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Chapter 9: Normal Distribution

Chapter 9: Normal Distribution. Motivating Example. Pizza Pit* Delivery They have a 30 minute “or it’s free” delivery guarantee The manager wants to know how often they can expect to give away free pizzas. *Pizza Pit is an actual restaurant in Ames, Iowa.

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Chapter 9: Normal Distribution

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  1. Chapter 9:Normal Distribution

  2. Motivating Example • Pizza Pit* Delivery • They have a 30 minute “or it’s free” delivery guarantee • The manager wants to know how often they can expect to give away free pizzas *Pizza Pit is an actual restaurant in Ames, Iowa. Logo is taken from http://www.pizzapitames.com/pizzapit/

  3. Motivating Example: Historical Data • Pizza Pit has been in operation for 20 years • The owner has kept track of the amount of time required for every delivery ever made • The owner creates a histogram (Figure 1, next slide) • The distribution is symmetric • The mean, median, and mode are 25 minutes • The standard deviation is 5 minutes

  4. Motivating Example: Histogram Figure 1. Histogram for Pizza Pit Delivery Times

  5. The owner creates a frequency polygon (Figure 2) based on the histogram, which he recognizes as a normal distribution Motivating Example: Frequency Polygon Figure 2. Frequency Polygon for Pizza Pit Delivery Times

  6. Normal Distribution Properties • Shape: Bell-shaped frequency polygon • Central Tendency: The mean, median, and mode coincide at the middle of the distribution • Symmetric: Half the cases fall above, and half fall below, the mean • Population Distribution: The distribution is based on a population

  7. Normal Distribution Properties • Mean and Standard Deviation: The normal distribution is characterized by two numbers • Mean, μ (mu – what a kitty says) • Standard Deviation, σ (sigma) • The Symbols Have Changed?!? Before we used the symbols and S for the mean and standard deviation • These represent characteristics of a sample • The normal distribution is based on the population • Importance of the Normal Distribution • The “normality assumption” is an important part of inferential statistics (we will cover this later in the semester) • It is also encountered in everyday life (“grading on a curve”)

  8. Normal Distribution Properties • How Do We Get μ and σ? • Textbook examples are often made up • Information might be available from a population (e.g., Census) • Historical data • Information from a sample is used to provide an estimate of μ and σ • Bottom Line: In this course, you will be given μ and σ

  9. Area Under the Normal Curve • The area under the normal curve between two values of a variable represents 3 things • Probability: The probability of being between the two values in the population • Proportion: The proportion of the population between the two values • Percent: When the probability or proportion is multiplied by 100, it represents the percent of the population between the two values

  10. Area Under the Normal Curve • Area Under the Entire Curve • Area, proportion, or probability is 1 • Percent is 100% • Area Above the Mean • Area, proportion, or probability is 0.50 • Percent is 50% • Area Below the Mean • Area, proportion, or probability is 0.50 • Percent is 50%

  11. Special Percentages • 68.26% of the cases are within 1 standard deviation of the mean (μ ± 1σ) • 34.13% are between the mean and 1 standard deviation above the mean • 34.13% are between the mean and 1 standard deviation below the mean • 95.46% of the cases are within 2 standard deviations of the mean (μ ± 2σ) • 99.72% of the cases are within 3 standard deviations of the mean (μ ± 3σ)

  12. Area Under the Normal Curve 99.72% 95.46% 68.26% 2.13% 13.60% 34.13% 34.13% 13.60% 2.13% -3σ-2σ -1σμ +1σ+2σ +3σ

  13. Pizza Pit Example: What Do We Know So Far? • Recall: Mean μ = 25 and standard deviation σ = 5 • 50% of the delivery times were more than 25 minutes • The probability that any given delivery will be more than 25 minutes is 0.50 • The proportion of delivery times more than 25 minutes is 0.50 • 50% of the delivery times were less than 25 minutes • The probability that any given delivery will be less than 25 minutes is 0.50 • The proportion of delivery times less than 25 minutes is 0.50

  14. Pizza Pit Example: What Do We Know So Far? • 68.26% of delivery times were between 20 and 30 minutes (within 1 standard deviation of the mean) • 34.13% of delivery times were between 20 and 25 minutes • 34.13% of delivery times were between 25 and 30 minutes • 95.46% of delivery times were between 15 and 35 minutes (within 2 standard deviations of the mean) • 99.72% of delivery times were between 10 and 40 minutes (within 3 standard deviations of the mean)

  15. Definition: The number of standard deviations that a given value of a variable is above or below the mean A negative sign indicates a value below the mean A positive sign indicates a value above the mean Computation: Y = Value of a variable μ = Mean of the variable σ = Standard deviation of the variable Z-Scores

