Question

3. In San Francisco, 30% of workers take public transportation daily (USA Today, December 21, 2005). In a sample of 10 workers, Let X represents the number of workers taking public transportation daily. [5+5+5+5+10 = 30 points] (a) What is the distribution of X? Give the name and corresponding parameters. (b) Write down the probability mass function of X. (c) In a sample of 10 workers, what is the expected number of workers taking public transportation daily? (d) In a sample of 10 workers, what is the probability that exactly 3 workers take public transportation daily? (e) In a sample of 10 workers, Compute manually the probability that at least 3 workers take public transportation daily

Answer 1

a)

X follows binomial distribution with parameter n=10 and p=0.30

b)

probability mass function of X =P(X=x)=¹⁰C_x(0.3)^x(0.7)^10-x

c) expected number of workers taking public transportation daily =np=10*0.3=3

d) probability that exactly 3 workers take public transportation daily =¹⁰C₃(0.3)³(0.7)^10-3

=0.2668

e)

probability that at least 3 workers take public transportation daily =P(X>=3)

=1-P(X<=2)=1-(P(X=0)+P(X=1)+P(X=2))

=1-(¹⁰C₃(0.3)³(0.7)^10-3+¹⁰C₃(0.3)³(0.7)^10-3+¹⁰C₃(0.3)³(0.7)^10-3)

=1-(0.0282+0.1211+0.2335)=0.6172