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Chapter 3 – The Normal Distributions

= .6061. 40 out of 66 obs fall between 4 - 7. Chapter 3 – The Normal Distributions. Density Curves vs. Histograms:. Histogram of 66 Observations. 16. - Histogram displays count of obs in a given category…. 14. 12. 10. 16 out of 66 obs fall between 5-6. 8. =1. 6. 4. 16/66 = .2424. 2.

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Chapter 3 – The Normal Distributions

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  1. = .6061 40 out of 66 obs fall between 4 - 7 Chapter 3 – The Normal Distributions Density Curves vs. Histograms: Histogram of 66 Observations 16 - Histogram displays count of obs in a given category… 14 12 10 16 out of 66 obs fall between 5-6 8 =1 6 4 16/66 = .2424 2 40/66 = .6061 1 2 3 4 5 6 7 8 9 10 - Density curves describe what proportion of the observations fall into each category (not the count of the obs…)

  2. A Density Curve is a curve that: - Is always on or above the horizontal axis, and - has area exactly 1 underneath it. • Describes the overall pattern of the distribution • Areas under the curve and above any range of values on the x-axis represent the proportion of total observations taking those values… Mean and Median of a Density Curve • The Median of a density curve is the = areas point… ½ of the area on one side and ½ on the other.. • The Mean of a density curve is the point at which it would balance if made of solid material…

  3. 1Q 3Q Symmetric Density Curve .5 .5 Mx

  4. x M

  5. Notation Computed Values Density Curve Values Mean x Std.Dev s - Density Curves are “idealized descriptions” of the data distribution… so we need to distinguish between the actually computed values of the mean and standard deviation and the mean and standard deviation of a density curve…they MAY be different… m (mu) s (sigma) ** English = Computed Values / Greek = Density Curve **

  6. Stats 1.3 Continued • Normal Curves: • Symmetric • Single-Peaked • Bell shaped • Describes a ‘Normal’ Distribution • All normal distributions have the same overall shape • Described by giving the mean () and std. deviation () asN() ex: N(25, 4.7) • Mean = Median

  7. More Steep Less Steep  Locating 

  8. 68% 95% 99.7% **Applies to all normal distributions** 68 - 95 - 99.7 Rule

  9. m  68% 95% 99.7% ex: Heights of women 18-24: N(64.5, 2.5) (inches) 2 = 2 x (2.5) = 5 inches 3 = 3 x (2.5) = 7.5 inches  57 59.5 2 69.5 72

  10. Why is normal important? • Represents some distributions of real data (ex: test scores, biological populations, etc…) • Provides good approximations to chance outcomes (ex: coin tosses) • Statistical Inference procedures based on normal distributions work well for ‘roughly’ symmetric distributions.

  11. WARNING! WARNING!MANY DISTRIBUTIONS ARENOT NORMAL!!

  12. x -  Z =  1.3 cont’d - The Standard Normal Distribution • Standardized Observations / “Standardizing” • Theory: All normal distributions are the same if we measure in units of size  from  as the center… •  = std. deviations / common scale • Changing to these units is called “standardizing”… Notation / Formula for Standardizing an observation: (Z = “z-score” or “z-number”) • Z can be negative - tells us how many std. dev’s. away from the mean we are AND in which direction (+/-)

  13. Height - 64.5 Z = 2.5 68 - 64.5 Z = = 1.4 2.5 60 - 64.5 Z = = -1.8 2.5 ex: if x > , Z = (+) / if x < , Z = (-) ex: Heights of women 18-24 = N(64.5, 2.5) ex: z-score for height of 68 inches… ex: z-score for height of 60 inches…

  14. 1400 - 1200 Z = 200 1100 - 900 Z = 100 ex: Weights of different species… South American vs. African Crocodiles South American: N(1200, 200) African: N(900, 100) For a South American croc at 1400 lbs and an African croc at 1100 lbs, which is the greater anomaly? South American z-score: = 1.0 African z-score: = 2.0 ** The African croc at +2 is more of an anomaly **

  15. 1.3 cont’d - Normal Distribution Calculations • Calculating area under a density curve • Area = proportion of the observations in a distribution • Because all normal distributions are the same after we standardize, we can use one normal curve to compute areas for ANY normal distribution. • The one curve we use is called the “Standard Normal Distribution”… • Can use the calculator OR Table-A inside front cover of book to find areas under the std. normal curve.

  16. Z = 1.4 68 - 64.5 Z = = 1.4 2.5 ex: What proportion of all women 18-24 are less than 68 in. tall? Area = .9192

  17. Highlight and Press ‘ENTER’ 3) 1) 4) 5) 6) Finding Normal Curve Areas - Calculator Steps 2)

  18. 4) 5) 6) .00 .01 z 1.3 .9032 .9049 1.4 .9192 .9207 1.5 .9332 .9345 Finding Normal Curve Areas - Using Table -A 2) Find correct hundreths place column 1) Find z-value to tenths place 3) Find intersection of z-value and hundreths place

  19. 3) 240 - 170 2) z = = 2.33 30 Finding Normal Proportions 1) State the problem in terms of the desired variable. 2) Standardize / Draw pic 3) Use Calculator / Tbl-A ex: Blood cholesterol levels distribute normally in people of the same age and sex. For 14 yr old males the distribution is N(170, 30) (mg/dl). Levels above 240 will require medical attention. Find the proportion of 14 yr old males that have a blood cholesterol level above 240. 1) x = level of cholesterol

  20. A to the left = .9 A to the right = .1 Z = ? = .0098 --> about 1% Finding a value given a proportion ex: SAT scores distribute normally N(430, 100). What score would be necessary to be in the top 10% of all scores?

  21. A<, ,  ENTER Finding value given a proportion - Calculator 2nd VARS (DISTR) invNorm(area to the left of value, mean, std.dev) = 558 or greater to be in top 10%

  22. .08 .09 z 1.1 .8810 .8830 1.2 .8997 .9015 1.3 .9162 .9177 Finding a value given a proportion - Reverse Table-A lookup 1) Scan table for area value closest to what is needed… 2) Trace left and up to obtain z-number to two places… For area = .9 to the left, z = 1.28 Plug z-value into z-formula solved for x:

  23. z = x -  x = z +  Solving z formula for x: x -  z =  For N(430, 100) and z = 1.28, x = (1.28)(100) + 430 x = 558

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