  16. Z-Scores: Application to Pizza Pit Example • Z-score for a delivery time of 30 minutes? • Calculation: • Interpretation: 30 minutes is 1 standard deviation above the mean • Z-score for a delivery time of 18 minutes? • Calculation: • Interpretation: 18 minutes is 1.4 standard deviations below the mean • Z-score for a delivery time of 25 minutes? • Calculation: • Interpretation: 25 minutes is the mean

  17. In-Class Exercise: Pregnancy • Pregnancy Duration: Number of days from time of conception to time of birth has a normal distribution with mean μ = 266 days and standard deviation σ = 16 days • Question 1: Calculateand interpret the z-scores for women having the following pregnancy durations: • Rachael was pregnant for 266 days. • Monica was pregnant for 300 days. • Phoebe was pregnant for 255 days.

  18. Comparisons Using Z-Scores • “Common Scale”: Z-scores place everybody on a “common scale” called standard deviation units • Comparisons: Z-scores can be used to compare people on variables having different ranges of values and/or means and standard deviations • Restriction: This only applies to variables having a normal distribution

  19. Comparing Relative Performance on Exams • Cindy • The Soc 4880 final exam in Spring 2006 was worth 75 points • The scores in class had a normal distribution • The mean μ = 56 • The standard deviation σ = 6 • Cindy earned 70 points • Bobby • The Soc 4880 final exam in Fall 2006 was worth 100 points • The scores in class had a normal distribution • The mean μ = 75 • The standard deviation σ = 9 • Bobby earned 70 points

  20. Comparing Relative Performance on Exams • Who Did Better? If we know the means, standard deviations, and can assume a normal distribution, then comparisons can be made using z-scores • Cindy: • Bobby: • Conclusion? • Cindy was 2.33 standard deviations above the mean while Bobby was 0.56 standard deviations below the mean • Cindy did better than Bobby

  21. In-Class Exercise: Pregnancy • Question 2: Karla has been pregnant for 254 days and Garkel, a Yarkian woman, has been pregnant for 360 days (a) Calculate the z-score for Karla’s pregnancy. (Earth women have a normally-distributed pregnancy duration with mean μ = 266 days and standard deviation σ = 16 days.) (b) Compute the z-score for Garkel’s pregnancy. (Yarkian women have a normally-distributed pregnancy duration with mean μ = 412 days and standard deviation σ = 24 days.) (c) Use the z-scores from parts (a) and (b) to decide whether Karla or Garkel is further along in her pregnancy.

  22. Standard Normal Table (Appendix B) • Column A: Lists z-scores from 0 to +4 • The standard normal is symmetric • Proportions for positive values are the same as proportions for negative values • Column B: Lists the proportion of cases between the mean and z-score listed in Column A • When the z-score is positive, the proportion is to the right • When the z-score is negative, the proportion is to the left • Column C: Lists the proportion of cases beyond the z-score listed in Column A • When the z-score is positive, the proportion is to the right • When the z-score is negative, the proportion is to the left

  23. Standard Normal Table (Appendix B)

  24. Standard Normal Table (Appendix B) • How Will We Use Appendix B?: To find proportions, probabilities, and percentages between values of a variable • Pizza Pit Example • Let Y represent values of the variable (delivery time) • Recall that μ = 25 and σ = 5

  25. Proportion/Probability/PercentageBetween Mean and Positive Z-Score • Question: What is the probability that a delivery time will be between 25 and 33 minutes? • Procedure • Convert 33 to a z-score: • Look up 1.60 in Column A of Appendix B • Read across to Column B: 0.4452 • The probability is 0.4452 that a delivery time will be between 25 and 33 minutes

  26. Proportion/Probability/PercentageBetween Mean and Positive Z-Score

  27. Proportion/Probability/PercentageBetween Mean and Negative Z-Score • Question: What percentage of delivery times are between 10 and 25 minutes? • Procedure • Convert 10 to a z-score: • Look up 3.00 (ignore the negative sign) in Column A of Appendix B • Read across to Column B: 0.4986 • 49.86% of delivery times are between 10 and 25 minutes

  28. Proportion/Probability/PercentageBetween Mean and Negative Z-Score

  29. In-Class Exercise: Pregnancy • Pregnancy duration has a normal distribution with mean μ = 266 days and standard deviation σ = 16 days • Question 3: Calculate the percent of pregnancies that have a duration between 266 and 270 days • Question 4: Compute the proportion of pregnancies that have a duration between 220 and 266 days

