Module 6 Lecture Notes
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This document covers key concepts about the uniform and normal probability distributions including: - The uniform distribution is defined by minimum and maximum values and is rectangular in shape. - Examples are given of uniform distributions for times students wait for a bus and times patients wait to see a doctor. - The normal distribution is characterized by its mean and standard deviation. - Applications of the normal distribution include approximating binomial distributions and solving for values like x, σ, and μ given probabilities or z-scores.
Module 6 Lecture Notes
- 1. Module 6: Continuous Probability Distributions
- 2. The Uniform Distribution The uniform probability distribution is perhaps the simplest distribution for a continuous random variable. This distribution is rectangular in shape and is defined by minimum and maximum values.
- 3. The Uniform Distribution – Mean and Standard Deviation
- 4. Southwest Arizona State University provides bus service to students while they are on campus. A bus arrives at the North Main Street and College Drive stop every 30 minutes between 6 A.M. and 11 P.M. during weekdays. Students arrive at the bus stop at random times. The time that a student waits is uniformly distributed from 0 to 30 minutes. 1. Draw a graph of this distribution. 2. Show that the area of this uniform distribution is 1.00. 3. How long will a student “typically” have to wait for a bus? In other words, what is the mean waiting time? What is the standard deviation of the waiting times? 4. What is the probability a student will wait more than 25 minutes? 5. What is the probability a student will wait between 10 and 20 minutes?
- 5. The length of time patients must wait to see a doctor in a local clinic is uniformly distributed between 15 minutes and 2 1/2 hours. • What is the probability that a patient would have to wait between ¾ of an hour and 2 hours? • Calculate the probability that a patient would have to wait over 2 hours. The travel time for a college student traveling between her home and her college is uniformly distributed between 40 and 90 minutes. • What is the probability that she will finish her trip in 80 minutes or less? • What is the probability that her trip will take longer than 60 minutes? • What is the Standard Deviation of the distribution?
- 6. Characteristics of Normal Curve
- 7. Characteristics of Normal Curve
- 8. Areas Under the Curve
- 9. Applications of Normal Distribution • A plant manager knows that the number of boxes of supplies received weekly is normally distributed with a mean of 200 and a standard deviation of 20. – What percentage of the time will the number of boxes received weekly be greater than 210 or Less than 180? – What percentage of the time will the number of boxes received weekly be greater than 200 – What percentage of the time will the number of boxes received weekly be less than 160? What percentage of the time will the number of boxes received weekly be greater than 210 or Less than 180? – What percentage of the time will the number of boxes received weekly be greater than 175
- 10. Normal Approximation to Binomial: Continuity Correction Factor The “Test” … 푛 푥 푝 > 5 퐴푁퐷 푛 푥 푞 > 5 Apply a Continuity Correction Factor of 0.50 to convert a Discrete Variable to a Continuous Variable
- 11. Normal Approximation to the Binomial • The probability that a call to an emergency help line is answered in less than 10 seconds is 0.8. Assume that the calls are independent of each other. Using the normal approximation for binomial with a correction for continuity, the probability that at least 75 of 100 calls are answered within 10 seconds is approximately _______ • Assuming an equal chance of a new baby being a boy or a girl we would like to find the probability of 40 or more of the next 100 births at a local hospital will be boys. Use the normal approximation for binomial with a correction for continuity.
- 12. The “Magic Formula” for Normal Distribution!
- 13. Solving for X • Suppose you are working with a data set that is normally distributed with a mean of 400 and a standard deviation of 20. Determine the value of x such that 60% of the values are greater than x. • Suppose x is a normal random variable with mean 60 and standard deviation 2. A z score was calculated for a number, and the z score is 3.4. What is x?
- 14. Solving for σ and/or μ • The weight of a product is normally distributed with a mean 5 ounces. A randomly selected unit of this product weighs 7.1 ounces. The probability of a unit weighing more than 7.1 ounces is .0014. The production supervisor has lost files containing various pieces of information regarding this process including the standard deviation. Determine the value of standard deviation for this process.
- 15. Solving for σ and/or μ • If the mileage per gallon for a car is normally distributed, 32 mpg has a z-score of 1.2, and 24 mpg has a z-score of -.4, what is the mean mpg of the distribution? • Given the length an athlete throws a hammer is a normal random variable with a standard deviation of 5 feet. If the probability he throws it no less than 55 feet is 15.87%, what is the average distance an athlete can throw a hammer?