Probability Quantitative Methods in HPELS 6210 Agenda Introduction
Probability Quantitative Methods in HPELS 6210
Agenda Introduction n Probability and the Normal Distribution n Probability and the Binomial Distribution n Inferential Statistics n
Introduction n Recall: ¨ Inferential statistics: Sample statistic PROBABILITY population parameter n Marbles Example
Assume: N = 100 marbles 50 black, 50 white 90 black, 10 white What is the probability of drawing a black marble?
Introduction n n 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 n What is probability? ¨ n 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 n n 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 2. Probability is constant (selection with replacement) 1.
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 n Probability and the Normal Distribution n Probability and the Binomial Distribution n Inferential Statistics n
Probability Normal Distribution n Recall Normal Distribution: ¨ Symmetrical ¨ Unified n 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 n Assume the following about adult height: n ¨µ = 68 inches ¨ = 6 inches n Probability Question: ¨ What is the probability of selecting an adult with a height greater than 80 inches? ¨ p (X > 80) = ?
Probability Questions n n Example 6. 2: Process: Draw a sketch: 2. Compute Z-score: 3. Use normal distribution to determine probability 1.
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? n Normal Table Figure 6. 6, p 170 n
Column A: Z-score Column C: Tail = smaller side Column B: Body = larger side Column D: 0. 50 – p(Z)
Using the Normal Table n Several applications: Determining a probability from a specific Zscore 2. Determining a Z-score from a specific probability or probabilities 3. Determining a probability between two Zscores 4. Determining a raw score from a specific probability or Z-score 1.
Determining a probability from a specific Z-score n Process: Draw a sketch 2. Locate the probability from normal table 1. n Examples: Figure 6. 7, p 171
p(X > 1. 00) = ? p(X < 1. 50) = ? p(X < -0. 50) = ? Tail or Body? p(X > 0. 50) = ? p = 15. 87% p = 93. 32% Tail or Body? p = 30. 85%
Using the Normal Table n Several applications: Determining a probability from a specific Zscore 2. Determining a Z-score from a specific probability or probabilities 3. Determining a probability between two Zscores 4. Determining a raw score from a specific probability or Z-score 1.
Determining a Z-score from a specific probability n Process: Draw a sketch 2. Locate Z-score from normal table 1. n Examples: Figure 6. 8 a and b, p 173
What Z-score is associated with a raw score that has 90% of the population below and 10% above? 20% (0. 200) 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 Column C (tail) p = 0. 200 Z = 1. 28 Z = 0. 84 and -0. 84 Column C (tail) p = 0. 100 Column D (0. 500 – p(Z)) 0. 300 Z = 1. 28 Z = 0. 84 and – 0. 84 30% (0. 300)
Using the Normal Table n Several applications: Determining a probability from a specific Zscore 2. Determining a Z-score from a specific probability or probabilities 3. Determining a probability between two Zscores 4. Determining a raw score from a specific probability or Z-score 1.
Determining a probability between two Z-scores n Process: Draw a sketch 2. Calculate Z-scores 3. Locate probabilities normal table 4. Calculate probability that falls between Zscores 1. n 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: Step 2: Locate probabilities Z=X-µ/ Z = -0. 30 (column D) = 0. 1179 Z = 55 – 58/10 Z = 65 – 58/10 Z = 0. 70 (column D) = 0. 2580 Z = -0. 30 Z = 0. 70 Step 4: Calculate probabilities between Z-scores p = 0. 1179 + 0. 2580 = 0. 3759
Using the Normal Table n Several applications: Determining a probability from a specific Zscore 2. Determining a Z-score from a specific probability or probabilities 3. Determining a probability between two Zscores 4. Determining a raw score from a specific probability or Z-score 1.
Determining a raw score from a specific probability or Z-score n Process: Draw sketch 2. Locate Z-score from normal table 3. Calculate raw score from Z-score equation 1. n 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 Step 3: Calculate raw score from Z-score equation p = 0. 150 (column D) Z=X-µ/ X=µ+Z Z = 1. 04 X = 500 + 1. 04(100) X = 604
Agenda Introduction n Probability and the Normal Distribution n Probability and the Binomial Distribution n Inferential Statistics n
Probability Binomial Distribution n Binomial distribution? ¨ Literally means “two names” ¨ Variable measured with scale consisting of: Two categories or n Two possible outcomes n n Examples: ¨ Coin flip ¨ Gender
Probability Questions Binomial Distribution Binomial distribution is predictable n Probability questions are possible n Statistical notation: n ¨A and B: Denote the two categories/outcomes ¨ p = p(A) = probability of A occurring ¨ q = p(B) = probability of B occurring n Example 6. 13, p 185
Heads Tails p = p(A) = ½ = 0. 50 If you flipped the coin twice (n=2), how many combinations are possible? Heads Tails q = p(B) = ½ = 0. 50 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) n Parameters of a normal binomial distribution: n ¨ Mean: µ = pn ¨ SD: = √npq n Therefore: ¨Z = X – pn / √npq
Normal Approximation Binomial Distribution To maximize accuracy, use REAL LIMITS n Recall: n ¨ 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
Normal Approximation Binomial Distribution n n Example: 6. 22, p 189 Assume: ¨ Population: Psychology Department ¨ Males (A) = ¼ of population ¨ Females (B) = ¾ of population n 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 Distribution n Process: Draw a sketch 2. Confirm normality of binomial distribution 3. Calculate population µ and : 1. § § µ = pn = √npq Calculate Z-scores for upper and lower real limits 5. Locate probabilities in normal table 6. Calculate probability between real limits 4.
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 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
Normal Approximation Binomial Distribution n n Example extended What is the probability of selecting more than 14 males in a sample (n=48)? ¨ n p(A>14) p(A>14. 5) = ? Process: 1. 2. 3. 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 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 n Probability and the Normal Distribution n Probability and the Binomial Distribution n Inferential Statistics n
Looking Ahead Inferential Statistics n PROBABILITY links the sample to the population Figure 6. 24, p 191
Textbook Assignment n Problems: 2, 6, 8, 12, 16, 18
- Slides: 48