![Page 1: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/1.jpg)
![Page 2: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/2.jpg)
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?
![Page 3: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/3.jpg)
Binomial Distribution
Ingredients:N = total number of eventsp = the probability of a success on any one trialq = (1 – p) = the probability of a failure on any one trialX = number of successful events
)(
)!(!
!)( XNXqp
XNX
NXp
![Page 4: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/4.jpg)
Binomial Distribution
Ingredients:N = total number of eventsp = the probability of a success on any one trialq = (1 – p) = the probability of a failure on any one trialX = number of successful events
)810(8 502.498.)!810(!8
!10)(
Xp
![Page 5: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/5.jpg)
Binomial Distribution
Ingredients:N = total number of eventsp = the probability of a success on any one trialq = (1 – p) = the probability of a failure on any one trialX = number of successful events
)810(8 502.498.)!810(!8
!10)(
Xp
p = .0429
![Page 6: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/6.jpg)
![Page 7: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/7.jpg)
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
![Page 8: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/8.jpg)
Probability of Winning Blackjack
• p = .498, N = 10Number of Wins p
0
1
2
3
4
5
6
7
8
9
10
![Page 9: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/9.jpg)
Probability of Winning Blackjack
• p = .498, N = 10Number of Wins p
0 .001
1
2
3
4
5
6
7
8
9
10
![Page 10: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/10.jpg)
Probability of Winning Blackjack
• p = .498, N = 10Number 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
![Page 11: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/11.jpg)
Probability of Winning Blackjack
• p = .498, N = 10Number 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
![Page 12: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/12.jpg)
Binomial Distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4 5 6 7 8 9 10
Games Won
p
![Page 13: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/13.jpg)
Hypothesis Testing
• You wonder if winning at least 7 games of blackjack is significantly (.05) better than what would be expected due to chance.
• H1= Games won > 6 • H0= Games won < or equal to 6
• What is the probability of winning 7 or more games?
![Page 14: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/14.jpg)
Binomial Distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4 5 6 7 8 9 10
Games Won
p
![Page 15: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/15.jpg)
Binomial Distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4 5 6 7 8 9 10
Games Won
p
![Page 16: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/16.jpg)
Probability of Winning Blackjack
• p = .498, N = 10Number 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
![Page 17: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/17.jpg)
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
77 .115.115
88 .044.044
99 .009.009
1010 .001.001
1.00
![Page 18: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/18.jpg)
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.
![Page 19: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/19.jpg)
Probability of Winning SlotNumber 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
![Page 20: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/20.jpg)
Binomial Distribution
00.050.1
0.150.2
0.250.3
0.350.4
0.450.5
0 1 2 3 4 5 6 7 8 9 10
Games Won
p
![Page 21: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/21.jpg)
Probability of Winning Slot
• p of winning at least 4 games
• .005+.001+.000 . . . .000 = .006
• p< .05
• Winning at least 4 games is significantly better than chance
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
![Page 22: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/22.jpg)
Binomial Distribution
• These distributions can be described with means and SD.
• Mean = Np• SD = Npq
![Page 23: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/23.jpg)
Binomial Distribution
• Black Jack; p = .498, N =10
• M = 4.98
• SD = 1.59
![Page 24: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/24.jpg)
Binomial Distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4 5 6 7 8 9 10
Games Won
p
![Page 25: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/25.jpg)
Binomial Distribution
• Statistical Slot Machine; p = .08, N = 10
• M = .8
• SD = .86
![Page 26: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/26.jpg)
Binomial Distribution
00.050.1
0.150.2
0.250.3
0.350.4
0.450.5
0 1 2 3 4 5 6 7 8 9 10
Games Won
p
Note: as N gets bigger, distributions will approach normal
![Page 27: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/27.jpg)
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?
