1 / 129

Chapter 6 Probability Distributions

Chapter 6 Probability Distributions. Learn …. To analyze how likely it is that sample results will be “close” to population values How probability provides the basis for making statistical inferences. Inferential Statistics.

shawn
Download Presentation

Chapter 6 Probability Distributions

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 6Probability Distributions • Learn …. To analyze how likely it is that sample results will be “close” to population values How probability provides the basis for making statistical inferences

  2. Inferential Statistics • Use sample data to make decisions and predictions about a population

  3. Section 6.1 How Can We Summarize Possible Outcomes and Their Probabilities?

  4. Randomness • The numerical values that a variable assumes are the result of some random phenomenon: • Selecting a random sample for a population or • Performing a randomized experiment

  5. Random Variable • A random variable is a numerical measurement of the outcome of a random phenomenon.

  6. Random Variable • Use letters near the end of the alphabet, such as x, to symbolize variables. • Use a capital letter, such as X, to refer to the random variable itself. • Use a small letter, such as x, to refer to a particular value of the variable.

  7. Probability Distribution • The probability distribution of a random variable specifies its possible values and their probabilities.

  8. Discrete Random Variable • The possible outcomes are a set of separate numbers: (0, 1,2, …).

  9. Probability Distribution of a Discrete Random Variable • A discrete random variable X takes a set of separate values (such as 0,1,2,…) • Its probability distribution assigns a probability P(x) to each possible value x: • For each x, the probability P(x) falls between 0 and 1 • The sum of the probabilities for all the possible x values equals 1

  10. Example: How many Home Runs Will the Red Sox Hit in a Game? • What is the estimated probability of at least three home runs?

  11. Example: How many Home Runs Will the Red Sox Hit in a Game?

  12. Parameters of a Probability Distribution • Parameters: numerical summaries of a probability distribution.

  13. The Mean of a Probability Distribution • The mean of a probability distribution is denoted by the parameter, µ.

  14. The Mean of a Discrete Probability Distribution • The mean of a probability distribution for a discrete random variable is where the sum is taken over all possible values of x.

  15. Expected Value of X • The mean of a probability distribution of a random variable X is also called the expected value of X. • The expected value reflects not what we’ll observe in a single observation, but rather that we expect for the average in a long run of observations.

  16. Example: What’s the Expected Number of Home Runs in a Baseball Game? • Find the mean of this probability distribution.

  17. Example: What’s the Expected Number of Home Runs in a Baseball Game? • The mean: = 0(0.23) + 1(0.38) + 2(0.22) + 3(0.13) + 4(0.03) + 5(0.01) = 1.38

  18. The Standard Deviation of a Probability Distribution • The standard deviation of a probability distribution, denoted by the parameter, σ, measures its spread. • Larger values of σ correspond to greater spread.

  19. Continuous Random Variable • A continuous random variable has an infinite continuum of possible values in an interval. • Examples are: time, age and size measures such as height and weight.

  20. Probability Distribution of a Continuous Random Variable • A continuous random variable has possible values that from an interval. • Its probability distribution is specified by a curve. • Each interval has probability between 0 and 1. • The interval containing all possible values has probability equal to 1.

  21. Continuous Variables are Measured in a Discrete Manner because of Rounding.

  22. Which Wager do You Prefer? • You are given $100 and told that you must pick one of two wagers, for an outcome based on flipping a coin: A. You win $200 if it comes up heads and lose $50 if it comes up tails. B. You win $350 if it comes up head and lose your original $100 if it comes up tails. • Without doing any calculation, which wager would you prefer?

  23. You win $200 if it comes up heads and lose $50 if it comes up tails. Find the expected outcome for this wager. • $100 • $25 • $50 • $75

  24. You win $350 if it comes up head and lose your original $100 if it comes up tails. Find the expected outcome for this wager. • $100 • $125 • $350 • $275

  25. Section 6.2 How Can We Find Probabilities for Bell-Shaped Distributions?

  26. Normal Distribution • The normal distribution is symmetric, bell-shaped and characterized by its mean µ and standard deviationσ. • The probability of falling within any particular number of standard deviations of µ is the same for all normal distributions.

