Extra Brownie Points Lottery To Win choose the
Extra Brownie Points! • Lottery • To Win: • choose the 5 winnings numbers – from 1 to 49 • AND • Choose the "Powerball" number – from 1 to 42 • What is the probability you will win? – Use combinations to answer this question
• p of winning jackpot • Total number of ways to win / total number of possible outcomes
Total Number of Outcomes Different 5 number combinations Different Powerball outcomes Thus, there are 1, 906, 884 * 42 = 80, 089, 128 ways the drawing can occur
Total Number of Ways to Win Only one way to win –
Remember • Playing perfect black jack – the probability of winning a hand is. 498 • What is the probability that you will win 8 of the next 10 games of blackjack?
Binomial Distribution Ingredients: N = total number of events p = the probability of a success on any one trial q = (1 – p) = the probability of a failure on any one trial X = number of successful events
Binomial Distribution Ingredients: N = total number of events p = the probability of a success on any one trial q = (1 – p) = the probability of a failure on any one trial X = number of successful events
Binomial Distribution p =. 0429 Ingredients: N = total number of events p = the probability of a success on any one trial q = (1 – p) = the probability of a failure on any one trial X = number of successful events
Binomial Distribution • What if you are interested in the probability of winning at least 8 games of black jack? • To do this you need to know the distribution of these probabilities
Probability of Winning Blackjack Number of Wins • p =. 498, N = 10 0 1 2 3 4 5 6 7 8 9 10 p
Probability of Winning Blackjack • p =. 498, N = 10 Number of Wins p 0 . 001 1 2 3 4 5 6 7 8 9 10
Probability of Winning Blackjack • p =. 498, N = 10 Number of Wins p 0 . 001 1 . 010 2 . 045 3 . 119 4 . 207 5 . 246 6 . 203 7 . 115 8 . 044 9 . 009 10 . 001
Probability of Winning Blackjack • p =. 498, N = 10 Number of Wins p 0 . 001 1 . 010 2 . 045 3 . 119 4 . 207 5 . 246 6 . 203 7 . 115 8 . 044 9 . 009 10 . 001 1. 00
Binomial Distribution p Games Won
Hypothesis Testing • You wonder if winning at least 7 games of blackjack is significantly (. 05) better than what would be expected due to chance. • H 1= Games won > 6 • H 0= Games won < or equal to 6 • What is the probability of winning 7 or more games?
Binomial Distribution p Games Won
Binomial Distribution p Games Won
Probability of Winning Blackjack • p =. 498, N = 10 Number of Wins p 0 . 001 1 . 010 2 . 045 3 . 119 4 . 207 5 . 246 6 . 203 7 . 115 8 . 044 9 . 009 10 . 001 1. 00
Probability of Winning Blackjack • p =. 498, N = 10 • p of winning 7 or more games • . 115+. 044+. 009+. 001 =. 169 • p >. 05 • Not better than chance Number of Wins p 0 . 001 1 . 010 2 . 045 3 . 119 4 . 207 5 . 246 6 . 203 7 . 115 8 . 044 9 . 009 10 . 001 1. 00
Practice • The probability at winning the “Statistical Slot Machine” is. 08. • Create a distribution of probabilities when N = 10 • Determine if winning at least 4 games of slots is significantly (. 05) better than what would be expected due to chance.
Probability of Winning Slot Number of Wins p 0 . 434 1 . 378 2 . 148 3 . 034 4 . 005 5 . 001 6 . 000 7 . 000 8 . 000 9 . 000 10 . 000 1. 00
Binomial Distribution p Games Won
Probability of Winning Slot Number of Wins p 0 . 434 1 . 378 2 . 148 3 . 034 4 . 005 5 . 001 6 . 000 7 . 000 8 . 000 9 . 000 10 . 000 • p of winning at least 4 games • . 005+. 001+. 000 =. 006 • p<. 05 • Winning at least 4 games is significantly better than chance 1. 00
Binomial Distribution • These distributions can be described with means and SD. • Mean = Np • SD =
Binomial Distribution • Black Jack; p =. 498, N =10 • M = 4. 98 • SD = 1. 59
Binomial Distribution p Games Won
Binomial Distribution • Statistical Slot Machine; p =. 08, N = 10 • M =. 8 • SD =. 86
Binomial Distribution Note: as N gets bigger, distributions will approach normal p Games Won
Next Step • You think someone is cheating at BLINGOO! • p =. 30 of winning • You watch a person play 89 games of blingoo and wins 39 times (i. e. , 44%). • Is this significantly bigger than. 30 to assume that he is cheating?
