Chapter 6 - Section 6.5 - The Normal Approximation to the Binomial Distribution - Exercises - Page 262: 6.51

N=100, p=.80 , q= .20, μ=np =100*.80=80, σ = $\sqrt npq$=$\sqrt 100*.80*.20$=4 a. $P(x=75) $=$ P(74.5\leq x \leq 75.5) $ For x=74.5, z = $\frac{74.5-80}{4}$=-1.375 For x=75.5, z = $\frac{75.5-80}{4}$=-1.125 $P(74.5 \leq x \leq 75.5)$= $P(-1.375\leq z \leq -1.125)$ =0.1303-0.0846= 0.0457 b. $P(x \leq 73)$ $P(x \leq 73) $: z = $\frac{73-80}{4}$ =-1.75 $P(z \leq -1.75)$ =0.0401 c. $P(74\leq x \leq 85) $ For x=74, z = $\frac{74-80}{4}$=-1.5 For x=85, z =$ \frac{85-80}{4}$=1.25 $P(74 \leq x \leq 85 )$ = $P(-1.5\leq z \leq 1.25)$ =0.8944-0.0668= 0.8275

Using the $z$ values in the tables, we find: N=100, p=.80 , q= .20, μ=np =100*.80=80, σ = $\sqrt npq$=$\sqrt 100*.80*.20$=4 a. $P(x=75) $=$ P(74.5\leq x \leq 75.5) $ For x=74.5, z = $\frac{74.5-80}{4}$=-1.375 For x=75.5, z = $\frac{75.5-80}{4}$=-1.125 $P(74.5 \leq x \leq 75.5)$= $P(-1.375\leq z \leq -1.125)$ =0.1303-0.0846= 0.0457 b. $P(x \leq 73)$ $P(x \leq 73) $: z = $\frac{73-80}{4}$ =-1.75 $P(z \leq -1.75)$ =0.0401 c. $P(74\leq x \leq 85) $ For x=74, z = $\frac{74-80}{4}$=-1.5 For x=85, z =$ \frac{85-80}{4}$=1.25 $P(74 \leq x \leq 85 )$ = $P(-1.5\leq z \leq 1.25)$ =0.8944-0.0668= 0.8275