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# Steps on calculating the probability under the standard normal distribution

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1. The average annual rainfall in a certain region is normally distributed with a mean of 31.3 inches and a standard deviation of 7.2 inches.

A. In any given year, what is the probability that the amount of rain will exceed 24.82 inches?
B In any given year, what is the probability that the amount of rain will be between 31.3 and 51.172 inches?
C. Find a value 'a' such that the probability is .0495 that the amount of rain in any given year will exceed 'a' inches.

2. Suppose a consumer agency provides the following information in comparing two brands of automobile batteries: brand A has an average life expectancy of 44.1 months with a standard deviation of 6.7 months; brand B has an average life expectancy of 53.9 months, with a standard deviation of 14.5 months. Assuming the life expectancies for the two brands of batteries are normally distributed, which of the two brands would provide fewer batteries with a life expectancy of less than 30.7 months?

3. A vending machine sells coffee in 6-ounce cups. If the machine is set to dispense 5 ounces of coffee, the amount of coffee dispensed per cup is normally distributed with a mean of 5 ounces and a standard deviation of .25 ounce. Assume that the standard deviation is .25 regardless of the mean amount dispensed.
A. If the machine is set to dispense a mean of 5.75 ounces, what is the probability the dispensed coffee will overflow the 6-ounce cup?
B. If the machine is set to dispense a mean of 5 ounces, what percentage of the customers will get only 4.5 ounces or less?
C. Where would we set the mean amount dispensed so that only 2.5% of cups would overflow?
D. Suppose you decide that overflows of hot coffee are worse for business than giving the customer a short cup. You set the machine to dispense a mean of 5.0 ounces and the next cup overflows. Would you conclude that the machine isn't working properly? Why?

4. Travel time to the Atlanta airport from the north side of town is normally distributed and depends on which route you take:
I-75 route I-285 route
Mean time is 30 minutes Mean time is 35 minutes
Standard deviation is 10 minutes Standard deviation is 5 minutes

A. If you have 35 minutes to reach the airport, which route is best? Why?
B. If you have 40 minutes to reach the airport, which route is best? Why?
C. If you have 45 minutes to reach the air-port, which route is best? Why?

https://brainmass.com/statistics/random-variables/steps-on-calculating-the-probability-under-the-standard-normal-distribution-590321

#### Solution Preview

1. The average annual rainfall in a certain region is normally distributed with a mean of 31.3 inches and a standard deviation of 7.2 inches.

A. In any given year, what is the probability that the amount of rain will exceed 24.82 inches?
We know that z=(x-mean)/sd=(x-31.3)/7.2. So P(X>24.82)=P(Z>(24.82-31.3)/7.2)=P(Z>-0.9)=0.8159 from standard normal table.

B In any given year, what is the probability that the amount of rain will be between 31.3 and 51.172 inches?
We know that z=(x-mean)/sd=(x-31.3)/7.2. So P(31.3<X<51.172)=P((31.3-31.3)/7.2<Z<(51.172-31.3)/7.2)=P(0<Z<2.76)=0.4971 from standard normal table.

C. Find a value 'a' such that the probability is .0495 that the amount of rain in any given year will exceed 'a' inches.
We need to find value of 'a' such that P(Z>a)=0.0495. From standard normal table, a=1.65.

2. Suppose a consumer agency provides the following information in comparing two brands of automobile batteries: brand A has an average life expectancy of 44.1 months with a standard deviation of 6.7 months; brand B has an average life expectancy of 53.9 months, with a standard deviation of 14.5 months. Assuming ...

#### Solution Summary

The solution gives detailed steps on calculating the probability under the standard normal distribution.

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