Answer to Question #232548 in Statistics and Probability for Hamza

Question

Suppose that a market research analyst for a cell phone company conducts a study of their customers who exceeds the time allowance included on their basic cell phone contract. The analyst finds that for those people who exceed the time included in their basic contract, the excess time used follows an exponential distribution with a mean of 30 minutes. Consider a random sample of 65 customers who exceed the time allowance included in their basic cell phone contract. In an exponential distribution mean and standard deviation values are same.

Find the probability that the mean excess time used by the 65 customers in the sample is in between 1500 and 1680 seconds.

Answer 1

Solution:

Let X be the excess time in minutes used by one individual cell phone customer who exceeds his contracted time allowance.

$X ∼ Exp(\frac1{30})$

$n=65,\mu=30,\sigma=30$

Using CLT, $\bar X\sim N(30,\dfrac{30}{\sqrt{65}})$

$P(1500\ sec\le\bar X\le1680 \ sec)=P(25 \min\le\bar X\le28 \ \min) \\=P(\bar X\le 28)-P(\bar X\le 25) \\=P(z\le \dfrac{28-30}{\dfrac{30}{\sqrt{65}}})-P(z\le \dfrac{25-30}{\dfrac{30}{\sqrt{65}}}) \\=P(z\le-0.54)-P(z\le-1.34) \\=1-P(z\le0.54)-[1-P(z\le1.34)] \\=P(z\le1.34)-P(z\le0.54) \\=0.90988-0.70540 \\=0.20448$