practice
DESCRIPTION
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?” - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/1.jpg)
![Page 2: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/2.jpg)
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?”
![Page 3: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/3.jpg)
Results
Yes No
Yes 152 561
No 61 159
Marijuana ?
Dea
th
Pen
alty
?
![Page 4: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/4.jpg)
Step 1: State the Hypothesis
• H1: There is a relationship between opinions about the death penalty and the legalization of marijuana
• H0:Opinions about the death penalty and the legalization of marijuana are independent of each other
![Page 5: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/5.jpg)
Step 2: Create the Data Table
Yes No Total
Yes 152 561 713
No 61 159 220
Total 213 720 933
Marijuana ?
Dea
th
Pen
alty
?
![Page 6: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/6.jpg)
Step 3: Find 2 critical
• df = (R - 1)(C - 1)
• df = (2 - 1)(2 - 1) = 1 = .01
2 critical = 6.64
![Page 7: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/7.jpg)
Step 4: Calculate the Expected Frequencies
Yes No Total
Yes 152(162.77)
561(550.23)
713
No 61(50.23)
159(169.78)
220
Total 213 720 933
Marijuana ?
Dea
th
Pen
alty
?
![Page 8: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/8.jpg)
Step 5: Calculate 2
O E O - E (O - E)2 (O - E)2
E152 162.77 -10.77 115.99 .71
61 50.23 10.77 115.99 2.31
561 550.23 10.77 115.99 .21
159 169.78 -10.78 115.99 .682 = 3.91
![Page 9: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/9.jpg)
Step 6: Decision
• Thus, if 2 > than 2critical
– Reject H0, and accept H1
• If 2 < or = to 2critical
– Fail to reject H0
![Page 10: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/10.jpg)
Step 6: Decision
• Thus, if 2 > than 2critical
– Reject H0, and accept H1
• If If 22 < or = to < or = to 22criticalcritical
– Fail to reject HFail to reject H00
2 = 3.91
2 crit = 6.64
![Page 11: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/11.jpg)
Step 7: Put it answer into words
• H0: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
![Page 12: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/12.jpg)
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.
![Page 13: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/13.jpg)
Phi
• Use with 2x2 tables
N
2 N = sample size
![Page 14: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/14.jpg)
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?”
![Page 15: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/15.jpg)
Results
Yes No
Yes 152 561
No 61 159
Marijuana ?
Dea
th
Pen
alty
?
![Page 16: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/16.jpg)
Step 6: Decision
• Thus, if 2 > than 2critical
– Reject H0, and accept H1
• If If 22 < or = to < or = to 22criticalcritical
– Fail to reject HFail to reject H00
2 = 3.91
2 crit = 6.64
![Page 17: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/17.jpg)
Phi
• Use with 2x2 tables
06.933
91.3
![Page 18: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/18.jpg)
Bullied Example
Height Yes No Total
Short 42 50 92
Not short 30 87 117
Total 72 137 209
Ever Bullied
![Page 19: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/19.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 20: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/20.jpg)
Phi
• Use with 2x2 tables
21.209
13.9
![Page 21: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/21.jpg)
Practice
• Practice– Page 170 #6.10– How strong is the relationship?
![Page 22: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/22.jpg)
Results
Remed Reg
No 22 (28.37)
187 (180.62)
209
Yes 61 (12.63)
159 (80.37)
93
41 261 302
English
AD
D
X2 = 5.38
X2 crit = 3.83
![Page 23: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/23.jpg)
Phi
• Use with 2x2 tables
13.302
38.5
![Page 24: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/24.jpg)
![Page 25: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/25.jpg)
![Page 26: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/26.jpg)
Practice
• In the 1930’s 650 boys participated in the Cambridge-Somerville Youth Study. Half of the participants were randomly assigned to a delinquency-prevention pogrom and the other half to a control group. At the end of the study, police records were examined for evidence of delinquency. In the prevention program 114 boys had a police record and in the control group 101 boys had a police record. Analyze the data and write a conclusion.
![Page 27: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/27.jpg)
• Chi Square = 1.17
• Chi Square observed = 3.84
• Phi = .04– Note the results go in the opposite direction
that was expected!
![Page 28: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/28.jpg)
![Page 29: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/29.jpg)
2 as a test for goodness of fit
• But what if:
• You have a theory or hypothesis that the frequencies should occur in a particular manner?
