Answer
$P(X = 70) = \frac{85!}{70!(85-70)!} \times 0.8^{70} \times (1 - 0.8)^{(85-70)}= 0.0970$
$P(z < 0.68) - P(z < 0.41) = 0.0927$
Work Step by Step
Here we have: n = 85, p = 0.8, x = 70
Using the binomial probability formula:
$P(X = 70) = \frac{85!}{70!(85-70)!} \times 0.8^{70} \times (1 - 0.8)^{(85-70)}= 0.0970$
Check whether the normal distribution can be used as an approximation for the binomial distribution:
$np(1-p) = 85 x 0.8 (1 - 0.8) = 13.6 \gt 10$
Hence, the normal distribution can be used.
$μ_{x} = np = 85 \times 0.8 = 68$
$σ_{x} = \sqrt {np(1-p)} = \sqrt {68 (0.2)} = 3.69$
$z = \frac{x - μ_{x}}{σ_{x}} = \frac{69.5 - 68}{3.69} = 0.41$
$z = \frac{x - μ_{x}}{σ_{x}} = \frac{70.5 - 68}{3.69} = 0.68$
$P(X = 70) = P(69.5 < X < 70.5) = P(0.41 < z < 0.68)$
$P(z < 0.68) - P(z < 0.41) = 0.7517 - 0.6591 = 0.0927$