![Page 28: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/28.jpg)
Hypothesis
• H1= .44 > .30
• H0= .44 < or equal to .30
• Or
• H1= 39 wins > 26.7 wins
• H0= 39 wins < or equal to 26.7 wins
![Page 29: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/29.jpg)
Distribution
• Mean = 26.7
• SD = 4.32
• X = 39
![Page 30: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/30.jpg)
Z-score
![Page 31: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/31.jpg)
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)
![Page 32: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/32.jpg)
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%
![Page 33: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/33.jpg)
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
![Page 34: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/34.jpg)
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
![Page 35: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/35.jpg)
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
![Page 36: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/36.jpg)
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
![Page 37: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/37.jpg)
Practice
• In the past you have had a 5% success rate at getting someone to accept a date from you.
• What is the probability that at least 1 of the next 10 people you ask out will accept?
• Note: N isn’t big enough in these problems to use the Z-score formula
![Page 38: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/38.jpg)
![Page 39: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/39.jpg)
Bullied as a child?
![Page 40: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/40.jpg)
Are you tall or short?
6’ 4”
5’ 10”
4’
2’ 4”
![Page 41: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/41.jpg)
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)
![Page 42: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/42.jpg)
Results
Height Yes No
Short 42 50
Not short 30 87
Ever Bullied
![Page 43: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/43.jpg)
Results
Height Yes No Total
Short 42 50 92
Not short 30 87 117
Total 72 137 209
Ever Bullied
![Page 44: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/44.jpg)
Results
Height Yes No Total
Short 42 50 92
Not short 30 87 117
Total 34% 66% 209
Ever Bullied
![Page 45: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/45.jpg)
Results
Height Yes No Total
Short 42 50 44%
Not short 30 87 56%
Total 72 137 209
Ever Bullied
![Page 46: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/46.jpg)
Results
Height Yes No Total
Short 42 50 92
Not short 30 87 117
Total 72 137 209
Ever Bullied
![Page 47: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/47.jpg)
Results
Height Yes No Total
Short 45% 55% 92
Not short 26% 74% 117
Total 72 137 209
Ever Bullied
![Page 48: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/48.jpg)
Is this difference in proportion due to chance?
• To test this you use a Chi-Square (2)
• Notice you are using nominal data
![Page 49: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/49.jpg)
Hypothesis
• H1: There is a relationship between the two variables– i.e., a persons size is related to if they were
bullied
• H0:The two variables are independent of each other– i.e., there is no relationship between a persons
size and if they were bullied
![Page 50: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/50.jpg)
Logic
• 1) Calculate an observed Chi-square
• 2) Find a critical value
• 3) See if the the observed Chi-square falls in the critical area
![Page 51: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/51.jpg)
Chi-Square
O = observed frequency
E = expected frequency
![Page 52: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/52.jpg)
Results
Height Yes No Total
Short 42 50 92
Not short 30 87 117
Total 72 137 209
Ever Bullied
![Page 53: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/53.jpg)
Observed Frequencies
Height Yes No Total
Short 42 50 92
Not short 30 87 117
Total 72 137 209
Ever Bullied
![Page 54: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/54.jpg)
Expected frequencies
• Are how many observations you would expect in each cell if the null hypothesis was true
– i.e., there there was no relationship between a persons size and if they were bullied
![Page 55: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/55.jpg)
Expected frequencies
• To calculate a cells expected frequency:
• For each cell you do this formula
![Page 56: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/56.jpg)
Expected Frequencies
Height Yes No Total
Short 42 50 92