![Page 30: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/30.jpg)
Example
• M&Ms claim that of their candies:
• 30% are brown
• 20% are red
• 20% are yellow
• 10% are blue
• 10% are orange
• 10% are green
![Page 31: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/31.jpg)
Example
• Based on genetic theory you hypothesize that in the population:
• 45% have brown eyes
• 35% have blue eyes
• 20% have another eye color
![Page 32: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/32.jpg)
To solve you use the same basic steps as before (slightly
different order)• 1) State the hypothesis
• 2) Find 2 critical
• 3) Create data table
• 4) Calculate the expected frequencies
• 5) Calculate 2
• 6) Decision
• 7) Put answer into words
![Page 33: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/33.jpg)
Example
• M&Ms claim that of their candies:
• 30% are brown• 20% are red• 20% are yellow• 10% are blue• 10% are orange• 10% are green
![Page 34: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/34.jpg)
Example
• Four 1-pound bags of plain M&Ms are purchased
• Each M&Ms is counted and categorized according to its color
• Question: Is M&Ms “theory” about the colors of M&Ms correct?
![Page 35: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/35.jpg)
Observed
Brown 602
Red 396
Yellow 379
Blue 227
Orange 242
Green 235
Total 2081
![Page 36: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/36.jpg)
Step 1: State the Hypothesis
• H0: The data do fit the model
– i.e., the observed data does agree with M&M’s theory
• H1: The data do not fit the model
– i.e., the observed data does not agree with M&M’s theory
– NOTE: These are backwards from what you have done before
![Page 37: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/37.jpg)
Step 2: Find 2 critical
• df = number of categories - 1
![Page 38: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/38.jpg)
Step 2: Find 2 critical
• df = number of categories - 1
• df = 6 - 1 = 5 = .05
2 critical = 11.07
![Page 39: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/39.jpg)
Observed
Brown 602
Red 396
Yellow 379
Blue 227
Orange 242
Green 235
Total 2081
Step 3: Create the data table
![Page 40: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/40.jpg)
Observed ExpectedProp.
Brown 602 .30
Red 396 .20
Yellow 379 .20
Blue 227 .10
Orange 242 .10
Green 235 .10
Total 2081
Step 3: Create the data tableAdd the expected proportion of each category
![Page 41: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/41.jpg)
Observed ExpectedProp.
Brown 602 .30
Red 396 .20
Yellow 379 .20
Blue 227 .10
Orange 242 .10
Green 235 .10
Total 2081
Step 4: Calculate the Expected Frequencies
![Page 42: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/42.jpg)
Observed ExpectedProp.
ExpectedFreq
Brown 602 .30
Red 396 .20
Yellow 379 .20
Blue 227 .10
Orange 242 .10
Green 235 .10
Total 2081
Step 4: Calculate the Expected Frequencies
Expected Frequency = (proportion)(N)
![Page 43: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/43.jpg)
Observed ExpectedProp.
ExpectedFreq
Brown 602 .30 624.30
Red 396 .20
Yellow 379 .20
Blue 227 .10
Orange 242 .10
Green 235 .10
Total 2081
Step 4: Calculate the Expected Frequencies
Expected Frequency = (.30)(2081) = 624.30
![Page 44: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/44.jpg)
Observed ExpectedProp.
ExpectedFreq
Brown 602 .30 624.30
Red 396 .20 416.20
Yellow 379 .20
Blue 227 .10
Orange 242 .10
Green 235 .10
Total 2081
Step 4: Calculate the Expected Frequencies
Expected Frequency = (.20)(2081) = 416.20
![Page 45: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/45.jpg)
Observed ExpectedProp.
ExpectedFreq
Brown 602 .30 624.30
Red 396 .20 416.20
Yellow 379 .20 416.20
Blue 227 .10
Orange 242 .10
Green 235 .10
Total 2081
Step 4: Calculate the Expected Frequencies
Expected Frequency = (.20)(2081) = 416.20
![Page 46: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/46.jpg)
Observed ExpectedProp.
