Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16...
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Transcript of Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16...
![Page 1: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/1.jpg)
Chapter 3 – The Normal Distributions
Density Curves vs. Histograms:
1 102 3 4 5 6 7 8 9
2
4
6
8
10
12
14
16
Histogram of 66 Observations
- Histogram displays count of obs in a given category…
40 out of 66 obs fall between 4 - 7
16/66 = .2424
40/66 = .6061
- Density curves describe what proportion of the observations fall into each category (not the count of the obs…)
= .6061=116 out of 66 obs fall
between 5-6
![Page 2: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/2.jpg)
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…
![Page 3: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/3.jpg)
Symmetric Density Curve
Mx
.5.5
1Q 3Q
![Page 4: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/4.jpg)
M x
![Page 5: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/5.jpg)
- 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 Values Density Curve Values
Mean
Std.Dev
x
s sigma
** English = Computed Values / Greek = Density Curve **
![Page 6: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/6.jpg)
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
![Page 7: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/7.jpg)
Locating
Less Steep
More Steep
![Page 8: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/8.jpg)
68 - 95 - 99.7 Rule
68%
95%
99.7%
**Applies to all normal distributions**
![Page 9: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/9.jpg)
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
69.5 72259.557
68%
95%
99.7%
![Page 10: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/10.jpg)
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.
![Page 11: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/11.jpg)
WARNING! WARNING!MANY DISTRIBUTIONS
ARENOT NORMAL!!
![Page 12: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/12.jpg)
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 (+/-)
![Page 13: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/13.jpg)
ex: if x > , Z = (+) / if x < , Z = (-)
ex: Heights of women 18-24 = N(64.5, 2.5)
Z = Height - 64.5
2.5
ex: z-score for height of 68 inches…
Z = 68 - 64.5
2.5= 1.4
ex: z-score for height of 60 inches…
Z = 60 - 64.5
2.5= -1.8
![Page 14: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/14.jpg)
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 = 1400 - 1200
200= 1.0
African z-score: Z = 1100 - 900
100= 2.0
** The African croc at +2 is more of an anomaly **
![Page 15: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/15.jpg)
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.
![Page 16: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/16.jpg)
ex: What proportion of all women 18-24 are less than 68 in. tall?
Z = 68 - 64.5
2.5= 1.4
Area = .9192
Z = 1.4
![Page 17: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/17.jpg)
Finding Normal Curve Areas - Calculator Steps
1)
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Highlight and Press ‘ENTER’
![Page 18: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/18.jpg)
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Finding Normal Curve Areas - Using Table -A
z .00 .01
1.3
1.4
1.5
.9032
.9332
.9192
.9049
.9207
.9345
1) Find z-value to tenths place
2) Find correct hundreths place column
3) Find intersection of z-value and hundreths place
![Page 19: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/19.jpg)
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
2) z = 240 - 170
30= 2.33
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![Page 20: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/20.jpg)
QuickTime™ and aPNG decompressor
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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 = .1A to the right = .1
![Page 21: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/21.jpg)
Finding value given a proportion - Calculator
2nd VARSQuickTime™ and aPNG decompressor
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invNorm(area to the left of value, mean, std.dev)
ENTERQuickTime™ and aPNG decompressor
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(DISTR)
A<, ,
![Page 22: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/22.jpg)
Finding a value given a proportion - Reverse Table-A lookup
z .08 .09
1.1
1.2
1.3
.8810
.9162
.8997
.8830
.9015
.9177
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:
![Page 23: Chapter 3 – The Normal Distributions Density Curves vs. Histograms: 11023456789 2 4 6 8 12 14 16 Histogram of 66 Observations - Histogram displays count.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649ebd5503460f94bc7752/html5/thumbnails/23.jpg)
x -
z = z = x - x = z +
For N(430, 100) and z = 1.28, x = (1.28)(100) + 430
x = 558
Solving z formula for x: