Let S100 be the number of heads that turn up in 100 tosses of a fair coin. Use

the Central Limit Theorem to estimate

(a) \(P(S_{100} \leq 45)\)

\[ E(x) = np = 100 \cdot 0.5 = 50 \\ \sigma^2 = \sqrt{npq} = \sqrt{100 \cdot 0.5 \cdot 0.5} = \sqrt{25} = 5 \]

\[ \begin{align} P(S_{100} \leq 45) &= P(S_{n}^* \leq \frac{45 + \frac{1}{2} - 50}{5}) \\ &= P(S_{n}^* \leq \frac{45.5 - 50}{5}) \\ &= P(S_{n}^* \leq \frac{-4.5}{5}) \\ &= NA(-\infty,-0.9) \end{align} \]

p <- pnorm(-0.9)
p

## [1] 0.1840601

\[ P(S_{100} \leq 45) = 0.1840601 \]

(b) \(P(45 < S_{100} < 55)\)

\[ \begin{align} P(45 < S_{100} < 55) &= P(\frac{45 - \frac{1}{2} - 50}{5} < S_{n}^* < \frac{55 + \frac{1}{2} - 50}{5}) \\ &= P(\frac{44.5 - 50}{5} < S_{n}^* < \frac{55.5 - 50}{5}) \\ &= P(\frac{-5.5}{5} < S_{n}^* < \frac{5.5}{5}) \\ &= NA(-1.1,1.1) \\ &= 2NA(0,1.1) \end{align} \]

p <- pnorm(1.1) - pnorm(-1.1)
p

## [1] 0.7286679

\[ P(45 < S_{100} < 55) = 0.7286679 \]

(c) \(P(S_{100} > 63)\)

\[ \begin{align} P(S_{100} > 63) &= 1 - P(S_{100} < 63) \\ &= 1 - P(S_{n}^* < \frac{63 + \frac{1}{2} - 50}{5}) \\ &= 1 - P(S_{n}^* < \frac{63.5 - 50}{5}) \\ &= 1 - P(S_{n}^* < \frac{13.5}{5}) \\ &= 1- NA(-\infty,2.7) \end{align} \]

p <- 1- pnorm(2.7)
p

## [1] 0.003466974

\[ P(S_{100} > 63) = 0.003467 \]

(d) \(P(S_{100} < 57)\)

\[ \begin{align} P(P(S_{100} < 57) &= P(S_{n}^* < \frac{57 + \frac{1}{2} - 50}{5}) \\ &= P(S_{n}^* < \frac{57.5 - 50}{5}) \\ &= P(S_{n}^* < \frac{7.5}{5}) \\ &= NA(-\infty,1.5) \end{align} \]

p <- pnorm(1.5)
p

## [1] 0.9331928

\[ P(P(S_{100} < 57) = 0.9331928 \]

Chapter 9.1 Exercise 2