  30. Proportion/Probability/Percentage Between Two Z-Scores on Opposite Sides of the Mean • Question: What proportion of delivery times are between 22 and 27 minutes? • Procedure • Convert 22 and 27 to z-scores: • Look up 0.60 and 0.40 in Column A of Appendix B • Read across to Column B: 0.2257 and 0.1554 • Add the values (0.2257 + 0.1554) to get .3811 • The proportion of delivery times between 22 and 27 minutes is 0.3811

  31. Proportion/Probability/Percentage Between Two Z-Scores on Opposite Sides of the Mean

  32. Proportion/Probability/Percentage Between Two Z-Scores on the Same Side of the Mean • Question: What is the probability that a delivery time will be between 15 and 20 minutes? • Procedure • Convert 15 and 20 to z-scores: • Look up 2.00 and 1.00 in Column A of Appendix B • Read across to Column B: 0.4772 and 0.3413 • Subtract the smaller value from the larger value (.4772 - .3413) to get 0.1359 • The probability is 0.1359 that a delivery time will be between 15 and 20 minutes

  33. Proportion/Probability/Percentage Between Two Z-Scores on the Same Side of the Mean

  34. Summary of Rules (So Far…)

  35. Summary of Rules (So Far…)

  36. In-Class Exercise: Pregnancy • Pregnancy duration has a normal distribution with mean μ = 266 days and standard deviation σ = 16 days • Question 5: My mother-in-law Phyllis (a maternity nurse) claims that most births occur between 252 and 273 days. Determine the percent of pregnancies with durations between 252 and 273 days, and explain how I should response to Phyllis’ claim. • Question 6: Calculate the probability that a pregnancy will have a duration between 269 and 292 days.

  37. Proportion/Probability/PercentageAbove a Positive Z-Score • Question: What percentage of deliveries take more than 30 minutes? • Procedure • Convert 30 to a z-score: • Look up 1.00 in Column A of Appendix B • Read across to Column C: 0.1587 • 15.87% of deliveries take more than 30 minutes

  38. Proportion/Probability/PercentageAbove a Positive Z-Score

  39. Proportion/Probability/Percentage Above a Positive Z-Score: Special Example • Question: A customer complains that it took 45 minutes for a delivery. How likely does this seem? • Procedure • Convert 45 to a z-score: • Look up 4.00 in Column A of Appendix B • Read across to Column C: <0.0001 • This does not seem very likely (the probably is less than 0.0001). Either the customer is wrong or the delivery person was having a bad day

  40. Proportion/Probability/Percentage Above a Positive Z-Score: Special Example

  41. Proportion/Probability/PercentageBelow a Negative Z-Score • Question: What proportion of delivery times are less than 13 minutes? • Procedure • Convert 13 to a z-score: • Look up 2.40 in Column A of Appendix B • Read across to Column C: 0.0082 • The proportion of delivery times less than 13 minutes is 0.0082

  42. Proportion/Probability/PercentageBelow a Negative Z-Score

  43. In-Class Exercise: Pregnancy • Pregnancy duration has a normal distribution with mean μ = 266 days and standard deviation σ = 16 days • Question 7: In an article published in the National Inquirer, Kathleen claims to have been pregnant for 365 days. How likely does this seem? • Question 8: Births prior to 252 days are considered premature. Compute the percent of births that are premature.

  44. Question: What is the probability that a delivery will take less than 37 minutes? Procedure Convert 37 to a z-score: Look up 2.40 in Column A of Appendix B Add 0.50 to the value in Column B (0.4918): 0.4918 + 0.50 = 0.9918 OR subtract the value in Column C (0.0082) from 1: 1-0.0082 = 0.9918 The probability that a delivery will take less than 37 minutes is 0.9918 Proportion/Probability/PercentageBelow a Positive Z-Score

  45. Proportion/Probability/PercentageBelow a Positive Z-Score

  46. Question: What percentage of delivery times are more than 24 minutes? Procedure Convert 24 to a z-score: Look up 0.20 in Column A of Appendix B Add 0.50 to the value in Column B (0.0793): 0.0793 + 0.50 = 0.5793 OR subtract the value in Column C (0.4207) from 1: 1-0.4207 = 0.5793 57.93% of delivery times are more than 24 minutes Proportion/Probability/PercentageAbove a Negative Z-Score

  47. Proportion/Probability/PercentageAbove a Negative Z-Score

  48. Summary of Additional Rules

  49. Summary of Additional Rules

  50. In-Class Exercise: Pregnancy • Pregnancy duration has a normal distribution with mean μ = 266 days and standard deviation σ = 16 days • Question 9: Determine the proportion of pregnancies that are more than 250 days • Question 10: Compute the percent of pregnancies that are less than 280 days

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