ExpectedFreq
Brown 602 .30 624.30
Red 396 .20 416.20
Yellow 379 .20 416.20
Blue 227 .10 208.10
Orange 242 .10 208.10
Green 235 .10 208.10
Total 2081
Step 4: Calculate the Expected Frequencies
Expected Frequency = (.10)(2081) = 208.10
![Page 47: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/47.jpg)
Step 5: Calculate 2
O = observed frequency
E = expected frequency
![Page 48: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/48.jpg)
2
O E O - E (O - E)2 (O - E)2
E
![Page 49: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/49.jpg)
2
O E O - E (O - E)2 (O - E)2
E602 624.30
396 416.20
379 416.20
227 208.10
242 208.10
235 208.10
![Page 50: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/50.jpg)
2
O E O - E (O - E)2 (O - E)2
E602 624.30 -22.3
396 416.20 -20.2
379 416.20 -37.2
227 208.10 18.9
242 208.10 33.9
235 208.10 26.9
![Page 51: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/51.jpg)
2
O E O - E (O - E)2 (O - E)2
E602 624.30 -22.3 497.29
396 416.20 -20.2 408.04
379 416.20 -37.2 1383.84
227 208.10 18.9 357.21
242 208.10 33.9 1149.21
235 208.10 26.9 723.61
![Page 52: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/52.jpg)
2
O E O - E (O - E)2 (O - E)2
E602 624.30 -22.3 497.29 .80
396 416.20 -20.2 408.04 .98
379 416.20 -37.2 1383.84 3.32
227 208.10 18.9 357.21 1.72
242 208.10 33.9 1149.21 5.22
235 208.10 26.9 723.61 3.48
![Page 53: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/53.jpg)
2
O E O - E (O - E)2 (O - E)2
E602 624.30 -22.3 497.29 .80
396 416.20 -20.2 408.04 .98
379 416.20 -37.2 1383.84 3.32
227 208.10 18.9 357.21 1.72
242 208.10 33.9 1149.21 5.22
235 208.10 26.9 723.61 3.48
15.52
![Page 54: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/54.jpg)
Step 6: Decision
• Thus, if 2 > than 2critical
– Reject H0, and accept H1
• If 2 < or = to 2critical
– Fail to reject H0
![Page 55: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/55.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 = 15.52
2 crit = 11.07
![Page 56: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/56.jpg)
Step 7: Put it answer into words
• H1: The data do not fit the model
• M&M’s color “theory” did not significantly (.05) fit the data
![Page 57: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/57.jpg)
Practice
• Among women in the general population under the age of 40:
• 60% are married• 23% are single• 4% are separated• 12% are divorced• 1% are widowed
![Page 58: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/58.jpg)
Practice
• You sample 200 female executives under the age of 40
• Question: Is marital status distributed the same way in the population of female executives as in the general population ( = .05)?
![Page 59: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/59.jpg)
Observed
Married 100
Single 44
Separated 16
Divorced 36
Widowed 4
Total 200
![Page 60: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/60.jpg)
Step 1: State the Hypothesis
• H0: The data do fit the model
– i.e., marital status is distributed the same way in the population of female executives as in the general population
• H1: The data do not fit the model
– i.e., marital status is not distributed the same way in the population of female executives as in the general population
![Page 61: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/61.jpg)
Step 2: Find 2 critical
• df = number of categories - 1
![Page 62: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/62.jpg)
Step 2: Find 2 critical
• df = number of categories - 1
• df = 5 - 1 = 4 = .05
2 critical = 9.49
![Page 63: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/63.jpg)
Step 3: Create the data table
Observed ExpectedProp.
Married 100 .60
Single 44 .23
Separated 16 .04
Divorced 36 .12
Widowed 4 .01
Total 200
![Page 64: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/64.jpg)
Step 4: Calculate the Expected Frequencies
Observed ExpectedProp.
ExpectedFreq.
Married 100 .60 120
Single 44 .23 46
Separated 16 .04 8
Divorced 36 .12 24
Widowed 4 .01 2
Total 200
![Page 65: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/65.jpg)
Step 5: Calculate 2
O = observed frequency
E = expected frequency
![Page 66: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/66.jpg)
2
O E O - E (O - E)2 (O - E)2
E100 120 -20 400 3.33
44 46 -2 4 .09
16 8 8 64 8
36 24 12 144 6
4 2 2 4 2
19.42
![Page 67: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/67.jpg)
Step 6: Decision
• Thus, if 2 > than 2critical
– Reject H0, and accept H1
• If 2 < or = to 2critical
– Fail to reject H0
![Page 68: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/68.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 = 19.42
2 crit = 9.49
![Page 69: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/69.jpg)
Step 7: Put it answer into words
• H1: The data do not fit the model
• Marital status is not distributed the same way in the population of female executives as in the general population ( = .05)
![Page 70: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/70.jpg)
![Page 71: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/71.jpg)
Practice
• In the past you have had a 20% success rate at getting someone to accept a date from you.
• What is the probability that at least 2 of the next 10 people you ask out will accept?
![Page 72: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/72.jpg)
Practice
• p zero will accept = .11
• p one will accept = .27
• p zero OR one will accept = .38
• p two or more will accept = 1 - .38 = .62
![Page 73: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/73.jpg)
![Page 74: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/74.jpg)
Practice
• IQ– Mean = 100– SD = 15
• What is the probability that the stranger you just bumped into on the street has an IQ between 95 and 110?
