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# Normal distribution, Binomial distribution

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A)The mean of a normal distribution is 400 pounds. The standard deviation is 10 pounds. a) what is the area between 415 pounds and the mean of 400 pounds? b) what is the area between the mean and 395 pounds? c) what is the probability of selecting a value at random and discovering that it has a value of less than 395 pounds?

B)Shorty's Muffler advertises they can install a new muffler in 30 minutes or less. However, the work standards department at corporate headquarters recently conducted a study and found that 20 percent of the mufflers were not installed in 30 minutes or less. The Maumee branch installed 50 last month. If the corporate report is correct: a) how many of the installations at the maumee branch would you expect more than 30 minutes? b) what is the likelihood that fewer than eight installations took more than 30 minutes? c) what is the liklihood that eight or fewer installations took more than 30 minutes? d) what is the liklihood that exactly 8 of the 50 installations took more than 30 minutes?

C)A study of long distance phone calls made from the corporate offices of the Pepsi Bottling Group, Inc., in Somers, New YOrk, showed the calls follow the normal distribution. The mean length of time per call was 4.2 minutes and the standard deviation was o.60 minutes. a) what fraction of the calls last between 4.2 and 5 minutes? b) what fraction of the calls last more thatn 5 minutes? c) what fraction of the calls last between 5 and 6 minutes? d) what fraction of the calls last between 4 and 6 minutes? e) as part of her report tot he president, the Director of Communications owuld like to report the length of the longest (in duration) 4 percent of the calls. What is this time?

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#### Solution Preview

Probability values corresponding to z can be read from normal distribution tables for problems a) and c)

A)The mean of a normal distribution is 400 pounds. The standard deviation is 10 pounds. a) what is the area between 415 pounds and the mean of 400 pounds? b) what is the area between the mean and 395 pounds? c) what is the probability of selecting a value at random and discovering that it has a value of less than 395 pounds?

a) what is the area between 415 pounds and the mean of 400 pounds?

Mean=M = 400 pounds

Standard deviation =s= 10 pounds

x1= 400 pounds

x2= 415 pounds

z1=(x1-M )/s= 0 =(400-400)/10

z2=(x2-M )/s= 1.5 =(415-400)/10

Cumulative Probability corresponding to z1= 0 is= 0.5 0r= 50.00%

Cumulative Probability corresponding to z2= 1.5 is= 0.9332 0r= 93.32%

Therefore probability that the value of x will be between x1= 400 and x2= 415

is = 43.32% =93.32%-50.%

b) what is the area between the mean and 395 pounds?

Mean=M = 400 pounds

Standard deviation =s= 10 pounds

x1= 395 pounds

x2= 400 pounds

z1=(x1-M )/s= -0.5 =(395-400)/10

z2=(x2-M )/s= 0 =(400-400)/10

Cumulative Probability corresponding to z1= -0.5 is= 0.3085 0r= 30.85%

Cumulative Probability corresponding to z2= 0 is= 0.5 0r= 50.00%

Therefore probability that the value of x will be between x1= 395 and x2= 400

is = 19.15% =50.%-30.85%

c) what is the probability of selecting a value at random and discovering that it has a value of less than 395 pounds?

Mean=M = 400 pounds

Standard deviation =s= 10 pounds

x= 395 pounds

z=(x-M )/s= -0.5 =(395-400)/10

Cumulative Probability corresponding to z= -0.5 is= 0.3085

Or Probability corresponding to x< 395.00 is Prob(Z)= 0.3085 or 30.85%

B)Shorty's Muffler advertises they can install a new muffler in 30 minutes or less. However, the work standards department at corporate headquarters recently conducted a study and found that 20 percent of the mufflers were not installed in 30 minutes or less. The Maumee branch installed 50 last month. If the corporate report is correct: a) how many of the installations ...

#### Solution Summary

Answers 3 questions on probability calculations using Normal distribution, Binomial distribution.

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