Answer
$P(X = 20) = 0.0616$
$P(z < -0.92) - P(z < -1.19) = 0.0618$
Work Step by Step
Here we have: n = 60, p = 0.4, x = 20
Using the binomial probability formula:
$P(X = 20) = \frac{60!}{20!(60-20)!} \times 0.4^{20} \times (1 - 0.4)^{(60-20)}$
= 0.0616
Check whether the normal distribution can be used as an approximation for the binomial distribution:
$np(1-p) = 60 x 0.4 (1 - 0.4) = 14.4 \ge 10$
Hence, the normal distribution can be used.
$μ_{x} = np = 60 \times 0.4 = 24$
$σ_{x} = \sqrt {np(1-p)} = \sqrt {24 (0.6)} = 3.79$
$z = \frac{x - μ_{x}}{σ_{x}} = \frac{19.5 - 24}{3.79} = -1.19$
$z = \frac{x - μ_{x}}{σ_{x}} = \frac{20.5 - 24}{3.79} = -0.92$
$P(X = 20) = P(19.5 < X < 21.5) = P(-1.19 < z < -0.92)$
$P(z < -0.92) - P(z < -1.19) = 0.1788 - 0.1170 = 0.0618$
