# Continuous Probability Distributions and Interval Estimation

Albany Automotive is a specialized auto and body shop and mostly serves residents in the Eastern part of Albany. The chief mechanic has determined that the time to paint automobiles is uniformly distributed and that the required time ranges between 45 minutes to 1 1/2 hours.

a. Write the function for the probability density function.

b. What is the probability that the painting time will be less than or equal to one hour?

c. What is the probability that the painting time will be more than 50 minutes?

d. Determine the expected painting time and its standard deviation.

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Scores on a recent standardized test are believed to be normally distributed with a mean of 80 and a standard deviation of 6.

a. What is the probability that a randomly selected exam will have a score of at least 71?

b. What percentage of exams will have scores between 89 and 92?

c. If the top 2.5% of test scores receive merit awards, what is the lowest score eligible for an award?

d. If there were 334 exams with scores of at least 89, how many students took the exam?

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A random sample of 81 credit sales in a department store showed an average sale of $68.00. From past data, it is known that the standard deviation of the population is $27.00.

a. Determine the standard error of the mean.

b. With a 0.95 probability, what can be said about the size of the margin of error?

c. What is the 95% confidence interval of the population mean?

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You are given the following information obtained from a sample of 5 observations taken from a population that has a normal distribution. (think carefully on the approach to use).

94 72 93 54 77

Develop a 98% confidence interval estimate for the mean of the population.

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#### Solution Preview

Albany Automotive is a specialized auto and body shop and mostly serves residents in the Eastern part of Albany. The chief mechanic has determined that the time to paint automobiles is uniformly distributed and that the required time ranges between 45 minutes to 1 1/2 hours.

a. Write the function for the probability density function.

Let X be time to paint automobiles in minutes, then X~Uniform(45,90). So f(x)=1/45 if 45<x<90

b. What is the probability that the painting time will be less than or equal to one hour?

P(X<=60)=(60-45)/45=1/3=0.33

c. What is the probability that the painting time will be more than 50 minutes?

P(X>50)=P(90-50)/45=8/9=0.89

d. Determine the expected painting time and its standard deviation.

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#### Solution Summary

The solution gived detailed steps on solving various questions on the normally distributed data. All formula and calcuations are shown and explained.