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Probability

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Probability

Quantitative Methods in HPELS

440:210

- Introduction
- Probability and the Normal Distribution
- Probability and the Binomial Distribution
- Inferential Statistics

- Recall:
- Inferential statistics: Sample statistic PROBABILITY population parameter

- Marbles Example

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?

- 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?

- 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%

- 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)

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?

- Introduction
- Probability and the Normal Distribution
- Probability and the Binomial Distribution
- Inferential Statistics

- 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)

With either definition, the predictability of the Normal Distribution allows you to answer 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) = ?

- Example 6.2:
- Process:
- Draw a sketch:
- Compute Z-score:
- 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.

- What if Z-score is not 0.0, 1.0 or 2.0?
- Normal Table Figure 6.6, p 170

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

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

- 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

- Process:
- Draw a sketch
- 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%

- 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

Process:

Draw a sketch

Locate Z-score from normal table

Examples: Figure 6.8a and b, p 173

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)

- 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

- 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?

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

- 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

- 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%?

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

- Introduction
- Probability and the Normal Distribution
- Probability and the Binomial Distribution
- Inferential Statistics

- Binomial distribution?
- Literally means “two names”
- Variable measured with scale consisting of:
- Two categories or
- Two possible outcomes

- Examples:
- Coin flip
- Gender

- 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

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

- 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

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

- 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:
- 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

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

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:
- Draw a sketch
- Calculate Z-score for upper real limit
- Locate probability in normal table

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

- Introduction
- Probability and the Normal Distribution
- Probability and the Binomial Distribution
- Inferential Statistics

- PROBABILITY links the sample to the population Figure 6.24, p 191

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