints:

Let $X=\sum_{i=1}^{200}X_i$, where $X_i=\{\text{No.i person votes for the democratic candidate}\}$, so that $$X_i\sim \begin{pmatrix} 1 & 0\\ p & 1-p\\ \end{pmatrix}$$

You have already calculated $E(X)$ and $D(X)$. So according to CLT, $X\sim?$

Answer 2

$c)$ Let $X\sim Bin(200,.45)$ that counts the number of votes for the democratic candidate so $X=\sum_{i=1}^{200}\xi_i$ where $\xi_i\sim Bernoulli(.45)$ and $\xi_i$ are i.i.d. so by central limit theorem we have that: $$X\sim^{\star}N(np,np(1-p))$$ $$X\sim^{\star}N(90,49.5)$$ So: $$P[X> 100]=1-P[X\leq100]\approx $$ $$1-P\left[\dfrac{X-90}{\sqrt{49.5}}\leq \dfrac{100-90}{\sqrt{49.5}}\approx1.4213\right]$$ $$=1-.07885=.92115$$