Answer to Question #264201 in Statistics and Probability for Nathan

Question

The amount of time customers spend in a bank is assumed to follow an exponential distribution with a mean of 10 minutes. The bank manager wants to ensure that there are enough service points in the bank to reduce the time their customers spend waiting for service. He randomly selects a sample of 45 customers and records the time that each customer spent in the bank. What is the probability that their average time in the bank exceeds 12 minutes?

Answer 1

Since we have large sample size (≥ 30), we can use the central limit theorem

$X$ ~ $N(m,{\frac {\sigma^2} {\sqrt{n}}})=m+{\frac {\sigma} {\sqrt {n}}}N(0,1)$ , where X - sample mean, m - distribution mean, $\sigma$- distribution standard deviation, n - sample size. In the given case we have m = 10, $\sigma=10$, n = 45

$P(X>12)=P(10+{\frac {10} {6.7}}N(0,1)>12)=P(N(0,1)>1.34)=0.09012$