Upload
ashley-muriel-lindsey
View
233
Download
0
Tags:
Embed Size (px)
Citation preview
Some probability distributionSome probability distributionThe Normal DistributionThe Normal Distribution
هـ9/4/1435 Noha Hussein Elkhidir
ObjectivesObjectivesIntroduce the Normal Distribution
Properties of the Standard Normal Distribution
Introduce the Central Limit Theorem
هـ9/4/1435 Noha Hussein Elkhidir
Normal DistributionNormal DistributionWhy are normal distributions so important?
Many dependent variables are commonly assumed to be normally distributed in the population
If a variable is approximately normally distributed we can make inferences about values of that variable
Example: Sampling distribution of the mean
هـ9/4/1435 Noha Hussein Elkhidir
Normal DistributionNormal Distribution
Symmetrical, bell-shaped curveAlso known as Gaussian
distributionPoint of inflection = 1 standard
deviation from meanMathematical formula
هـ9/4/1435 Noha Hussein Elkhidir
f (X ) 1
2(e)
(X )2
2 2
Since we know the shape of the curve, we can calculate the area under the curve
The percentage of that area can be used to determine the probability that a given value could be pulled from a given distribution
◦The area under the curve tells us about the probability- in other words we can obtain a p-value for our result (data) by treating it as a normally distributed data set.
هـ9/4/1435 Noha Hussein Elkhidir
Key Areas under the CurveKey Areas under the Curve
For normal distributions
+ 1 SD ~ 68%+ 2 SD ~ 95%
+ 3 SD ~ 99.9%
هـ9/4/1435 Noha Hussein Elkhidir
Example IQ mean = 100 s Example IQ mean = 100 s = 15= 15
هـ9/4/1435 Noha Hussein Elkhidir
Problem :◦Each normal distribution with its own
values of and would need its own calculation of the area under various
points on the curve
هـ9/4/1435 Noha Hussein Elkhidir
Normal Probability Normal Probability DistributionsDistributions
Standard Normal Distribution – N(0,1)Standard Normal Distribution – N(0,1)We agree to use the
standard normal distribution
Bell shaped=0=1
Note: not all bell shaped distributions
are normal distributions
هـ9/4/1435 Noha Hussein Elkhidir
Normal Probability Normal Probability DistributionDistribution
Can take on an infinite number of
possible values.The probability of
any one of those values occurring is
essentially zero.Curve has area or
probability = 1
هـ9/4/1435 Noha Hussein Elkhidir
Normal DistributionNormal DistributionThe standard normal distribution
will allow us to make claims about the probabilities of values
related to our own dataHow do we apply the standard
normal distribution to our data?
هـ9/4/1435 Noha Hussein Elkhidir
Z-scoreZ-score
If we know the population mean and population standard
deviation, for any value of X we can compute a z-score by
subtracting the population mean and dividing the result by the population standard deviation
هـ9/4/1435 Noha Hussein Elkhidir
zX
Important z-score infoImportant z-score infoZ-score tells us how far above or below
the mean a value is in terms of standard deviations
It is a linear transformation of the original scores
◦Multiplication (or division) of and/or addition to (or subtraction from) X by a constant
◦Relationship of the observations to each other remains the same
Z = (X-)/then
X = Z + [equation of the general form Y = mX+c]
هـ9/4/1435 Noha Hussein Elkhidir
Probabilities and z scores: z Probabilities and z scores: z tablestables
Total area = 1Only have a probability from width
◦For an infinite number of z scores each point has a probability of 0 (for the single
point)Typically negative values are not
reported◦Symmetrical, therefore area below
negative value = Area above its positive value
Always helps to draw a sketch!هـ9/4/1435 Noha Hussein Elkhidir
Probabilities are depicted by areas under the Probabilities are depicted by areas under the curvecurve
Total area under the curve is 1
The area in red is equal to p(z > 1)
The area in blue is equal to p(-1< z <0)
Since the properties of the normal distribution are known, areas can
be looked up on tables or calculated on
computer.
هـ9/4/1435 Noha Hussein Elkhidir
Strategies for finding probabilities Strategies for finding probabilities for the standard normal random for the standard normal random
variablevariable..
