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Normal distribution probability questions

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Problem 1. The commuter trains on the Red Line for the Regional Transit Authority (RTA) in Cleveland, OH, have a waiting time during peak rush hour periods of twelve (12) minutes.

(a) What is the random variable?
(b) Find the height of this uniform distribution.
(c) Find the probability of waiting between four and five minutes.
(d) Find the probability of waiting between three and eight minutes.
(e) Find the probability of waiting five minutes exactly.

Problem 2. According to a recent study the mean blood pressure for people in China is 128 mmHg with a standard deviation of 23 mmHg. Assume that blood pressure is normally distributed.
(a) What is the random variable?
(b) Find the probability that a person in China has blood pressure of 140 mmHg or more.
(c) Find the probability that a person in China has blood pressure of 135 mmHg or less.
(d) Is it unusual for a person in China to have a blood pressure of 135 mmHg or less? Why or why not?
(e) Find the probability that a person in China has blood pressure between 120 and 125 mmHg.
(f) What is the 90th percentile blood pressure for the people in China? (That is, what is the blood pressure which 90 percent of the people have less than?
Continuing with this problem, suppose now a sample of 15 people are chosen from the above normal distribution and we are interested in the mean blood pressure of such a sample.
(g) Describe the distribution of the sample means: What is its shape? What is its mean? What is its standard deviation?
(h) Find the probability that the sample mean blood pressure of 15 people in China is 140 mmHg or more.
(i) Would it be unusual to find a sample mean blood pressure of 15 people in China of 140 mmHg or more? Why or why not?

Problem 3. A Maytag dishwasher has a mean life of 12 years with an estimated standard deviation of 1.25 years. Assume the life of a dishwasher is normally distributed.

(a) What is the random variable?
(b) Find the probability that a dishwasher will last more than 15 years.
(c) Find the probability that a dishwasher will last less than 6 years.
(d) Find the probability that a dishwasher will last between 8 and 10 years.
(e) If you found a dishwasher that lasted less than 6 years, would you think that you have a problem with the manufacturing process? Why or why not?
(f) Maytag only wants to replace free of charge 5% of all dishwashers. How long should the manufacturer make the warranty period?

Continuing with this problem, suppose now a sample of 10 dishwashers are chosen from the above normal distribution.

(g) Describe the distribution of the sample means: What is its shape? What is its mean? What is its standard deviation?
(h) Find the probability that the sample mean of the dishwashers is less than 6 years.
(i) If you found the sample mean life of the 10 dishwashers to be less than 6 years, would you think that you have a problem with the manufacturing process? Why or why not?

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Solution Summary

Calculations for each problem are shown. Where appropriate, reasons are given for whether the results are unusual.

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Problem 1. The commuter trains on the Red Line for the Regional Transit Authority (RTA) in Cleveland, OH, have a waiting time during peak rush hour periods of twelve (12) minutes.

(a)What is the random variable?
Waiting times during peak rush hours.

(b) Find the height of this uniform distribution.
f(x) = 1/(b - a)
= 1/(12-0)
= 1/12

(c) Find the probability of waiting between four and five minutes.
P(4 < x < 5) = (5 - 4) * (1/12) = 1/12

(d) Find the probability of waiting between three and eight minutes.
P(3 < x < 8) = (8 - 3) * (1/12) = 5/12

(e) Find the probability of waiting five minutes exactly.
P(x = 5) = 0 since this is a continuous distribution.

Problem 2. According to a recent study the mean blood pressure for people in China is 128 mmHg with a standard deviation of 23 mmHg. Assume that blood pressure is normally distributed.

(a)What is the random variable?
Blood pressure for people in China.

(b) Find the probability that a person in China has blood pressure of 140 mmHg or more.
Z = (140 - 128) / 23 = 0.52
From a standard normal z distribution table,
P(Z >= 0.52) = 0.3015

(c) Find the probability that a person in China has blood pressure of 135 mmHg or less.
Z = (135 - 128) / 23 = 0.30
From a standard normal z distribution table,
P(Z <= 0.30) = 0.6179

(d) Is it unusual for a person in China to have a blood ...

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  • MSc, California State Polytechnic University, Pomona
  • MBA, University of California, Riverside
  • BSc, California State Polytechnic University, Pomona
  • BSc, California State Polytechnic University, Pomona
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