Question

A school newspaper reporter decides to randomly survey 9 students to see if they will attend Tet festivi knows that 40% of students attend Tet festivities. we are interested in the number of students who will attend the festivities. ties this year. Based on past years, she Where applicable, your entries must be either fractions in "p/q" format or decimal values accurate to nearest 0.01 a dS%) i e denotes the random vanable we wish to measure, then X b. (15%) X takes on integer values from [(pick one") to [(pick one) e (15%) is the number of students (pick one) X is a random variable that follows al(pick one)· distribution. d. (15%) How many of the N students do we expect to attend the festivities? EIX)- e. (20%) Find the probability that at most 4 students will attend. P(X 4)- f. (20%) Find the probability that more than 3 students will attend, P(X > 3)-

Answer 1

a) X is the number of students which attend the Tet Festivities this year.

b) As sample size is 9, So X can take values from 0 to 9

c) As probability is constant 0.40, and one student is independent of the other, So X has Binomial distribution

d) Using Binomial distribution we have $E(X)=9\times 0.40=3.6$

e) Using Binomial distribution we have

$P(X=r)=\binom{9}{r}(0.40)^r(1-0.40)^{9-r}$

So

$P(X\leq 4)=P(X=0)+ P(X=1)+P(X=2)+P(X=3)+P(X=4)\\ ~~~~~~~~~~~~~~~~~~~ =\binom{9}{0}(0.40)^0(0.60)^{9-0}+ \binom{9}{1}(0.40)^1(0.60)^{9-1}+\binom{9}{2}(0.40)^2(0.60)^{9-2}+ \binom{9}{3}(0.40)^3(0.60)^{9-3}+\binom{9}{4}(0.40)^4(0.60)^{9-4}\\ ~~~~~~~~~~~~~~~~~~~=0.010078+0.060466+0.161243+0.250823+0.250823 \\~~~~~~~~~~~~~~~~~~~=0.733432$

f) The required probability is

$P(X>3)=1-[P(X=0)+ P(X=1)+P(X=2)+P(X=3)]\\ ~~~~~~~~~~~~~~~~~~~ =1-[0.010078+0.060466+0.161243+0.250823] \\~~~~~~~~~~~~~~~~~~~=0.51739$