![Page 75: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/75.jpg)
Step 1: Sketch out question
-3 -2 -1 1 2 3
11095
?
![Page 76: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/76.jpg)
Step 2: Calculate Z scores for both values
• Z = (X - ) /
• Z = (95 - 100) / 15 = -.33
• Z = (110 - 100) / 15 = .67
![Page 77: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/77.jpg)
Step 3: Look up Z scores
-3 -2 -1 1 2 3
.67-.33
.1293 .2486
![Page 78: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/78.jpg)
Step 4: Add together the two values
-3 -2 -1 1 2 3
.67-.33
.3779
![Page 79: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/79.jpg)
![Page 80: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/80.jpg)
Practice
• A professor would like to determine if there has been a change in grading practices over the years. In the past, the overall grade distribution was 14% As, 26% Bs, 31% Cs, 19% Ds, and 10% Fs.
• A sample of 200 students this years had the following grades
![Page 81: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/81.jpg)
Practice
• A = 32• B = 61• C = 64• D = 31• F = 12
• Do the data indicate a significant change in the grade distribution? Test at the .05 level.
![Page 82: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/82.jpg)
Step 1: State the Hypothesis
• H0: The data do fit the model
– i.e., the grades are distributed the same
• H1: The data do not fit the model
– i.e., the grades are not distributed the same
![Page 83: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/83.jpg)
Practice
• A = 32 28• B = 61 52• C = 64 62• D = 31 38• F = 12 20
• Chi square = 6.68• Critical Chi square (4) = 9.49
![Page 84: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/84.jpg)
Step 6: Decision
• Thus, if 2 > than 2critical
– Reject H0, and accept H1
• If If 22 < or = to < or = to 22criticalcritical
– Fail to reject HFail to reject H00
2 = 6.68
2 crit = 9.49
![Page 85: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/85.jpg)
Step 7
• H0: The data do fit the model
– i.e., the grades are distributed the same
• There is no evidence that the grades have changed
![Page 86: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/86.jpg)
Practice
• An early hypothesis of schizophrenia was that it has a simple genetic cause. In accordance with the theory 25% of the offspring of a selected group of parents would be expected to be diagnosed as schizophrenic. Suppose that of 140 offspring, 19.3% were schizophrenic. Test this theory.
![Page 87: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/87.jpg)
• Goodness of fit chi-square
• Make sure you compute the Chi square with the frequencies.
• Chi square = 2.439• Observed = 3.84
• These data are consistent with the theory!
![Page 88: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/88.jpg)
![Page 89: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/89.jpg)
Practice
• In 1693, Samuel Pepys asked Isaac Newton whether it is more likely to get at least one ace in 6 rolls of a die or at least two aces in 12 rolls of a die. This problems is known a Pepys' problem.
![Page 90: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/90.jpg)
Binomial Distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 1 2 3 4 5 6Aces
p
p = .67
![Page 91: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/91.jpg)
Binomial Distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 1 2 3 4 5 6 7 8 9 10 11 12Aces
p
p = .62
![Page 92: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/92.jpg)
Practice
• In 1693, Samuel Pepys asked Isaac Newton whether it is more likely to get at least one ace in 6 rolls of a die or at least two aces in 12 rolls of a die. This problems is known a Pepys' problem.
• It is more likely to get at least one ace in 6 rolls of a die!
![Page 93: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/93.jpg)
![Page 94: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/94.jpg)
Practice
• Which is more likely: at least one ace with 4 throws of a fair die or at least one double ace in 24 throws of two fair dice? This is known as DeMere's problem, named after Chevalier De Mere.
• Blaise Pascal later solved this problem.
![Page 95: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/95.jpg)
Binomial Distribution
)04(0 8333.1667.)!04(!0
!4)(
Xp
p = .482 of zero aces
1 - .482 = .518 at least one ace will occur
![Page 96: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/96.jpg)
Binomial Distribution
)024(0 9722.0278.)!024(!0
!24)(
Xp
p = .508 of zero double aces
1 - .508 = .492 at least one double ace will occur
![Page 97: Practice](https://reader035.vdocuments.us/reader035/viewer/2022062721/5681389f550346895da05bdf/html5/thumbnails/97.jpg)
Practice
• Which is more likely: at least one ace with 4 throws of a fair die or at least one double ace in 24 throws of two fair dice? This is known as DeMere's problem, named after Chevalier De Mere.
• More likely at least one ace with 4 throws will occur