Answer
$P(x = 250) \approx 0.0029$
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 $P(x = 250)$
Apply the continuity correction: $P(x = 250)$ = $P(249.5 \leq x \leq 250.5)$
iv) Find $P(249.5 \leq x \leq 250.5)$
Convert 249.5 and 250.5 to z-scores
z = $\frac{249.5-225}{\sqrt{123.75}}= 2.20$
z = $\frac{250.5-225}{\sqrt{123.75}}= 2.29$
Therefore, $P(249.5 < x < 250.5)$ = $P(2.20 < z < 2.29)$
$= P(z < 2.29) - P(z < 2.20)$
$= 0.9890 - 0.9861$
$= 0.0029$
