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Understanding Continuous Probability Distributions

Learn the characteristics, calculations, and applications of continuous probability distributions, including the uniform and normal distributions.

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Understanding Continuous Probability Distributions

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  1. Chapter 7 Continuous Probability Distributions

  2. Goals • Understand the difference between discrete and continuous distributions • Compute the mean and the standard deviation for a uniform distribution • Compute probabilities using the uniform distribution

  3. Goals • List the characteristics of the: • Normal probability distribution • Standard normal probability distribution • Define and calculate z values • Use the standard normal probability distribution to find area: • Above the mean • Below the mean • Between two values • Above one value • Below one value • Use the normal distribution to approximate the binomial probability distribution

  4. Probability Distribution • A listing of all the outcomes of an experiment and the probability associated with each outcome • Probability distributions are useful for making probability statements concerning the values of a random variable • Our goal is to find probability between two values: • Example: What is the probability that the daily water usage will lie between 15 and 25 gallons? A: 68%

  5. Probability Distribution • Discrete Probability Distributions (Chapter 6) • Based On Discrete Random Variables • We looked at: • Binomial Probability Distribution • Continuous Probability Distribution (Chapter 7) • Based On Continuous Random Variables: • We will look at: • Uniform Probability Distribution • Normal Probability Distribution

  6. Continuous Probability Distributions • These continuous probability distributions will be all about  • Area!!!!

  7. Continuous Probability Distribution: • Uniform Probability Distribution • Within the interval 15 to 25 minutes, the time it takes to fill out a typical 1040EZ tax return at a VITA site tends to follow a uniform distribution • Random Variable is time (only possibilities within the interval) • Each value has same probability • Normal Probability Distribution • The weight distribution of a manufactured box of cereal tends to follow a normal distribution • Random Variable is box weight and will cover all possibilities • Each value has different probability

  8. Uniform Probability Distribution • Distributions shape is rectangular • Minimum value = a • Maximum value = b • a and b imply a range • Height of the distribution is constant (uniform) for all values between a and b • Implies all values in range are equally likely

  9. What is the mean wait time? Suppose the time that you wait on the telephone for a live representative of your phone company to discuss your problem with you is uniformly distributed between 5 and 25 minutes. a + b 2 m= 5+25 2 = = 15 What is the standard deviation of the wait time? (b-a)2 12 s = (25-5)2 12 = = 5.77

  10. What is the probability of waiting more than ten minutes? The area from 10 to 25 minutes is 15 minutes. Thus: P(10 < wait time < 25) = height*base = 1 (25-5) *15 = .75

  11. What is the probability of waiting between 15 and 20 minutes? The area from 15 to 20 minutes is 5 minutes. Thus: P(15 < wait time < 20) = height*base = 1 (25-5) *5 = .25

  12. Normal Probability Distribution Is All About Area!Total Area = 1.0

  13. Normal Probability Distribution Formula Awesome! No, that’s o.k., we can use Appendix or Excel functions!

  14. Characteristics of a NormalProbability Distribution(And Accompanying Normal Curve) • 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 located at the peak • Thus half the area under the curve is above the mean and half is below it • The normal probability distribution is symmetrical about its mean • If we cut the normal curve vertically at this center value, the two halves will be mirror images

  15. Characteristics of a NormalProbability Distribution(And Accompanying Normal Curve) • The normal probability distribution is asymptotic • The curve gets closer and closer to the X-axis but never actually touches it • The “tails” of the curve extend indefinitely in both directions • The Location of a normal distribution is determined by mean µ • The dispersion of a normal distribution is determined by the standard deviation  Now Let’s Look At Some Pictures That Will Show Relationships Amongst Various Means & Standard Deviations

  16. Characteristics of a Normal Distribution Normal curve is symmetrical Theoretically, curve extends to infinity Mean, median, and mode are equal There Is A Family Of Normal Probability Distributions

  17. Which One Is Normal?

  18. Equal Means, Unequal Standard Deviations

  19. Equal Means, Unequal Standard Deviations

  20. Equal Means, Unequal Standard Deviations

  21. Unequal Means, Equal Standard Deviations

  22. Unequal Means, Unequal Standard Deviations This Family Of Normal Probability Distributions Is Unlimited In Number! Luckily, One Of The Family Members May Be Used In All Circumstances Where The Normal Distribution Is Applicable Standard Normal Distribution

