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Probability. Quantitative Methods in HPELS 440:210. Agenda. Introduction Probability and the Normal Distribution Probability and the Binomial Distribution Inferential Statistics. Introduction. Recall: Inferential statistics: Sample statistic  PROBABILITY  population parameter

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Probability

Probability

Quantitative Methods in HPELS

440:210


Agenda

Agenda

  • Introduction

  • Probability and the Normal Distribution

  • Probability and the Binomial Distribution

  • Inferential Statistics


Introduction

Introduction

  • Recall:

    • Inferential statistics: Sample statistic  PROBABILITY  population parameter

  • Marbles Example


Probability

Assume:

N = 100 marbles

50 black, 50 white

Assume:

N = 100 marbles

90 black, 10 white

What is the probability of drawing a black marble?

What is the probability of drawing a black marble?


Introduction1

Introduction

  • Using information about a population to predict the sample is the opposite of INFERENTIAL statistics

  • Consider the following examples


Probability

While blindfolded, you choose n=4 marbles from one of the two jars

Which jar did you PROBABLY choose your sample?


Introduction2

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%


Introduction3

Introduction

  • This definition of probability assumes that the samples are obtained RANDOMLY

  • A random sample has two requirements:

    • Each outcome has equal chance of being selected

    • Probability is constant (selection with replacement)


Probability

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?


Agenda1

Agenda

  • Introduction

  • Probability and the Normal Distribution

  • Probability and the Binomial Distribution

  • Inferential Statistics


Probability normal distribution

Probability  Normal Distribution

  • Recall  Normal Distribution:

    • Symmetrical

    • Unified mean, median and mode

  • Normal distribution can be defined:

    • Mathematically (Figure 6.3, p 168)

    • Standard deviations (Figure 6.4, 168)


Probability

With either definition, the predictability of the Normal Distribution allows you to answer PROBABILITY QUESTIONS


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 questions1

Probability Questions

  • Example 6.2:

  • Process:

    • Draw a sketch:

    • Compute Z-score:

    • Use normal distribution to determine probability


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 questions2

Probability Questions

  • What if Z-score is not 0.0, 1.0 or 2.0?

  • Normal Table  Figure 6.6, p 170


Probability

Column A: Z-scoreColumn C: Tail = smaller side

Column B: Body = larger side Column D: 0.50 – p(Z)


Using the normal table

Using the Normal Table

  • Several applications:

    • Determining a probability from a specific Z-score

    • Determining a Z-score from a specific probability or probabilities

    • Determining a probability between two Z-scores

    • Determining a raw score from a specific probability or Z-score


Determining a probability from a specific z score

Determining a probability from a specific Z-score

  • Process:

    • Draw a sketch

    • Locate the probability from normal table

  • Examples: Figure 6.7, p 171


Probability

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 table1

Using the Normal Table

  • Several applications:

    • Determining a probability from a specific Z-score

    • Determining a Z-score from a specific probability or probabilities

    • Determining a probability between two Z-scores

    • Determining a raw score from a specific probability or Z-score


Determining a z score from a specific probability

Process:

Draw a sketch

Locate Z-score from normal table

Examples: Figure 6.8a and b, p 173

Determining a Z-score from a specific probability


Probability

20%

(0.200)

20%

(0.200)

What Z-score is associated with a raw score that has 90% of the population below and 10% above?

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 B (body)  p = 0.900

Z = 1.28

Column C (tail)  p = 0.100

Z = 1.28

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

30%

(0.300)

30%

(0.300)


Using the normal table2

Using the Normal Table

  • Several applications:

    • Determining a probability from a specific Z-score

    • Determining a Z-score from a specific probability or probabilities

    • Determining a probability between two Z-scores

    • Determining a raw score from a specific probability or Z-score


Determining a probability between two z scores

Determining a probability between two Z-scores

  • Process:

    • Draw a sketch

    • Calculate Z-scores

    • Locate probabilities normal table

    • 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?


Probability

Step 1: Sketch

Step 2: Calculate Z-scores:

Z = X - µ / Z = X - µ / 

Z = 55 – 58/10Z = 65 – 58/10

Z = -0.30Z = 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 table3

Using the Normal Table

  • Several applications:

    • Determining a probability from a specific Z-score

    • Determining a Z-score from a specific probability or probabilities

    • Determining a probability between two Z-scores

    • Determining a raw score from a specific probability or Z-score


Determining a raw score from a specific probability or z score

Determining a raw score from a specific probability or Z-score

  • Process:

    • Draw sketch

    • Locate Z-score from normal table

    • Calculate raw score from Z-score equation

  • Example: Figure 6.13, p 178

    • What SAT score is needed to score in the top 15%?


Probability

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


Agenda2

Agenda

  • Introduction

  • Probability and the Normal Distribution

  • Probability and the Binomial Distribution

  • Inferential Statistics


Probability binomial distribution

Probability  Binomial Distribution

  • Binomial distribution?

    • Literally means “two names”

    • Variable measured with scale consisting of:

      • Two categories or

      • Two possible outcomes

  • Examples:

    • Coin flip

    • Gender


Probability questions binomial distribution

Probability Questions  Binomial Distribution

  • Binomial distribution is predictable

  • Probability questions are possible

  • Statistical notation:

    • A and B: Denote the two categories/outcomes

    • p = p(A) = probability of A occurring

    • q = p(B) = probability of B occurring

  • Example 6.13, p 185


Probability

If you flipped the coin twice (n=2), how many combinations are possible?

Heads

p = p(A) = ½ = 0.50

Heads

Heads

Tails

q = p(B) = ½ = 0.50

Heads

Tails

Each outcome has an equal chance of occurring  ¼ = 0.25

Tails

Heads

What is the probability of obtaining at least one head in 2 coin tosses?

Figure 6.19, p 186

Tails

Tails


Normal approximation binomial distribution

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


Normal approximation binomial distribution1

Normal Approximation  Binomial Distribution

  • To maximize accuracy, use REAL LIMITS

  • Recall:

    • Upper and lower

    • Examples: Figure 6.21, p 188


Probability

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 distribution2

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) = ?


Normal approximation binomial distribution3

Normal Approximation  Binomial Distribution

  • Process:

    • Draw a sketch

    • Confirm normality of binomial distribution

    • Calculate population µ and :

      • µ = pn

      •  = √npq

    • Calculate Z-scores for upper and lower real limits

    • Locate probabilities in normal table

    • Calculate probability between real limits


Probability

Step 1: Draw a sketch

Step 2: Confirm normality

pn = 0.25(48) = 12 > 10

qn = 0.75(48) = 36 > 12

Step 3: Calculate µ and 

µ = pn  = √npq

µ = 0.25(48) = √48*0.25*0.75

µ = 12  = 3

Step 5: Locate probabilities

Z = 0.50 (column C) = 0.3085

Z = 0.83 (column C) = 0.2033

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


Probability

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


Normal approximation binomial distribution4

Normal Approximation  Binomial Distribution

  • 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:

    • Draw a sketch

    • Calculate Z-score for upper real limit

    • Locate probability in normal table


Probability

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

There is a 20.33% probability of selecting more than 14 males in a sample of n=48 from this population

Step 3: Locate probability

Z = 0.83 (column C) = 0.2033


Agenda3

Agenda

  • Introduction

  • Probability and the Normal Distribution

  • Probability and the Binomial Distribution

  • Inferential Statistics


Looking ahead inferential statistics

Looking Ahead  Inferential Statistics

  • PROBABILITY links the sample to the population  Figure 6.24, p 191


Textbook assignment

Textbook Assignment

  • Problems: 1, 3, 6, 8, 12, 15, 17, 27


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