Historical data shows that 63% of students enrolled in a Math class at a local u

Question

Historical data shows that 63% of students enrolled in a Math class at a local university during any term pass the course. A random sample of 24 students has been taken. Answer the following questions based on this scenario. (Show the formula or the calculator command that you use as appropriate). Round your answers to 3 decimal places.

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

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Find the probability that exactly 16 students in this section will pass this term. Would that be an unusual probability?

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Find the probability that at most 11 students pass the course this term.

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

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

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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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Explanation / Answer

Binomial(n,p) distribution applies to this problem.

There are fixed number of trials.

Probability is same for each trial .

p(x) = nCx px (1-p)n-x

1)

P(x = 16) = 24C16 0.6316 0.378

= 0.1591

If probability is less that 0.05 then it is unusual.

Therefore calculated probability that is 0.1591 is not unusual.

2)

p( x<=11) = P(X = 0) + P(X = 1) +  P(X = 2) +  P(X = 3) +  P(X = 4) +  P(X = 5) +  P(X = 6) +  P(X = 7) +  P(X = 8) +

P(X = 9) +  P(X = 10) + P(X = 11)

= 0.065

3)

p( x >=7) 1 - p( X <=6)

= 1 -{P(X = 0) + P(X = 1) +  P(X = 2) +  P(X = 3) +  P(X = 4) +  P(X = 5) +  P(X = 6) }

= 1 - 0.0002

= 0.9998

4)

Mean = np = 0.63 * 24 = 15.12

5)

Standard deviation = Sqrt(np(1-p))

= Sqrt( 24 * 0.63 * 0.37)

= 2.365

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