  23. Standard Normal Probability Distribution • The standard normal distribution (z distribution ) is a normal distribution with a mean of 0 and a standard deviation of 1 • The percentage of area between two z-scores in any normal distribution is the same! • Standard deviation & terms may be different, but area will be the same! • Normal distributions can be converted to the standard normal distribution using z-values…

  24. Define And Calculate z-values • Any normal distribution can be converted, or “standardized” to the standard normal distribution using z-values • Z-values: • Distance from the mean, measured in units of standard deviation • The Formula Is: • Z-values are also called: • Standard normal value • Z score • Z statistic • Standard normal deviate • Normal deviate Remember Your Algebra So That You Can Solve For Any One Of The Variables

  25. Standard Normal Probability Distribution 0 means that there is no deviation from the mean!

  26. Convert Value From A Normal Distribution To A Z-score Example 1 • 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 • What is the z-value for a salary of $2,200?

  27. Convert Value From A Normal DistributionTo A Z-score, Example 2 • What is the z-value of $1,700? • A z-value of 1 indicates that the value of $2,200 is one standard deviation above the mean of $2,000 (2000 + 1*200) • A z-value of –1.50 indicates that $1,700 is 1.5 standard deviation below the mean of $2000 (2000 – 1.5*200) Now we can look at a graph 

  28. What is the probability that a foreman’s salary will fall between 1,700 and $2,200?

  29. But It Is Really All About Area Under The CurveRemember: • Areas Under the Normal Curve • Empirical Rule (Normal Rule): • About 68% of the observations will lie within 1 σ of the mean • About 95% of the observations will lie within 2 σ of the mean • Virtually all the observations will be within 3 σ of the mean Hints: Many statistical chores can be solved with this normal curve Nevertheless, “The whole world does not fit into a normal curve”

  30. Empirical Rule (Normal Rule): Between what two values do about 95% of the values occur? What if you want to find the % of values that lie between z-scores 0 and 1.56?

  31. Table In Appendix Or On Inside Cover

  32. .475 z  0 1.96 Use The Standard Normal Probability Distribution To Find Area:(Table On Inside Back Cover)

  33. Example 1 • The daily water usage per person in New Providence, New Jersey is normally distributed • Mean = 20 gallons • Standard deviation = 5 gallons • About 68% of those living in New Providence will use how many gallons of water? • +/- 1 standard deviation will give us: • About 68% of the daily water usage will lie between 15 and 25 gallons

  34. Example 2 • What is the probability that a person from New Providence selected at random will use between 20 and 24 gallons per day?

  35. Use The Table In The Back Of The Book And Look Up .80 • 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 • See the following diagram:

  36. r a l i t r b u i o n : m = 0 , 0 . 4 0 . 3 0 . 2 x ( f 0 . 1 . 0 - 5 x Area =.2881 “28.81% of the residents use between 20 and 24 gallons of water per day” -4 -3 -2 -1 0 1 2 3 4 z

  37. Example 3 • What percent of the population use between 18 and 26 gallons per day?

  38. Example 3 • The area associated with a z-value of –0.40 is .1554 • Because the curve is symmetrical. look up .40 on the right • The area associated with a z-value of 1.20 is .3849 • .1554 + .3849 = .5403 • We conclude that 54.03 percent of the residents use between 18 and 26 gallons of water per day

  39. Example 4 • 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? • .50 - .15 = .35  This is the area under the curve! • You must look into table and find the value closest to .35

  40. Table In Appendix Or On Inside Cover

  41. Solve for X, The Score You Need To Get An A • The result is the score that separates students that earned an A from those that earned a B • Those with a score of 77.2 or more earn an A

  42. Example 5 What is the probability of selecting a shift foreman whose salary is between $790 & $1200?

  43. Example 5 • Find Z-scores • Look up area under the curve in the tables

  44. Table In Appendix Or On Inside Cover

  45. Example 5 • The probability of selecting a shift foreman whose salary is between $790 & $1200 is: • .4772 + .4821 = .9593

  46. Example 6 What is the probability of selecting a shift foreman whose salary is less than $790?

  47. Example 6 • Look up the area • The area = .4821 • .5 - .4821 = .0179 • The probability of selecting a shift foreman whose salary is less than $790 is .0179

  48. Summary

  49. Finding Area Under The Standard Normal Distribution – It’s All About Area!

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