Question

1. Forty-three students participated in a lottery for one of three free laptops. Fifteen of the students were in the same sorority. When all three of the winners were in the same sorority, several students were concerned that the drawing was not fair. Use a simulation of 10 trials to determine whether an all-sorority outcome could reasonably be expected if everyone had an equal opportunity to win one of the laptops.

1. Identify the component to be repeated: (2 points)
2. Explain how you will model the outcome: (3 points)
3. Identify the response variable: (2 points)
4. Run the ten trials and record your results. Use the random number table shown below beginning at Row 20. (5 points)
5. Summarize your results (1 point):
6. State your conclusion (5 points):

Row 20	39634 62349 74088 65564 16379 19713 39153 69459 17986 24537
	14595 35050 40469 27478 44526 67331 93365 54526 22356 93208
	30734 71571 83722 79712 25775 65178 07763 82928 31131 30196
	64628 89126 91254 24090 25752 03091 39411 73146 06089 15630
	42831 95113 43511 42082 15140 34733 68076 18292 69486 80468

Answer 1

a) We have to find the 3 students randomly selected from 43 students. We have to repeat sampling from 43 students with equal probability .

b) Let the students are numbered from 1, 2, ..,43. Let the 1,...,15 represents the sorority students. If we randomly selects 3 students and all are students from 1,..,15 we conclude that the 3 laptops were won by the sorority students.

c)  The response variable (also known as an outcome, target or dependent variable) is the set of 3 students drawn from 43.

d) The theoretical probability that all 3 laptops weree won by 15 students is

The R code for running the simulations is given below.

P <- rep(1/43,length=43)
S <- 1:43
S
N <- 1000
p <- 0
for (i in 1:N)
{
S3 <- sample(S, 3, prob = P, replace = FALSE)
if (0<S3[1]& S3[1]<16 & 0<S3[2]& S3[2]<16 & 0<S3[3]& S3[3]<16)
{
p <- p+1/N
}
}
p

The simulated probability is 0.037 close to theoretical 0.0369.

e) Since the probability 0.037 << 0.5, we can conclude that all-sorority outcome cannot reasonably be expected and the drawing was not fair. We at need at least a 50% (0.5) chance to say that the drawing was fair.