Draw a picture of standard normal distribution depicting the area of
interest.Re-express the area in terms of shapes
like the one on top of the Standard Normal Table
Look up the areas using the table.Do the necessary addition and
subtraction.
هـ9/4/1435 Noha Hussein Elkhidir
Suppose Z has standard normal Suppose Z has standard normal distribution Find p(0<Z<1.23)distribution Find p(0<Z<1.23)
هـ9/4/1435 Noha Hussein Elkhidir
Find p(-1.57<Z<0)Find p(-1.57<Z<0)
هـ9/4/1435 Noha Hussein Elkhidir
Find p(Z>.78)Find p(Z>.78)
هـ9/4/1435 Noha Hussein Elkhidir
Z is standard normalZ is standard normalCalculate p(-1.2<Z<.78)Calculate p(-1.2<Z<.78)
هـ9/4/1435 Noha Hussein Elkhidir
Table I: P(0<Z<z)Table I: P(0<Z<z)
z .00 .01 .02 .03 .04 .05 .06 0.0. 0000. 0040. 0080. 0120. 0160. 0199. 02390.1. 0398. 0438. 0478. 0517. 0557. 0596. 0636 0.2. 0793. 0832. 0871. 0910. 0948. 0987. 10260.3. 1179. 1217. 1255. 1293. 1331. 1368. 1404 0.4. 1554. 1591. 1628. 1664. 1700. 1736. 1772 0.5. 1915. 1950. 1985. 2019. 2054. 2088. 2123
… … … … … … … …1.0. 3413. 3438. 3461. 3485. 3508. 3531. 3554 1.1. 3643. 3665. 3686. 3708. 3729. 3749. 3770
ExamplesExamples
P(0<Z<1) =0.3413
Example P(1<Z<2)
=P(0<Z<2)–P(0<Z<1)=0.4772–0.3413
=0.1359
ExamplesExamples
P(Z≥1)
= 0.5–P(0<Z<1)= 0.5–0.3413
= 0.1587
ExamplesExamples
P(Z ≥ -1)=0.3413+0.50
=0.8413
ExamplesExamples
P(-2<Z<1)=0.4772+0.3413
=0.8185
ExamplesExamples
P(Z ≤ 1.87)=0.5+P(0<Z ≤ 1.87)
=0.5+0.4693=0.9693
ExamplesExamples
P(Z<-1.87) =P(Z>1.87) =0.5–0.4693
=0.0307
ExampleExampleData come from distribution: =
0, = 3What proportion fall beyond
X=13?Z = (13-10)/3 = 1
=normsdist(1) or table 0.158715.9% fall above 13
هـ9/4/1435 Noha Hussein Elkhidir
Example dataExample data::Mean of data is 100
Standard deviation of data is 15
هـ9/4/1435 Noha Hussein Elkhidir
The data are normally distributed with mean 100 The data are normally distributed with mean 100 and standard deviation 15. Find the probability and standard deviation 15. Find the probability that a randomly selected data between 100 and that a randomly selected data between 100 and 115115
هـ9/4/1435 Noha Hussein Elkhidir
(100 115)
(100 100 100 115 100)
100 100 100 115 100(
15 15 15(0 1) .3413
P X
P X
XP
P Z
Say we have GRE scores are normally distributed Say we have GRE scores are normally distributed with mean 500 and standard deviation 100. Find with mean 500 and standard deviation 100. Find the probability that a randomly selected GRE score the probability that a randomly selected GRE score is greater than 620is greater than 620..
We want to know what’s the probability of getting a score 620 or beyond.
p(z > 1.2)Result: The probability of randomly
getting a score of 620 is ~.12
هـ9/4/1435 Noha Hussein Elkhidir
620 5001.2
100z
homeworkhomework : :What is the area for scores less than z =
-1.5?What is the area between z =1 and 1.5?What z score cuts off the highest 30% of
the distribution?What two z scores enclose the middle
50% of the distribution?If 500 scores are normally distributed
with mean = 50 and SD = 10, and an investigator throws out the 20 most extreme scores, what are the highest and lowest scores that are retained?
هـ9/4/1435 Noha Hussein Elkhidir