Suppose that 30% of Tulane students will remain in New Orleans throughout the su

Question

Suppose that 30% of Tulane students will remain in New Orleans throughout the summer. Suppose we randomly select 20 Tulane students. 8) Referring to Tulane, find the expected number of students who will remain in New Orleans. 9) Referring to Tulane, find the standard deviation of the number of students who will remain in New Orleans. 10) Referring to Tulane, find the probability that at least 8 of the students in the sample will remain in New Orleans throughout the summer. 11) Referring to Tulane, should we use the normal approximation to the binomial (or the Central Limit Theorem for p) to approximate the probability computed in the preceding problem? Why or why not?

Explanation / Answer

BIONOMIAL DISTRIBUTION
pmf of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
where
k = number of successes in trials   
n = is the number of independent trials   
p = probability of success on each trial
Q8.
mean = np
where
n = total number of repetitions experiment is excueted
p = success probability
expeccted number = mean = 20 * 0.3
= 6

Q9.
variance = npq
where
n = total number of repetitions experiment is excueted
p = success probability
q = failure probability
variance = 20 * 0.3 * 0.7
= 4.2
III.
standard deviation = sqrt( variance ) = sqrt(4.2)
=2.04939

Q10.
P( X < 8) = P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 20 7 ) * 0.3^7 * ( 1- 0.3 ) ^13 + ( 20 6 ) * 0.3^6 * ( 1- 0.3 ) ^14 + ( 20 5 ) * 0.3^5 * ( 1- 0.3 ) ^15 + ( 20 4 ) * 0.3^4 * ( 1- 0.3 ) ^16 + ( 20 3 ) * 0.3^3 * ( 1- 0.3 ) ^17 + ( 20 2 ) * 0.3^2 * ( 1- 0.3 ) ^18 + ( 20 1 ) * 0.3^1 * ( 1- 0.3 ) ^19 + ( 20 0 ) * 0.3^0 * ( 1- 0.3 ) ^20   
= 0.772272
P( X > = 8 ) = 1 - P( X < 8) = 0.227728

Q11.
Points to pass for normal approximation:
1) experiment consistes of a sequence of n identical trials
2) only 2 outcomes are possible on each trail, success or failure
3) trials are independent & below conditions should satisfy
n*p>5, 20*0.3> 5 => 6>5
n*(1-p)>5, 20*0.7> 5 => 14>5
can use normal approximation

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