Question

On February 24^th, 2020, MU University Board of Visitors announced that Dr.Washington was selected as the university’s eighth president. In a Washington Post article, Dr. Washington mentioned that he was a first-generation college student. It is known that 39% of MU students are the first in their families to attend college. A random sample of ten STAT 250 students were asked the question, “Are you a first-generation college student?”

1. Check if this situation fits the binomial setting. Write four complete sentences addressing each requirement in one sentence each.

a. Assuming this situation is a binomial experiment, build the probability distribution in table form.  use the binomial calculator and calculate the probability of each of the values of the random variable from X = 0 to X = 10. present this table horizontally or vertically and leave the probabilities unrounded.

b. Calculate the probability that exactly four of the students in this sample are first-generation college students using the binomial calculator. Then, write one sentence to interpret the probability in context of the question.

c. Calculate the probability that at least two students in this sample are first-generation college students using the probability distribution table you created in (a). Show your work “by hand.” Then, verify your answer using the binomial calculator.

d. Calculate the probability that between four and seven (inclusive) of the students in this sample are first-generation college students the binomial calculator graph and include this image with values.

e. What is the average number of students you expect to respond “yes” to being a first-generation student? To answer this question, calculate the mean and standard deviation of this probability distribution. Show your work using the binomial mean and binomial standard deviation formulas and provide your answers. Round to two decimal place when necessary.

f. Imagine you repeated this sample of ten students 10,000 times. Produce a properly titled and labeled relative frequency bar plot.

g. Compare the height of the bar above four with your answer to part (b) and identify which type of probability each value is.

Answer 1

1.The following are the 4 conditions:

i. Fixed number of identical trials : Yes, there are 10 identical trials (asking students if they are first generation or not).

ii. Two outcomes possible for each trial : Yes, either each one is a first generation student or they are not.

iii. Same probability of success for each trial : Yes, p = 0.39.

iv. Independent trials : Yes, each student's past is unique and doesn't depend on any other student.

A. The probability distribution of a binomially distributed random variable with success of probability p is

where k is the number of successful trials and belongs to the set {0,1,2,....,n}.

Using a binomial calculator, we get the following probabilities :

k	P(X=k)
0	.00713342912
1	0.04560716976
2	0.13121407038
3	0.22370923474
4	0.2502976274
5	0.19203162233
6	0.10231192993
7	0.03737859735
8	0.0089616719
9	0.001273243
10	0.00008140406

B. From the above table, P(X=4) = 0.2502976274.

Basically, if a group of 10 students is asked whether they are first generation, there is around 25% chance that exactly 4 of them would say yes.

C. Probability that at least two students are first generation college students is equivalent to saying that it the complement of the probability that there are either no or exactly one student who is a first generation college student.

P(X>=2) = 1 - (P(X=0) + P(X=1)) = 1 - (0.00713342911+0.04560716976)

= 0.94725940112

Using a binomial calculator, we get the same value.

D. Basically, we have been asked to calculate

=

= 0.58201977701