1 / 26

Chapter 5 Normal Probability Distribution

Chapter 5 Normal Probability Distribution. GOALS List the characteristics of the normal probability distribution. Define and calculate z values . Determine the probability an observation will lie between two points using the standard normal distribution.

dmccowan
Download Presentation

Chapter 5 Normal Probability Distribution

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Chapter 5Normal Probability Distribution • GOALS • List the characteristics of the normal probability distribution. • Define and calculate z values. • Determine the probability an observation will lie between two points using the standard normal distribution.

  2. CharacteristicsNormal Probability Distribution • The normal curve is bell-shaped and has a single peak at the exact center of the distribution. • The arithmetic mean, median, and mode of the distribution are equal and locatedat thepeak. • Thus half the area under the curve is above the mean and half is below it.

  3. Characteristics Normal Probability Distribution • The normal probability distribution is symmetrical about its mean. • The normal probability distribution is asymptotic. • That is the curve gets closer and closer to the X-axis but never actually touches it

  4. Characteristics Normal curve is symmetrical Theoretically, curve extends to infinity Mean, median, and mode are equal

  5. The Standard Normal PD • The standard normal (or z) distribution is a normal distribution • Mean of 0; standard deviation of 1 • A z-value is the distance between a selected value, designated X, and the population mean µ, divided by the population standard deviation, σ.

  6. Standard Normal - Example • The bi-monthly starting salaries of recent MBA graduates follows the normal distribution with a mean of $2,000 and a standard deviation of $200.

  7. Standard Normal - Example • Mean $2000 / SD $200… • What is the z-value for a salary of $2,200? • What is the z-value of $1,700?

  8. Areas Under the Normal Curve • +/- … • 1 Standard Deviation: About 68 % • 2 Standard Deviations: About 95% • Practically all (99.7%) is within three • Applications • Estimates of population ranges/description • Translates to “probability”

  9. Area - Example • The daily water usage per person in New Providence, New Jersey is normally distributed with a mean of 20 gallons and a standard deviation of 5 gallons. • About 68 percent of those living in New Providence will use how many gallons of water?

  10. Area - Example • What is the probability that a person from New Providence selected at random will use between 20 and 24 gallons per day? • Translate to “z value” • Use the book (or Excel) to determine the area under the curve… “the probability”

  11. Z-Table

  12. Area - Example • The area under a normal curve between a z-value of 0 and a z-value of 0.80 is 0.2881. • We conclude that 28.81 percent of the residents use between 20 and 24 gallons of water per day.

  13. -4 -3 -2 -1 0 1 2 3 4 x Area - Example P(0<z<.8) =.2881 0 < x < 0.8

  14. Area - Example • What percent of the population use between 18 and 26 gallons per day?

  15. Area - Example • Area associated with a z-value of – 0.40 is ____ • Remember that the curve is symmetrical! • Area associated with a z-value of 1.20 is ______ • Adding these areas, the result is _______ • We conclude that…

  16. Area - Example • Professor Mann has determined that the scores in his statistics course are approximately normally distributed with a mean of 72 and a standard deviation of 5. He announces to the class that the top 15 percent of the scores will earn an A. • What is the lowest score a student can earn and still receive an A?

  17. Area - Example • To begin let X be the score that separates an A from a B. • If 15 percent of the students score more than X, then 35 percent must score between the mean of 72 and X. • The z-value associated corresponding to 35 percent is about _______.

  18. Area - Example • We let z equal 1.04 and solve the standard normal equation for X. • The result is the score that separates students that earned an A from those that earned a B.

  19. Normal Approximation to the Binomial • The normal distribution (a continuous distribution) yields a good approximation of the binomial distribution (a discrete distribution) for large values of n. • Generally a good approximation to the binomial probability distribution when… • n*p and n*(1 - p) are both greater than 5.

  20. The Normal Approximation continued • Recall for the binomial experiment: • There are only two mutually exclusive outcomes (success or failure) on each trial. • A binomial distribution results from counting the number of successes. • Each trial is independent. • The probability is fixed from trial to trial, and the number of trials n is also fixed.

  21. Continuity Correction Factor • The value .5 subtracted or added, depending on the problem, to a selected value when a binomial probability distribution (a discrete probability distribution) is being approximated by a continuous probability distribution (the normal distribution). • Rules (pg 246)

  22. Approximation - Example • A recent study by a marketing research firm showed that 15% of American households owned a video camera. • For a sample of 200 homes, how many of the homes would you expect to have video cameras?

  23. Approximation - Example • What is the variance? • What is the standard deviation?

  24. Approximation - Example • What is the probability that less than 40 homes in the sample have video cameras? • We use the correction factor… (pg 246) • So X is ________ • The value of z is _______

  25. Approximation - Example • From Appendix D the area between 0 and 1.88 on the z scale is .4699. • So the area to the left of 1.88 is .5000 + .4699 = .9699. • The likelihood that less than 40 of the 200 homes have a video camera is about 97%.

  26. z = 1.88 0 1 2 3 4 P(z < 1.88) = .5000 + .4699 = 0.9699

More Related