1a.
Binomial distribution: There are only 2 outcomes: purchase, or dont purchase, and decisions made by every customer is independent of each other (i.e. decision of customer A will not affect that of customer B).
1bi.
48/120 would purchase implies that the probability of a customer purchasing is 48/120=0.4.
Hence X~B(50, 0.4).
P(X=25)=0.040463=0.0405
1bii.
P(X>=25)=1-P(X<=24)=0.0978
1c.
Normal distribution. Key phrase: 'huge database of customers', hence by the central limit theorem, X can be approximate to a normal distribution.
1ci.
E(X)=50*0.4=20
Var(X)=np(1-p)=50*0.4*0.6=12
X~N(20,12)
P(X=25)=0.0406
1cii.
P(X>=25)=0.0745
1d.
This graph should give you an idea of the difference:
At exact values (part i of qns 1&2), the graph intersect so the ans would be about the same. Normal distribution graph is smoother as it consider the case of decimal value of X as well, while a binomial distribution only considers the integer values of X. Not really sure how to phrase this but you get the rough idea :p
1e.
Let n be the number of customers selected.
X~(n, 0.4)
P(X>20)<=0.01
1-P(X<=18)<=0.01
This will be a long series of algebras so i'd express it in the form of a loop :p
for i:=0 to 18 do
PTotal:=PTotal+(nCi)(0.4^i)(0.6^(n-i)) (formula given in most formula lists)
1-PTotal-0.01<=0
Solve the inequality with a graphic calculator (plot graph and find the x-intercepts).
2a.
Xi~Po(0.7)
P(Xi>=2)=1-P(Xi<=1)=0.156
2b.
E(10Xi)=0.7*10=7
2c.
hmm seems like ans for 2b should be here. What do they mean by 'simulating' in 2b?
2d.
SD=sqd root 7=2.65
2e.
Don't really understand 'simulation', I don't know about 'R Commander' so it may be related to that.
2f.
Expected loss=5^0.7=$3.09 k
2g.
No. The average cost is only ~$3k/year, which is significantly less than the insurance fee of $20k/year. Also from 2a, the probability that the company would pay more than $20k/year (i.e. >=2 earthquakes) is pretty low.
(On an economic point of view, a stable maintenance cost may allow the company to plan and project profits more accurately. It may also be more favorable for small companies, depending on the company's capacity to absorb losses and willingness to take risks, as a devastating year with several earthquakes could cause bankruptcy) Feel free to ask if you don't understand any part.