Question

the following percentages of students enrolling in college in the fall immediately following high school completion was 69.2 percent in 2015 (source national center for education statistics) a random sample of 500 students are surveyed shortly after graduation and asked if they are enrolled in college for fall. a. explain why this scenario is a binomial random variable setting. b. approximate the probability that exactly 346 graduates will be enrolled in college. c. determine the mean and the standard deviation of students who enrolled in college in fall. d approximate the probability that between 326 and 366 graduates will be enrolled in college. e. would you be surprised if more 375 graduates were enrolled in the fall? explain.

Answer 1

(a)

This scenario is a binomial variable setting because it satisfies all the following conditions.

(i) The Total number of trials = n = 500 is fixed

(ii) Each trial is independent of the others

(iii) There are only 2 outcomes: they are enrolled in college for fall OR they are not enrolled in college for fall

(iv) The probability = p = 0.692 of each outcome remains constant from trial to trial.

(b)

So,

Answer is:

0.0386

(c)

(i) the mean of students who enrolled in college in fall = = np = 500 X 0.692 = 346

(ii) the standard deviation of students who enrolled in college in fall =

(d)

To find

P(326 < X < 366):

Case 1: For X from 326 to mid value:
Z = (326 - 346)/10.3232

= - 1.94

Table gives area = 0.4738

Case 2: For X from mid value to 366

Z = (366 - 346)/10.3232

= 1.94

Table gives area = 0.4738

So,

P(326 < X < 366) = 2 X 0.4738 = 0.9476

So,

Answer is:

0.9476

(e)

To find P(X>375):

Z = (375 - 346)/10.3232

= 2.8092

Table gives area = 0.4975

So,

P(X>375) = 0.5 - 0.4975 = 0.0025

Since Z = 2.2092 > 2, the probability of X > 375 is very small. So, we would be surprised if more 375 graduates were enrolled in the fall.