Question

A​ bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. If 40 different applicants are randomly​ selected, find the probability that their mean is above 215. Round your answer to four decimal places. A. 0.4713 B. 0.3821 C. 0.0287 D. 0.1179

Answer 1

Answer:

Probability that their mean is above 215 is 0.0287.

Step-by-step explanation:

We are given that a bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50.

For this, 40 different applicants are randomly​ selected.

Let X = ratings for credit

So, X ~ N($\mu=200,\sigma^{2}=50^{2}$)

Now, the z score probability distribution for sample mean is given by;

         Z = $\frac{X-\mu}{\frac{\sigma}{\sqrt{n} } }$ ~ N(0,1)

where, $\mu$ = population mean = 200

           $\sigma$ = standard deviation = 50

           $\bar X$ = sample mean

           n = sample of applicants = 40

So, probability that their mean is above 215 is given by = P($\bar X$ > 215)

    P($\bar X$ > 215) = P( $\frac{X-\mu}{\frac{\sigma}{\sqrt{n} } }$ > $\frac{215-200}{\frac{50}{\sqrt{40} } }$ ) = P(Z > 1.897) = 1 - P(Z $\leq$ 1.897)

                                                         = 1 - 0.97108 = 0.0287

Therefore, probability that their mean is above 215 is 0.0287.