Based on last year\'s data, 65% of students in four-year institutions had an int

Question

Based on last year's data, 65% of students in four-year institutions had an international issue discussed in a class? A sample of 134 students was selected. a. Is this a valid Binomial experiment? b. What is the mean of the probability distribution? c. What is the standard deviation of the probability distribution? d. What is the probability that 10 students had an international issue discussed in a class? e. What is the probability that at least one student had an international issue discussed in a class? f. What is the probability that more than half of the students had an international issue discussed in a class? g. Show a graph depicting the shape of the distribution. h. Based on your answers to (b), (c) and (g), what is your conclusion? i. If this is a Binomial experiment, what are the 2 mutually exclusive outcomes? j. What is/are your assumptions to calculate the above answers?

Explanation / Answer

Based on last year's data, 65% of students in four-year institutions had an international issue discussed in a class? A sample of 134 students was selected.

a. Is this a valid Binomial experiment?

Yes, as trials are classified into just 2 categories, those with international issues and those without.

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b. What is the mean of the probability distribution?

As n = 134, p = 0.65,

u = mean = n p = 134*0.65 = 87.1 [ANSWER]

c. What is the standard deviation of the probability distribution?

Also,
  
s = standard deviation = sqrt(np(1-p)) =    5.521322305 [ANSWER]

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d. What is the probability that 10 students had an international issue discussed in a class?

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 =    134      
p = the probability of a success =    0.65      
x = the number of successes =    10      
          
Thus, the probability is          
          
P (    10   ) =    1.43033*10^-44 [ANSWER]

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e)

Note that P(at least 1) = 1 - P(0).

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 =    134      
p = the probability of a success =    0.65      
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    8.03744E-62

Thus, P(at least one) = 1 - P(0) =   1   [ANSWER, very close to 1]

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