Answer
a) $P(x < 50,000) = 0.2451$
b) P(x > 53,500) = 0.0132
Work Step by Step
$\mu$ =50,830, $\sigma$ = 8,520, n =50
to find probabilities when applying the central limit theorem use z = $\frac{x - \mu}{\sigma/\sqrt n}$ where $\mu$ = $\mu_{\bar{x}}$
$\mu$ = 50,830 and $\frac{\sigma}{\sqrt n}$ = $\frac{8520}{\sqrt 50}$ $\approx 1,204.910$
PART A
i) $P(x < 50,000) = P(z< \frac{50,000 - 50,830}{1,204.910}) \approx P(z < -0.69)$
ii) $P(x < 50,000)$= $P( z < -0.69) = 0.2451$
PART B
i) $P(x > 53,500) = P(z> \frac{53,500 - 50,830}{1,204.810}) \approx P(z > 2.22)$
ii) $P(z>2.22) = 1 - P(z<2.22) = 1-0.9868 = 0.0132$
