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

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…

Symmetric Density Curve MxMx.5 1Q3Q

M x

- 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…  mu  Notation Computed ValuesDensity Curve Values Mean Std.Dev x s  sigma  ** English = Computed Values / Greek = Density Curve **

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 (  ) as N(  ) ex: N(25, 4.7) Mean = Median

Locating  Less SteepMore Steep 

Rule 68% 95% 99.7% **Applies to all normal distributions**

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  2 % 95% 99.7%

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.

WARNING! WARNING! MANY DISTRIBUTIONS ARE NOT NORMAL!!

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 = x -   (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 (+/-)

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

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: Z = = 1.0 African z-score: Z = = 2.0 ** The African croc at +2  is more of an anomaly **

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.

ex: What proportion of all women are less than 68 in. tall? Z = = 1.4 Area =.9192 Z = 1.4

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

4)5)6) Finding Normal Curve Areas - Using Table -A z ) Find z-value to tenths place 2) Find correct hundreths place column 3) Find intersection of z-value and hundreths place

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 ) x = level of cholesterol 2) z = = )

= > 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? Z = ? A to the left =.9 A to the right =.1

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

Finding a value given a proportion - Reverse Table-A lookup z ) 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 = 1.28Plug z-value into z-formula solved for x:

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