Question 451595
The speed limit in the state of Ohio is 55 miles per hour. On a certain stretch of interstate highway the mean speed of traffic (as checked by radar) is 58 miles per hour with a standard deviation of 4 miles per hour. The speeds of vehicles on this section of highway are normally distributed. 
a) What percentage of vehicles are exceeding the speed limit?
z(55) = (55-58)/4 = -3/4
P(x > 55 mph) = P(z> -3/4) = -.7734
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b) Above what speed are the top 5% of the speeders travelling?
invNorm(0.95) = 1.645
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x = zs+u = 1.645*4+58 = 64.5 mph
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A cruise ship has 74% chance of having leftover rooms the week before the cruise. There are 25 cruise ships setting sail this month. 
a) Is it possible to approximate this binomial distribution using a normal approximation?
Check the critera in your text.
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b) What is the mean and the standard deviation for this approximation?
mean = np = 25*0.74 = 18.5
std = sqrt(npq) = sqrt[18.5*0.26] = 2.183
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c) What is the probability that exactly 20 cruise ships will have leftover rooms using the normal approximation?
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z(19.5) = (19.5-18.5)/2.183 = 0.4581
z(20.5) = (20.5-18.5)/2.183 = 0.9162
P(x = 20) = P(0.4581< z< 0.9162) = 0.1437
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d) How different is the normal approximation from the binomial distribution value? Calculate the probability that exactly 20 cruise ships will have leftover rooms using the binomial method?
Ans: 25C20(0.74)^20(0.26)^5 = 0.1531
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Cheers,
Stan H.