Answer
$P(x \leq 220) = 0.3446$
Work Step by Step
i) Verify that we can use the normal approximation to the binomial
$np(1-p) \geq 10$
= $500(0.45)(0.55) \geq$ 10
= $123.75 \geq 10$
Yes, we can use the normal distribution to approximate the binomial.
ii) Find mean and standard deviation of the data
$\mu = np = 500 \times 0.45 = 225$
$\sigma = \sqrt {np(1-p)} = \sqrt{123.75}$
iii) Want to find $P(x \leq 220)$
Apply the continuity correction: $P(x \leq 220)$ = $P(x \leq 220.5)$
iv) Find $P(x \leq 220.5)$
Convert 220.5 to a z score
z = $\frac{220.5-225}{\sqrt{123.75}}= -0.40$
Therefore, $P(x \leq 220.5)$= $P(z < -0.40)$ $= 0.3446$