Hypothesis • H 1=. 44 >. 30 • H 0=. 44 < or equal to. 30 • Or • H 1= 39 wins > 26. 7 wins • H 0= 39 wins < or equal to 26. 7 wins
Distribution • Mean = 26. 7 • SD = 4. 32 • X = 39
Z-score
Results • (39 – 26. 7) / 4. 32 = 2. 85 • p =. 0021 • p <. 05 • . 44 is significantly bigger than. 30. There is reason to believe the person is cheating! • Or – 39 wins is significantly more than 26. 7 wins (which are what is expected due to chance)
BLINGOO Competition • You and your friend enter at competition with 2, 642 other players • p =. 30 • You win 57 of the 150 games and your friend won 39. • Afterward you wonder how many people – A) did better than you? – B) did worse than you? – C) won between 39 and 57 games • You also wonder how many games you needed to win in order to be in the top 10%
Blingoo • M = 45 • SD = 5. 61 • A) did better than you? • (57 – 45) / 5. 61 = 2. 14 • p =. 0162 • 2, 642 *. 0162 = 42. 8 or 43 people
Blingoo • M = 45 • SD = 5. 61 • A) did worse than you? • (57 – 45) / 5. 61 = 2. 14 • p =. 9838 • 2, 642 *. 9838 = 2, 599. 2 or 2, 599 people
Blingoo • M = 45 • SD = 5. 61 • A) won between 39 and 57 games? • • (57 – 45) / 5. 61 = 2. 14 ; p =. 4838 (39 – 45) / 5. 61 = -1. 07 ; p =. 3577. 4838 +. 3577 =. 8415 2, 642 *. 8415 = 2, 223. 2 or 2, 223 people
Blingoo • M = 45 • SD = 5. 61 • You also wonder how many games you needed to win in order to be in the top 10% • Z = 1. 28 • 45 + 5. 61 (1. 28) = 52. 18 games or 52 games
Is a persons’ size related to if they were bullied • You gathered data from 209 children at Springfield Elementary School. • Assessed: • Height (short vs. not short) • Bullied (yes vs. no)
Results Ever Bullied
Results Ever Bullied
Results Ever Bullied
Results Ever Bullied
Results Ever Bullied
Results Ever Bullied
Is this difference in proportion due to chance? • To test this you use a Chi-Square ( 2) • Notice you are using nominal data
Hypothesis • H 1: There is a relationship between the two variables – i. e. , a persons size is related to if they were bullied • H 0: The two variables are independent of each other – i. e. , there is no relationship between a persons size and if they were bullied
Logic • 1) Calculate an observed Chi-square • 2) Find a critical value • 3) See if the observed Chi-square falls in the critical area
Chi-Square O = observed frequency E = expected frequency
Results Ever Bullied
Observed Frequencies Ever Bullied
Expected frequencies • Are how many observations you would expect in each cell if the null hypothesis was true – i. e. , there was no relationship between a persons size and if they were bullied
Expected frequencies • To calculate a cells expected frequency: • For each cell you do this formula
Expected Frequencies Ever Bullied
Expected Frequencies Ever Bullied
Expected Frequencies Row total = 92 Ever Bullied
Expected Frequencies Row total = 92 Column total = 72 Ever Bullied
Expected Frequencies Row total = 92 N = 209 Column total = 72 Ever Bullied
Expected Frequencies E = (92 * 72) /209 = 31. 69 Ever Bullied
Expected Frequencies Ever Bullied
Expected Frequencies Ever Bullied
Expected Frequencies E = (92 * 137) /209 = 60. 30 Ever Bullied
Expected Frequencies E = (117 * 72) / 209 = 40. 30 E = (117 * 137) / 209 = 76. 69 Ever Bullied
Expected Frequencies The expected frequencies are what you would expect if there was no relationship between the two variables! Ever Bullied
How do the expected frequencies work? Looking only at: Ever Bullied
How do the expected frequencies work? If you randomly selected a person from these 209 people what is the probability you would select a person who is short? Ever Bullied
How do the expected frequencies work? If you randomly selected a person from these 209 people what is the probability you would select a person who is short? 92 / 209 =. 44 Ever Bullied
How do the expected frequencies work? If you randomly selected a person from these 209 people what is the probability you would select a person who was bullied? Ever Bullied
How do the expected frequencies work? If you randomly selected a person from these 209 people what is the probability you would select a person who was bullied? 72 / 209 =. 34 Ever Bullied
How do the expected frequencies work? If you randomly selected a person from these 209 people what is the probability you would select a person who was bullied and is short? Ever Bullied
How do the expected frequencies work? If you randomly selected a person from these 209 people what is the probability you would select a person who was bullied and is short? (. 44) (. 34) =. 15 Ever Bullied
How do the expected frequencies work? How many people do you expect to have been bullied and short? Ever Bullied
How do the expected frequencies work? How many people would you expect to have been bullied and short? (. 15 * 209) = 31. 35 (difference due to rounding) Ever Bullied
Back to Chi-Square O = observed frequency E = expected frequency
2
2
2
2
2
2
2
Significance • Is a 2 of 9. 13 significant at the. 05 level? • To find out you need to know df
Degrees of Freedom • To determine the degrees of freedom you use the number of rows (R) and the number of columns (C) • DF = (R - 1)(C - 1)
Degrees of Freedom Rows = 2 Ever Bullied
Degrees of Freedom Rows = 2 Columns = 2 Ever Bullied
Degrees of Freedom • To determine the degrees of freedom you use the number of rows (R) and the number of columns (C) • df = (R - 1)(C - 1) • df = (2 - 1) = 1
Significance • Look on page 691 • df = 1 • =. 05 • 2 critical = 3. 84
Decision • Thus, if 2 > than 2 critical – Reject H 0, and accept H 1 • If 2 < or = to 2 critical – Fail to reject H 0
Current Example • 2 = 9. 13 • 2 critical = 3. 84 • Thus, reject H 0, and accept H 1
Current Example • H 1: There is a relationship between the two variables – A persons size is significantly (alpha =. 05) related to if they were bullied
Seven Steps for Doing 2 • • 1) State the hypothesis 2) Create data table 3) Find 2 critical 4) Calculate the expected frequencies 5) Calculate 2 6) Decision 7) Put answer into words
Example • With whom do you find it easiest to make friends? • Subjects were either male and female. • Possible responses were: “opposite sex”, “same sex”, or “no difference” • Is there a significant (. 05) relationship between the gender of the subject and their response?
