Answer to Question #246190 in Statistics and Probability for Pearl

Question

The Gauteng traffic department records show that 25% of all drivers wear seatbelts. In a random sample of 400 cars stopped at a roadblock in Gauteng, 152 of the drivers were wearing seatbelts. What is the probability that at most 18% of the drivers in Gauteng wear seatbelts?

Answer 1

$p=0.25 \\ n=400 \\ \hat{p} \sim (p, \frac{p(1-p)}{n}) \\ Z = \frac{\hat{p} -p}{\sqrt{\frac{p(1-p)}{n}}} \\ P(\hat{p} ≤0.18) = P(\frac{\hat{p} -p}{\sqrt{\frac{p(1-p)}{n}}} ≤ \frac{0.18-0.25}{\sqrt{\frac{0.25 \times 0.75}{400}}}) \\ = P(Z≤ -3.233) \\ = 0.0006$