Historical data shows that 63 of students enrolled in a Math

Historical data shows that 63% of students enrolled in a Math class at a local university during any term pass the course. A randomly selected section of this course has 24 students enrolled. Answer the following questions based on this scenario. (Show the formula or the calculator command that you use as appropriate)

a) What type of probability distribution applies to this problem? Show that it satisfies the criteria of this probability distribution.

b) Find the probability that exactly 16 students in this section will pass this term.

c) Find the probability that no more than 11 students pass the course this term.

d) Find the probability that at least 7 students pass the course.

e) Would it be unusual for 20 students to pass the course this term? Justify your answer.

f) Find the expected number of students who will pass the course this term. i.e. the expected value of this probability distribution.

g) Find the standard deviation of the number of students who will pass the course this term. i.e. the standard deviation of this probability distribution.

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Solution

(a) Binomial distribution with n=24 and p=0.63

P(X=x)=24Cx*(0.63^x)*(1-0.63)^(24-x)

A binomial experiment (also known as a Bernoulli trial) is a statistical experiment that has the following properties:

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(b) P(X=16) = 24C16*(0.63^16)*(1-0.63)^(24-16)

=0.1590846

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(c) P(X<=11) = P(X=0) + P(X=1)+...+P(X=11)

=24C0*(0.63^0)*(1-0.63)^(24-0)+...+24C11*(0.63^11)*(1-0.63)^(24-11)

=0.06499734

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(d) P(X>=7) =1-P(X=0)-P(X=1)-..-P(X=6)=0.9998272

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(e) P(X=20) = 24C20*(0.63^20)*(1-0.63)^(24-20) =0.01931914

Since the probability is less than 0.05, it would be unusual for 20 students to pass the course this term

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(f)expected number =n*p

=24*0.63

=15.12

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(g)standard deviation =sqrt(n*p*(1-p))

=sqrt(24*0.63 *(1-0.63))

=2.365248