Results
Step 1: State the Hypothesis • H 1: There is a relationship between gender and with whom a person finds it easiest to make friends • H 0: Gender and with whom a person finds it easiest to make friends are independent of each other
Step 2: Create the Data Table
Step 2: Create the Data Table Add “total” columns and rows
Step 3: Find 2 critical • df = (R - 1)(C - 1)
Step 3: Find 2 critical • df = (R - 1)(C - 1) • df = (2 - 1)(3 - 1) = 2 • =. 05 • 2 critical = 5. 99
Step 4: Calculate the Expected Frequencies • Two steps: • 4. 1) Calculate values • 4. 2) Put values on your data table
Step 4: Calculate the Expected Frequencies E = (73 * 137) /205 = 48. 79
Step 4: Calculate the Expected Frequencies E = (73 * 68) /205 = 24. 21
Step 4: Calculate the Expected Frequencies E = (29 * 137) /205 = 19. 38
Step 4: Calculate the Expected Frequencies
Step 5: Calculate 2 O = observed frequency E = expected frequency
2
2
2
2
2 8. 5
Step 6: Decision • Thus, if 2 > than 2 critical – Reject H 0, and accept H 1 • If 2 < or = to 2 critical – Fail to reject H 0
Step 6: Decision • Thus, if 2 > than 2 critical – Reject H 0, and accept H 1 • If 2 < or = to 2 critical – Fail to reject H 0 2 = 8. 5 2 crit = 5. 99
Step 7: Put it answer into words • H 1: There is a relationship between gender and with whom a person finds it easiest to make friends • A persons gender is significantly (. 05) related with whom it is easiest to make friends.
Practice • Is there a significant ( =. 01) relationship between opinions about the death penalty and opinions about the legalization of marijuana? • 933 Subjects responded yes or no to: • “Do you favor the death penalty for persons convicted of murder? ” • “Do you think the use of marijuana should be made legal? ”
Results Death Penalty ? Marijuana ?
Step 1: State the Hypothesis • H 1: There is a relationship between opinions about the death penalty and the legalization of marijuana • H 0: Opinions about the death penalty and the legalization of marijuana are independent of each other
Step 2: Create the Data Table Death Penalty ? Marijuana ?
Step 3: Find 2 critical • df = (R - 1)(C - 1) • df = (2 - 1) = 1 • =. 01 • 2 critical = 6. 64
Step 4: Calculate the Expected Frequencies Death Penalty ? Marijuana ?
Step 5: Calculate 2
Step 6: Decision • Thus, if 2 > than 2 critical – Reject H 0, and accept H 1 • If 2 < or = to 2 critical – Fail to reject H 0
Step 6: Decision • Thus, if 2 > than 2 critical – Reject H 0, and accept H 1 • If 2 < or = to 2 critical – Fail to reject H 0 2 = 3. 91 2 crit = 6. 64
Step 7: Put it answer into words • H 0: Opinions about the death penalty and the legalization of marijuana are independent of each other • A persons opinion about the death penalty is not significantly (p >. 01) related with their opinion about the legalization of marijuana
Effect Size • Chi-Square tests are null hypothesis tests • Tells you nothing about the “size” of the effect • Phi (Ø) – Can be interpreted as a correlation coefficient.
Phi • Use with 2 x 2 tables N = sample size
Practice • Is there a significant ( =. 01) relationship between opinions about the death penalty and opinions about the legalization of marijuana? • 933 Subjects responded yes or no to: • “Do you favor the death penalty for persons convicted of murder? ” • “Do you think the use of marijuana should be made legal? ”
Results Death Penalty ? Marijuana ?
Step 6: Decision • Thus, if 2 > than 2 critical – Reject H 0, and accept H 1 • If 2 < or = to 2 critical – Fail to reject H 0 2 = 3. 91 2 crit = 6. 64
Phi • Use with 2 x 2 tables
Bullied Example Ever Bullied
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Phi • Use with 2 x 2 tables
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