Analysis of Variance (ANOVA) Quantitative Methods in HPELS HPELS 6210.
Probability Quantitative Methods in HPELS HPELS 6210.
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Transcript of Probability Quantitative Methods in HPELS HPELS 6210.
Probability
Quantitative Methods in HPELS
HPELS 6210
Agenda
Introduction Probability and the Normal Distribution Probability and the Binomial Distribution Inferential Statistics
Introduction
Recall: Inferential statistics: Sample statistic
PROBABILITY population parameter Marbles Example
Assume:
N = 100 marbles
50 black, 50 white
What is the probability of drawing a black marble?
Assume:
N = 100 marbles
90 black, 10 white
What is the probability of drawing a black marble?
Introduction
Using information about a population to predict the sample is the opposite of INFERENTIAL statistics
Consider the following examples
While blindfolded, you choose n=4 marbles from
one of the two jars
Which jar did you PROBABLY choose your
sample?
Introduction What is probability?
The chance of any particular outcome occurring as a fraction/proportion of all possible outcomes
Example: If a hat is filled with four pieces of paper
lettered A, B, C and D, what is the probability of pulling the letter A?
p = # of “A” outcomes / # of total outcomes p = 1 / 4 = 0.25 or 25%
Introduction
This definition of probability assumes that the samples are obtained RANDOMLY
A random sample has two requirements:1. Each outcome has equal chance of being
selected2. Probability is constant (selection with
replacement)
What is probability of drawing Jack of Diamonds from 52 card deck? Ace of spades?
What is probability of drawing Jack of Spades if you do not replace the first selection?
Agenda
Introduction Probability and the Normal Distribution Probability and the Binomial Distribution Inferential Statistics
Probability Normal Distribution Recall Normal Distribution:
SymmetricalUnified mean, median and mode
Normal distribution can be defined:Mathematically (Figure 6.3, p 168)Standard deviations (Figure 6.4, 168)
With either definition, the predictability of the Normal Distribution allows you to answer PROBABILITY QUESTIONS
Probability Questions
Example 6.2 Assume the following about adult height:
µ = 68 inches = 6 inches
Probability Question:What is the probability of selecting an adult
with a height greater than 80 inches?p (X > 80) = ?
Probability Questions
Example 6.2: Process:
1. Draw a sketch:
2. Compute Z-score:
3. Use normal distribution to determine probability
Step 1: Draw a sketch for p(X>80)
Step 2: Compute Z-score:
Z = X - µ /
Z = 80 – 68/6
Z = 12/6 = 2.00
Step 3: Determine probability
There is a 2.28% probability that you would select a person with a height greater than 80 inches.
Probability Questions
What if Z-score is not 0.0, 1.0 or 2.0? Normal Table Figure 6.6, p 170
Column A: Z-score Column C: Tail = smaller side
Column B: Body = larger side Column D: 0.50 – p(Z)
Using the Normal Table Several applications:
1. Determining a probability from a specific Z-score
2. Determining a Z-score from a specific probability or probabilities
3. Determining a probability between two Z-scores
4. Determining a raw score from a specific probability or Z-score
Determining a probability from a specific Z-score
Process:1. Draw a sketch
2. Locate the probability from normal table Examples: Figure 6.7, p 171
p(X > 1.00) = ?
Tail or Body?
p = 15.87%
p(X < 1.50) = ?
Tail or Body?
p = 93.32%
p(X < -0.50) = ?
p(X > 0.50) = ?
Tail or Body?
p = 30.85%
Using the Normal Table Several applications:
1. Determining a probability from a specific Z-score
2. Determining a Z-score from a specific probability or probabilities
3. Determining a probability between two Z-scores
4. Determining a raw score from a specific probability or Z-score
Determining a Z-score from a specific probability
Process:1. Draw a sketch
2. Locate Z-score from normal table Examples: Figure 6.8a and b, p 173
What Z-score is associated with a raw score that has 90% of the population below and 10% above?
Column B (body) p = 0.900
Z = 1.28
Column C (tail) p = 0.100
Z = 1.28
What two Z-scores are associated with raw scores that have 60% of the population located between them and 40% located on the ends?
Column C (tail) p = 0.200
Z = 0.84 and -0.84
Column D (0.500 – p(Z)) 0.300
Z = 0.84 and – 0.84
20%
(0.200)
20%
(0.200)
30%
(0.300)
30%
(0.300)
Using the Normal Table Several applications:
1. Determining a probability from a specific Z-score
2. Determining a Z-score from a specific probability or probabilities
3. Determining a probability between two Z-scores
4. Determining a raw score from a specific probability or Z-score
Determining a probability between two Z-scores
Process:1. Draw a sketch
2. Calculate Z-scores
3. Locate probabilities normal table
4. Calculate probability that falls between Z-scores
Example: Figure 6.10, p 176 What proportion of people drive between the
speeds of 55 and 65 mph?