Not short 30 87 117
Total 72 137 209
Ever Bullied
![Page 57: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/57.jpg)
Expected Frequencies
Height Yes No Total
Short 42 50 92
Not short 30 87 117
Total 72 137 209
Ever Bullied
![Page 58: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/58.jpg)
Expected Frequencies
Height Yes No Total
Short 42 50 92
Not short 30 87 117
Total 72 137 209
Ever Bullied
Row total = 92
![Page 59: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/59.jpg)
Expected Frequencies
Height Yes No Total
Short 42 50 92
Not short 30 87 117
Total 72 137 209
Ever Bullied
Row total = 92
Column total = 72
![Page 60: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/60.jpg)
Expected Frequencies
Height Yes No Total
Short 42 50 92
Not short 30 87 117
Total 72 137 209
Ever Bullied
Row total = 92 N = 209
Column total = 72
![Page 61: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/61.jpg)
Expected Frequencies
Height Yes No Total
Short 42 50 92
Not short 30 87 117
Total 72 137 209
Ever Bullied
E = (92 * 72) /209 = 31.69
![Page 62: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/62.jpg)
Expected Frequencies
Height Yes No Total
Short 42(31.69)
50 92
Not short 30 87 117
Total 72 137 209
Ever Bullied
![Page 63: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/63.jpg)
Expected Frequencies
Height Yes No Total
Short 42(31.69)
50 92
Not short 30 87 117
Total 72 137 209
Ever Bullied
![Page 64: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/64.jpg)
Expected Frequencies
Height Yes No Total
Short 42(31.69)
50(60.30)
92
Not short 30 87 117
Total 72 137 209
Ever Bullied
E = (92 * 137) /209 = 60.30
![Page 65: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/65.jpg)
Expected Frequencies
Height Yes No Total
Short 42(31.69)
50(60.30)
92
Not short 30(40.30)
87(76.69)
117
Total 72 137 209
Ever Bullied
E = (117 * 72) / 209 = 40.30
E = (117 * 137) / 209 = 76.69
![Page 66: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/66.jpg)
Expected Frequencies
Height Yes No Total
Short 42(31.69)
50(60.30)
92
Not short 30(40.30)
87(76.69)
117
Total 72 137 209
Ever Bullied
The expected frequencies are what you would expect if there was no relationship between the two variables!
![Page 67: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/67.jpg)
How do the expected frequencies work?
Height Yes No Total
Short 42(31.69)
50(60.30)
92
Not short 30(40.30)
87(76.69)
117
Total 72 137 209
Ever Bullied
Looking only at:
![Page 68: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/68.jpg)
How do the expected frequencies work?
Height Yes No Total
Short 42(31.69)
50(60.30)
92
Not short 30(40.30)
87(76.69)
117
Total 72 137 209
Ever Bullied
If you randomly selected a person from these 209 people what is the probability you would select a person who is short?
![Page 69: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/69.jpg)
How do the expected frequencies work?
Height Yes No Total
Short 42(31.69)
50(60.30)
92
Not short 30(40.30)
87(76.69)
117
Total 72 137 209
Ever Bullied
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
![Page 70: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/70.jpg)
How do the expected frequencies work?
Height Yes No Total
Short 42(31.69)
50(60.30)
92
Not short 30(40.30)
87(76.69)
117
Total 72 137 209
Ever Bullied
If you randomly selected a person from these 209 people what is the probability you would select a person who was bullied?
![Page 71: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/71.jpg)
How do the expected frequencies work?
Height Yes No Total
Short 42(31.69)
50(60.30)
92
Not short 30(40.30)
87(76.69)
117
Total 72 137 209
Ever Bullied
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
![Page 72: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/72.jpg)
How do the expected frequencies work?
Height Yes No Total
Short 42(31.69)
50(60.30)
92
Not short 30(40.30)
87(76.69)
117
Total 72 137 209
Ever Bullied
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?
![Page 73: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/73.jpg)
How do the expected frequencies work?
Height Yes No Total
Short 42(31.69)
50(60.30)
92
Not short 30(40.30)
87(76.69)
117
Total 72 137 209
Ever Bullied
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
![Page 74: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/74.jpg)
How do the expected frequencies work?
Height Yes No Total
Short 42(31.69)
50(60.30)
92
Not short 30(40.30)
87(76.69)
117
Total 72 137 209
Ever Bullied
How many people do you expect to have been bullied and short?
![Page 75: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/75.jpg)
How do the expected frequencies work?
