Normal Probability Distributions REFERENCE BOOK: BIOSTATISTICS FOR THE BIOLOGICAL AND HEALTH SCIE NCES BY MARC M. TRIOLA & MARIO F. TRIOLA COPYRIGHT © 2010, 2007, 2004 PEARSON EDUCATION, INC. ALL RIGHTS RESERVED.
Normal Probability Distributions Overview The Standard Normal Distribution
Applications of Normal Distributions
Overview Chapter focus is on: Continuous random variables (infinite number of outcomes) Normal distributions: mathematically is described by following equation
f(x) =
-1 e2
2 ) ( x-
2p
The Standard Normal Distribution
Introduction This section presents the standard normal distribution which has three properties:
1. It is bell-shaped. 2. It has a mean equal to 0. 3. It has a standard deviation equal to 1.
It is extremely important to develop the skill to find areas (or probabilities or relative frequencies) corresponding to various regions under the graph of the standard normal distribution.
Definition
ď ś A continuous random variable has a uniform distribution if its values spread evenly over the range of probabilities. The graph of a uniform distribution results in a rectangular shape.
Definition A density curve is the graph of a continuous probability distribution. It must satisfy the following properties:
1. The total area under the curve must equal 1. 2. Every point on the curve must have a vertical height that is 0 or greater (That is, the curve cannot fall below the x-axis). Using Area to Find Probability
Definition ď ś The standard normal distribution is a probability distribution with mean equal to 0 and standard deviation equal to 1, and the total area under its density curve is equal to 1.
The Standard Normal Distribution A standard normal distribution has an arithmetic mean = 0 and variance = 1, and is written as: Z ~ N (0, 1) Where N = normal 0=µ 1 = σ2
Z ~ N (0, 1)
µ
σ2
Table A-2 - Example z Score
Distance along horizontal scale of the standard normal distribution; refer to the leftmost column and top row of Table A-2. Area Region under the curve; refer to the values in the body of Table A-2.
Reading Z Table 1.
P(Z<1)
Larger portion = 0.8413 (Positive Z-Score)
Reading Z Table Please refer to Z table and observe the following probabilities for Z = 1
2.
P(Z>1) Larger portion = 0.8413 (Positive Z-Score)
Smaller portion = 1 â&#x20AC;&#x201C; 0.8413 = 0.1587
Positive Z Score Z=1=0.8413 Z=0=0.5 So the area under the highlighted curve is 0.8413-0.5=0.3413 P(0 < Z < 1) = 0.3413
3. P(0 < Z < 1)
0.5
0.5
1
The Empirical Rule
Standard Normal Distribution: µ = 0 and = 1
The Empirical Rule
Standard Normal Distribution: Âľ = 0 and ď ł = 1
68% within 1 standard deviation
34%
x-s
34%
x
x+s
The Empirical Rule
Standard Normal Distribution: Âľ = 0 and ď ł = 1 95% within 2 standard deviations 68% within 1 standard deviation
34%
34%
13.5%
x - 2s
13.5%
x-s
x
x+s
x + 2s
The Empirical Rule
Standard Normal Distribution: Âľ = 0 and ď ł = 1 99.7% of data are within 3 standard deviations of the mean
95% within 2 standard deviations 68% within 1 standard deviation
34%
34%
2.4%
2.4%
0.1%
0.1% 13.5%
x - 3s
x - 2s
13.5%
x-s
x
x+s
x + 2s
x + 3s
Notation P(a < z < b) denotes the probability that the z score is between a and b.
P(z > a) denotes the probability that the z score is greater than a.
P(z < a) denotes the probability that the z score is less than a.
Finding z Scores When Given Probabilities 5% or 0.05
(z score will be positive)
Finding the 95th Percentile
5% or 0.05
Finding z Scores When Given Probabilities - cont 5% or 0.05
(z score will be positive)
1.645
Finding the 95th Percentile
Finding z Scores When Given Probabilities - cont
(One z score will be negative and the other positive) Finding the Bottom 2.5% and Upper 2.5%
Finding z Scores When Given Probabilities - cont
(One z score will be negative and the other positive) Finding the Bottom 2.5% and Upper 2.5%
Finding z Scores When Given Probabilities - cont
(One z score will be negative and the other positive) Finding the Bottom 2.5% and Upper 2.5%
Applications of Normal Distributions
Introduction This section presents methods for working with normal distributions that are not standard. That is, the mean is not 0 or the standard deviation is not 1, or both. The key concept is that we can use a simple conversion that allows us to standardize any normal distribution so that the same methods of the previous section can be used.
Conversion Formula
Formula 6-2
z=
x–µ
z= transformed value of X (Round z scores to 2 decimal places), X=defined value from the original data set µ = mean of the original data set
= standard deviation of the original data set
Converting to a Standard Normal Distribution x– z=
Example: Data transformation Given X ~ N (55, 25), transform each value of X in the distribution into Z
Based on given value where µ = 55 and σ = 5, draw the distribution Transform the 1st value of X (40) into Z:
Z =
40 55 5
= 3
Example: Calculate probabilities
Given mean and std. dev. Commitment score are 65 and 5, respectively, calculate the probability of scores: 1.
More than 72 P X 72 72 65 = P Z 5 = P Z 1.4 Refer Positive Z-Score = 1 0.9192 = 0.0808
2.
Less than 68 P X 68 68 65 = P Z 5 Refer Positive Z-Score = P Z 0.6 = 0.7257
3.
Continue….. Between 63 to 77
P 63 X 77 77 65 63 65 = P Z 5 5 = P 0.4 Z 2.4 = 0.9918 0.3446 = 0.6472 0.3446 Refer Positive Z-Score Refer Negative Z-Score
0.9918
Cautions to Keep in Mind 1. Donâ&#x20AC;&#x2122;t confuse z scores and areas. z scores are distances along the horizontal scale, but areas are regions under the normal curve. Table A-2 lists z scores in the left column and across the top row, but areas are found in the body of the table.
2. Choose the correct (right/left) side of the graph. 3. A z score must be negative whenever it is located in the left half of the normal distribution. 4. Areas (or probabilities) are positive or zero values, but they are never negative.
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