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

This is case of hypergeometric distribution a population consists of N items, k of which are successes. And a random sample drawn from that population consists of n items, x of which are successes.

Here the population consists of N = 33 students, k = 14 of which are successes (female students). And a random sample drawn from that population consists of n = 8 students, X of which are successes. So, X follows hypergeometric distribution.

d.

The mean of the hypergeometric distribution is equal to n * k / N .

So, expected value of X = 8 * 14 / 33 = 3.39

e.

The variance of the hypergeometric distribution is equal to n * k * ( N - k ) * ( N - n ) / [ N2 * ( N - 1 ) ]

So, variance of X = 8 * 14 * ( 33 - 14 ) * ( 33 - 8 ) / [ 332 * ( 33 - 1 ) ] = 1.527

standard deviation of X = sqrt(1.527) = 1.24

f.

If X is within 1 standard deviation of its expected value, then

3.39 - 1.23 < X < 3.39 + 1.23

or,

2.16 < X < 4.62

As X can take integer values, so X = 3 or 4

Hypergeometric probability is given as,

h(x; N, n, k) = [ kCx ] [ N-kCn-x ] / [ NCn ]

So, h(3; 33, 8, 14) = [ 14C3 ] [ 33-14C8-3 ] / [ 33C8 ] = [ 14C3 ] [ 19C5 ] / [ 33C8 ]

= 0.3048

h(4; 33, 8, 14) = [ 14C4 ] [ 33-14C8-4 ] / [ 33C8 ] = [ 14C4 ] [ 19C4 ] / [ 33C8 ]

= 0.2794

So, the probability that X is within 1 standard deviation of its expected value = 0.3048 + 0.2794 = 0.5842

g.

If we select the students in the class one by one until we select a female, then

So, probability that we have to select more than 3 students = 1 - probability that we have to select less than or equal to 3 students

= 1 - (0.4242 + 0.2519 + 0.1463) = 0.1776

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