Height Yes No Total
Short 42(31.69)
50(60.30)
92
Not short 30(40.30)
87(76.69)
117
Total 72 137 209
Ever Bullied
How many people would you expect to have been bullied and short? (.15 * 209) = 31.35 (difference due to rounding)
![Page 76: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/76.jpg)
Back to Chi-Square
O = observed frequency
E = expected frequency
![Page 77: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/77.jpg)
2
O E O - E (O - E)2 (O - E)2
E
![Page 78: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/78.jpg)
2
O E O - E (O - E)2 (O - E)2
E42
50
30
87
![Page 79: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/79.jpg)
2
O E O - E (O - E)2 (O - E)2
E42 31.69
50 60.30
30 40.30
87 76.69
![Page 80: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/80.jpg)
2
O E O - E (O - E)2 (O - E)2
E42 31.69 10.31
50 60.30 -10.30
30 40.30 -10.30
87 76.69 10.31
![Page 81: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/81.jpg)
2
O E O - E (O - E)2 (O - E)2
E42 31.69 10.31 106.30
50 60.30 -10.30 106.09
30 40.30 -10.30 106.09
87 76.69 10.31 106.30
![Page 82: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/82.jpg)
2
O E O - E (O - E)2 (O - E)2
E42 31.69 10.31 106.30 3.35
50 60.30 -10.30 106.09 1.76
30 40.30 -10.30 106.09 2.63
87 76.69 10.31 106.30 1.39
![Page 83: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/83.jpg)
2
O E O - E (O - E)2 (O - E)2
E42 31.69 10.31 106.30 3.35
50 60.30 -10.30 106.09 1.76
30 40.30 -10.30 106.09 2.63
87 76.69 10.31 106.30 1.392 = 9.13
![Page 84: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/84.jpg)
Significance
• Is a 2 of 9.13 significant at the .05 level?
• To find out you need to know df
![Page 85: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/85.jpg)
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)
![Page 86: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/86.jpg)
Degrees of Freedom
Height Yes No Total
Short 42(31.69)
50(60.30)
92
Not short 30(40.30)
87(76.69)
117
Total 72 137 209
Ever Bullied
Rows = 2
![Page 87: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/87.jpg)
Degrees of Freedom
Height Yes No Total
Short 42(31.69)
50(60.30)
92
Not short 30(40.30)
87(76.69)
117
Total 72 137 209
Ever Bullied
Rows = 2
Columns = 2
![Page 88: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/88.jpg)
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)(2 - 1) = 1
![Page 89: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/89.jpg)
Significance
• Look on page 736
• df = 1 = .05
2critical = 3.84
![Page 90: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/90.jpg)
Decision
• Thus, if 2 > than 2critical
– Reject H0, and accept H1
• If 2 < or = to 2critical
– Fail to reject H0
![Page 91: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/91.jpg)
Current Example
2 = 9.13
2critical = 3.84
• Thus, reject H0, and accept H1
![Page 92: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/92.jpg)
Current Example
• H1: There is a relationship between the the two variables– A persons size is significantly (alpha = .05)
related to if they were bullied
![Page 93: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/93.jpg)
![Page 94: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/94.jpg)
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
![Page 95: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/95.jpg)
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?
