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Binomial Probability Distribution and Poisson Probability Distribution

1. According to a Pew Research Center poll, 40% of American college graduates have outstanding student loans. Suppose that this is true for the current population of American college graduates.
 
a. Find the probability that exactly 4 American college students in a sample of 12 have outstanding student loans.
 
b. Find the probability that at least one college student in a random sample of 12 have outstanding student loans.
 
c. What is the mean and standard deviation of a sample size of 12 from this population.
 
 
 
2. An average of .8 accidents occurs per day in a particular day in a particular large city.
 
a. Find the probability that 2 accidents occur in this city on a given day.
 
b. Find the probability that 10 accidents occur in this city in a given week.
 
c. What is the mean and standard deviation of this distribution in part a.
 
 
 
 
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1 Answer

1. Use the binomial distribution B(12,0.4) with a calculator.
a) P(X=4)=binomialpdf(12,0.4,4)=.2128
b) P(X≥1)=1-P(X=0)=1-0.612=.9978
c) µ=np=12*0.4=4.8, σ²=np(1-p)=12*0.4*0.6=2.88, σ=1.697
 
2. Use the Poisson distribution Pois(0.8) for a day or Pois(5.6) for a week with a calculator.
a) P(X=2)=poissonpdf(0.8,2)=.1438
b) P(X=10)=poissonpdf(5.6,10)=0.0309
c) µ=σ²=0.8,  σ=0.8944

Comments

The conjugate of "at least one student has loans" is "no student has loans", so P(X=0)+P(X≥1)=1.