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Probability Quantitative Methods in HPELS HPELS 6210

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

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Introduction Recall: Inferential statistics: Sample statistic PROBABILITY population parameter Marbles Example

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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?

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Introduction Using information about a population to predict the sample is the opposite of INFERENTIAL statistics Consider the following examples

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While blindfolded, you choose n=4 marbles from one of the two jars Which jar did you PROBABLY choose your sample?

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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%

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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 selected 2. Probability is constant (selection with replacement)

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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?

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

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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)

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

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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) = ?

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Probability Questions Example 6.2: Process: 1. Draw a sketch: 2. Compute Z-score: 3. Use normal distribution to determine probability

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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.

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

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Column A: Z-scoreColumn C: Tail = smaller side Column B: Body = larger side Column D: 0.50 – p(Z)

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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

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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

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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%

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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

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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

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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)

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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

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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?

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Step 1: Sketch Step 2: Calculate Z-scores: 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

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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

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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%?

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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

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

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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

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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

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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 Tails HeadsTails 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

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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

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To maximize accuracy, use REAL LIMITS Recall: Upper and lower Examples: Figure 6.21, p 188 Normal Approximation Binomial Distribution

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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

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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) = ?

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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

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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

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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

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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

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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

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

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Looking Ahead Inferential Statistics PROBABILITY links the sample to the population Figure 6.24, p 191

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Textbook Assignment Problems: 2, 6, 8, 12, 16, 18

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