![Page 96: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/96.jpg)
Results
Opposite Sex Same Sex No Difference
Females 58 16 63
Males 15 13 40
![Page 97: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/97.jpg)
Step 1: State the Hypothesis
• H1: There is a relationship between gender and with whom a person finds it easiest to make friends
• H0:Gender and with whom a person finds it easiest to make friends are independent of each other
![Page 98: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/98.jpg)
Step 2: Create the Data Table
Opposite Sex Same Sex No Difference
Females 58 16 63
Males 15 13 40
![Page 99: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/99.jpg)
Step 2: Create the Data Table
Opposite Sex Same Sex No Difference Total
Females 58 16 63 137
Males 15 13 40 68
Total 73 29 103 205
Add “total” columns and rows
![Page 100: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/100.jpg)
Step 3: Find 2 critical
• df = (R - 1)(C - 1)
![Page 101: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/101.jpg)
Step 3: Find 2 critical
• df = (R - 1)(C - 1)
• df = (2 - 1)(3 - 1) = 2 = .05
2 critical = 5.99
![Page 102: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/102.jpg)
Step 4: Calculate the Expected Frequencies
• Two steps:
• 4.1) Calculate values
• 4.2) Put values on your data table
![Page 103: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/103.jpg)
Step 4: Calculate the Expected Frequencies
Opposite Sex Same Sex No Difference Total
Females 58(48.78)
16 63 137
Males 15 13 40 68
Total 73 29 103 205
E = (73 * 137) /205 = 48.79
![Page 104: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/104.jpg)
Step 4: Calculate the Expected Frequencies
Opposite Sex Same Sex No Difference Total
Females 58(48.78)
16 63 137
Males 15(24.21)
13 40 68
Total 73 29 103 205
E = (73 * 68) /205 = 24.21
![Page 105: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/105.jpg)
Step 4: Calculate the Expected Frequencies
Opposite Sex Same Sex No Difference Total
Females 58(48.78)
16(19.38)
63 137
Males 15(24.21)
13 40 68
Total 73 29 103 205
E = (29 * 137) /205 = 19.38
![Page 106: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/106.jpg)
Step 4: Calculate the Expected Frequencies
Opposite Sex Same Sex No Difference Total
Females 58(48.78)
16(19.38)
63(68.83)
137
Males 15(24.21)
13(9.62)
40(34.17)
68
Total 73 29 103 205
![Page 107: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/107.jpg)
Step 5: Calculate 2
O = observed frequency
E = expected frequency
![Page 108: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/108.jpg)
2
O E O - E (O - E)2 (O - E)2
E58 48.79
15 24.21
16 19.38
13 9.62
63 68.83
40 34.17
![Page 109: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/109.jpg)
2
O E O - E (O - E)2 (O - E)2
E58 48.79 9.21 84.82 1.74
15 24.21
16 19.38
13 9.62
63 68.83
40 34.17
![Page 110: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/110.jpg)
2
O E O - E (O - E)2 (O - E)2
E58 48.79 9.21 84.82 1.74
15 24.21 -9.21 84.82 3.50
16 19.38
13 9.62
63 68.83
40 34.17
![Page 111: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/111.jpg)
2
O E O - E (O - E)2 (O - E)2
E58 48.79 9.21 84.82 1.74
15 24.21 -9.21 84.82 3.50
16 19.38 -3.38 11.42 .59
13 9.62 3.38 11.42 1.19
63 68.83 -5.83 33.99 .49
40 34.17 5.83 33.99 .99
![Page 112: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/112.jpg)
2
O E O - E (O - E)2 (O - E)2
E58 48.79 9.21 84.82 1.74
15 24.21 -9.21 84.82 3.50
16 19.38 -3.38 11.42 .59
13 9.62 3.38 11.42 1.19
63 68.83 -5.83 33.99 .49
40 34.17 5.83 33.99 .99
8.5
![Page 113: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/113.jpg)
Step 6: Decision
• Thus, if 2 > than 2critical
– Reject H0, and accept H1
• If 2 < or = to 2critical
– Fail to reject H0
![Page 114: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/114.jpg)
Step 6: Decision
• Thus, if Thus, if 22 > than > than 22criticalcritical
– Reject HReject H00, and accept H, and accept H11
• If 2 < or = to 2critical
– Fail to reject H0
2 = 8.5
2 crit = 5.99
![Page 115: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/115.jpg)
Step 7: Put it answer into words
• H1: 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.
![Page 116: 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?](https://reader035.vdocument.in/reader035/viewer/2022081603/5697bf8e1a28abf838c8ccf1/html5/thumbnails/116.jpg)
Practice
• 6.15
• 6.16