Step 1: Sketch
Step 2: Calculate Z-scores:
Z = X - µ / Z = X - µ /
Z = 55 – 58/10 Z = 65 – 58/10
Z = -0.30 Z = 0.70
Step 2: Locate probabilities
Z = -0.30 (column D) = 0.1179
Z = 0.70 (column D) = 0.2580
Step 4: Calculate probabilities between Z-scores
p = 0.1179 + 0.2580 = 0.3759
Using the Normal Table Several applications:
1. Determining a probability from a specific Z-score
2. Determining a Z-score from a specific probability or probabilities
3. Determining a probability between two Z-scores
4. Determining a raw score from a specific probability or Z-score
Determining a raw score from a specific probability or Z-score
Process:1. Draw sketch
2. Locate Z-score from normal table
3. Calculate raw score from Z-score equation Example: Figure 6.13, p 178
What SAT score is needed to score in the top 15%?
Step 1: Sketch
Step 2: Locate Z-score
p = 0.150 (column D)
Z = 1.04
Step 3: Calculate raw score from Z-score equation
Z = X - µ / X = µ + Z
X = 500 + 1.04(100)
X = 604
Agenda
Introduction Probability and the Normal Distribution Probability and the Binomial Distribution Inferential Statistics
Probability Binomial Distribution
Binomial distribution?Literally means “two names”Variable measured with scale consisting of:
Two categories or Two possible outcomes
Examples:Coin flipGender
Probability Questions Binomial Distribution
Binomial distribution is predictable Probability questions are possible Statistical notation:
A and B: Denote the two categories/outcomesp = p(A) = probability of A occurringq = p(B) = probability of B occurring
Example 6.13, p 185
Heads
Tails
p = p(A) = ½ = 0.50
q = p(B) = ½ = 0.50
If you flipped the coin twice (n=2), how many combinations are possible?
Heads Heads
Heads Tails
HeadsTails
Tails Tails
Each outcome has an equal chance of occurring ¼ = 0.25
What is the probability of obtaining at least one head in 2 coin tosses?
Figure 6.19, p 186
Normal Approximation Binomial Distribution
Binomial distribution tends to be NORMAL when “pn” and “qn” are large (>10)
Parameters of a normal binomial distribution:Mean: µ = pn SD: = √npq
Therefore:Z = X – pn / √npq
To maximize accuracy, use REAL LIMITS Recall:
Upper and lower Examples: Figure 6.21, p 188
Normal Approximation Binomial Distribution
Note: The binomial distribution is a histogram, with each bar extending to its real limits
Note: The binomial distribution approximates a normal distribution under certain conditions
Normal Approximation Binomial Distribution
Example: 6.22, p 189 Assume:
Population: Psychology Department Males (A) = ¼ of population Females (B) = ¾ of population
What is the probability of selecting 14 males in a sample (n=48)? p(A=14) p(13.5<A<14.5) = ?
Process:1. Draw a sketch
2. Confirm normality of binomial distribution
3. Calculate population µ and : µ = pn = √npq
4. Calculate Z-scores for upper and lower real limits
5. Locate probabilities in normal table
6. Calculate probability between real limits
Normal Approximation Binomial Distribution
Step 1: Draw a sketch
Step 3: Calculate µ and
µ = pn = √npq
µ = 0.25(48) = √48*0.25*0.75
µ = 12 = 3
Step 2: Confirm normality
pn = 0.25(48) = 12 > 10
qn = 0.75(48) = 36 > 12
Step 4: Calculate real limit Z-scores
Z = X–pn/√npq Z = X-pn/√npq
Z = 13.5-12/3 Z = 14.5-12/3
Z = 0.50 Z = 0.83
Step 5: Locate probabilities
Z = 0.50 (column C) = 0.3085
Z = 0.83 (column C) = 0.2033
Z = 0.50 (column C) = 0.3085
Z = 0.83 (column C) = 0.2033
Step 6: Calculate probability between the real limits
p = 0.3085 – 0.2033
p = 0.1052
There is a 10.52% probability of selecting 14 males from a sample of n=48 from this population
Example extended What is the probability of selecting more than
14 males in a sample (n=48)? p(A>14) p(A>14.5) = ?
Process:1. Draw a sketch
2. Calculate Z-score for upper real limit
3. Locate probability in normal table
Normal Approximation Binomial Distribution
Step 1: Draw a sketch
Step 2: Calculate Z-score of upper real limit
Z = X–pn/√npq
Z = 14.5 – 12 / 3
Z = 0.83
Step 3: Locate probability
Z = 0.83 (column C) = 0.2033
There is a 20.33% probability of selecting more than 14 males in a sample of n=48 from this population
Agenda
Introduction Probability and the Normal Distribution Probability and the Binomial Distribution Inferential Statistics
Looking Ahead Inferential Statistics
PROBABILITY links the sample to the population Figure 6.24, p 191
Textbook Assignment
Problems: 2, 6, 8, 12, 16, 18