  27. Normal Distribution

  28. Z-Score • Recall: The z-score for an observation is the number of standard deviations that it falls from the mean.

  29. Z-Score • For each fixed number z, the probability within z standard deviations of the mean is the area under the normal curve between

  30. Z-Score • For z = 1: 68% of the area (probability) of a normal distribution falls between:

  31. Z-Score • For z = 2: 95% of the area (probability) of a normal distribution falls between:

  32. Z-Score • For z = 3: Nearly 100% of the area (probability) of a normal distribution falls between:

  33. The Normal Distribution: The Most Important One in Statistics • It’s important because… • Many variables have approximate normal distributions. • It’s used to approximate many discrete distributions. • Many statistical methods use the normal distribution even when the data are not bell-shaped.

  34. Finding Normal Probabilities for Various Z-values • Suppose we wish to find the probability within, say, 1.43 standard deviations of µ.

  35. Z-Scores and the Standard Normal Distribution • When a random variable has a normal distribution and its values are converted to z-scores by subtracting the mean and dividing by the standard deviation, the z-scores have the standard normal distribution.

  36. Example: Find the probability within 1.43 standard deviations of µ

  37. Example: Find the probability within 1.43 standard deviations of µ • Probability below 1.43σ = .9236 • Probability above 1.43σ = .0764 • By symmetry, probability below -1.43σ = .0764 • Total probability under the curve = 1

  38. Example: Find the probability within 1.43 standard deviations of µ

  39. Example: Find the probability within 1.43 standard deviations of µ • The probability falling within 1.43 standard deviations of the mean equals: 1 – 0.1528 = 0.8472, about 85%

  40. How Can We Find the Value of z for a Certain Cumulative Probability? • Example: Find the value of z for a cumulative probability of 0.025.

  41. Example: Find the Value of z For a Cumulative Probability of 0.025 Example: Find the Value of z For a Cumulative Probability of 0.025 • Look up the cumulative probability of 0.025 in the body of Table A. • A cumulative probability of 0.025 corresponds to z = -1.96. • So, a probability of 0.025 lies below µ - 1.96σ.

  42. Example: Find the Value of z For a Cumulative Probability of 0.025

  43. Example: What IQ Do You Need to Get Into Mensa? • Mensa is a society of high-IQ people whose members have a score on an IQ test at the 98th percentile or higher.

  44. Example: What IQ Do You Need to Get Into Mensa? • How many standard deviations above the mean is the 98th percentile? • The cumulative probability of 0.980 in the body of Table A corresponds to z = 2.05. • The 98th percentile is 2.05 standard deviations above µ.

  45. Example: What IQ Do You Need to Get Into Mensa? • What is the IQ for that percentile? • Since µ = 100 and σ 16, the 98th percentile of IQ equals: µ + 2.05σ = 100 + 2.05(16) = 133

  46. Z-Score for a Value of a Random Variable • The z-score for a value of a random variable is the number of standard deviations that x falls from the mean µ. • It is calculated as:

  47. Example: Finding Your Relative Standing on The SAT • Scores on the verbal or math portion of the SAT are approximately normally distributed with mean µ = 500 and standard deviation σ = 100. The scores range from 200 to 800.

  48. Example: Finding Your Relative Standing on The SAT • If one of your SAT scores was x = 650, how many standard deviations from the mean was it?

  49. Example: Finding Your Relative Standing on The SAT • Find the z-score for x = 650.

  50. Example: Finding Your Relative Standing on The SAT • What percentage of SAT scores was higher than yours? • Find the cumulative probability for the z-score of 1.50 from Table A. • The cumulative probability is 0.9332.

More Related