There are 33 students in JJHSs 9th grade Algebra class. 14 of these students are

Question

There are 33 students in JJHSs 9th grade Algebra class. 14 of these students are female and 19 are male. 8 students from the class are randomly selected for automatic As. Let X = the number of selected students who are female.  

d. What is the expected value of X?
e. What is the standard deviation of X?
f. What is the probability that X is within 1 standard deviation of its expected value?
g. If we select the students in the class one by one until we select a female then what is the probability that we have to select more than 3 students?

Explanation / Answer

THe given distribution is hypergeometric distribution

where N = 33, n = 8 , K = 14 ,

here

d. Expected Value of X E(X) = 8 * (14/33) = 3.4

e. Standard deviation of X = Sqrt(Var(X))

Var(X) = n (K/N) [(N-K)/N] [ (N-n)/(N-1) ]

Var(X) = 8 * (14/33) * (19/33) * (25/32) = 1.5266

STD(X) = 1.2356

(f) Number of students within 1 standard deviation of expected values are

+- = 3.4 +- 1.2356 = ( 2.16, 4.64) so only X = 3 and X =4 are the values under one standard deviation.

so Pr(X < +- ) = Pr(X = 3) +Pr(X = 4) = 14C419C4 /33C8 + 14C319C5/ 33C8

= 0.3049 + 0.2794 = 0.5843

(g) Pr(it take more than 3students to select first female) = 1 - [Pr(it require less than 3 students)]

Let X is the number of students require to obtain first female.

Pr(X >3) = 1 - [ Pr(X = 1) + Pr(X =2) + Pr(X =3)]

=1 - [14/33 + 19/33 * 14/32 + 19/33 * 18/32 * 14/31]

= 1 - 0.8224

= 0.1776

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