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AP Statistics

AP Statistics . Section 2.2 The Normal Distribution. Objective: To be able to calculate percentiles using the normal distribution. Normal Distributions: Bell-shaped density curve. All basically the same shape. Identified by the mean μ and standard deviation σ . SIDE:

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AP Statistics

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  1. AP Statistics Section 2.2 The Normal Distribution

  2. Objective: To be able to calculate percentiles using the normal distribution. Normal Distributions: • Bell-shaped density curve. • All basically the same shape. • Identified by the mean μ and standard deviation σ. SIDE: • Use Greek letters to represent parameters. (population) • Use standard letters for statistics. (sample)

  3. Notation: X ~ N ( μ, σ) Sketch: Inflection point: a point on the graph where the curve changes concavity.

  4. 68-95-99.7 Rule: In a normal distribution, approximately • 68% of all observations lie within _____ standard deviation of the mean. • 95% of all observations lie within _____ standard deviations of the mean. • 99.7% of all observations lie within _____ standard deviations of the mean.

  5. Ex. Adult males weights are normally distributed with a mean of 190 pounds and a standard deviation of 30 pounds. Find the proportion of adult males whose weights fall in the following regions. • X < 190 b. X < 160 c. X > 250 d. X = 250 • 160 < X < 250 f. X > 280 g. 100 < X < 250 h. X < 130 or X > 250

  6. Q: What does changing μ but not σ do to the distribution? Q: What does changing σ but not μ do to the distribution? Why do we use the Normal Distribution: • Good model for real world data. • Easy to approximate percentiles. • Many statistical inference procedures are based on normality. Equation for the Normal Distribution:

  7. Standard Normal Distributions: Z ~ N(0,1) Diagram: Using the z-table, find the proportion of observations such that: (Area to the left) Z < -1 Z < 2.06 Z < .56

  8. (Area to the right) Z > 1.53 Z > -1.05 (Area in between two values) -1 < Z < 1 -2.54 < Z < -.26 Working backwards with the table: What z-score represents the 40th percentile?

  9. What z-score represents the first quartile? What z-score represent the upper 10 percent of the area? Using X = weight of an adult male and X~N(190,30), find the proportion of observations such that: X < 145 X < 213

  10. X > 245 X > 153 122 < X < 200 X < 132 or X > 205

  11. Working backwards from a percentile to a value of X. Q: What weight represents the 85th percentile? Q: What body weight represents the heaviest 5% of adult males? **If a z-score falls outside the range on the z-table, then it is approximately 0.

  12. Example: In 2011 Jose Reyes had a batting average of .337. During that season X ~ N(.266,.028). In 2008 Chipper Jones had a batting average of .364. During that season X ~ N(.280, .037). Which player had the better season? What batting average represents the 70th %-ile in 2008?

  13. ASSESSING NORMALITY: • Construct a histogram or stem and leaf plot and look for a bell-shaped pattern. • Good for large data sets. • Mark the x-axis with and observe how closely the observations follow the 68-95-99.7 Rule. Ex. FLIP 50 Program or Pulse Data

  14. Normal Probability Plot (Normal Quantile Plot) NPP • Most common method for assessing normality. • It is a plot of • IF THE PLOT APPEARS FAIRLY LINEAR, THEN WE CAN ASSUME THAT THE DATA FOLLOWS A NORMAL DISTRIBUTION. • If most of the points are above the line y=x , then the data is skewed right. • If most of the points are below the line y=x , then the data is skewed left. • This graph will be very important later in the course! Ex. FLIP 50 results

  15. How a NPP works: (extra) • Rank the data from min to max. • Calculate the percentile for each point such that %-ile is calculated for each . • Calculate the z-score for each percentile. • Plot all ordered pairs .

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