Historical data shows that 65 of students enrolled in a writ

Historical data shows that 65% of students enrolled in a writing class during any term pass the course. A randomly selected section of this course hs 20 students enrolled.   What type of probability distribution applies to this problem?   Find the probabilty that exactly 15 students in this section will pass the term?   Find the probabilty that no more than 10 students pass the course. Find the probabilty that at least 8 students pass the course. Would it be unusual for 17 students to pass the course this term?   Why?   Find the expected number of students who will pass the course this term, the expected value of this probabilty distribution. Find the standard deviation of the number of students who will pass the course i.e. the standard deviation of this probabilty distribution.

Solution

What type of probability distribution applies to this problem?   

BINOMIAL PROBABILITY DISTRIBUTION.

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Find the probabilty that exactly 15 students in this section will pass the term?

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    20      
p = the probability of a success =    0.65      
x = the number of successes =    15      
          
Thus, the probability is          
          
P (    15   ) =    0.127199186 [ANSWER]

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Find the probabilty that no more than 10 students pass the course.

Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    20      
p = the probability of a success =    0.65      
x = the maximum number of successes =    10      
          
Then the cumulative probability is          
          
P(at most   10   ) =    0.121780586 [ANSWER]

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Find the probabilty that at least 8 students pass the course.

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    20      
p = the probability of a success =    0.65      
x = our critical value of successes =    8      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   7   ) =    0.00601527
          
Thus, the probability of at least   8   successes is  
          
P(at least   8   ) =    0.99398473 [ANSWER]

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Would it be unusual for 17 students to pass the course this term? Why?   

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    20      
p = the probability of a success =    0.65      
x = our critical value of successes =    17      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   16   ) =    0.955624397
          
Thus, the probability of at least   17   successes is  
          
P(at least   17   ) =    0.044375603

As this probability is less than 0.05, YES, IT IS UNUSUAL. [ANSWER]

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Find the expected number of students who will pass the course this term, the expected value of this probabilty distribution.

u = mean = np =    13 [ANSWER]

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Find the standard deviation of the number of students who will pass the course i.e. the standard deviation of this probabilty distribution.

s = standard deviation = sqrt(np(1-p)) =    2.133072901 [ANSWER]