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

Opposite Sex Same Sex No Difference

Females 58 16 63

Males 15 13 40

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

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

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

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

Step 5: Calculate 2

O = observed frequency

E = expected frequency

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

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

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

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

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

Step 6: Decision

• Thus, if 2 > than 2critical

– Reject H0, and accept H1

• If 2 < or = to 2critical

– Fail to reject H0

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

Step 7: Put 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.

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

Yes No

Yes 152 561

No 61 159

Marijuana ?

Dea

th

Pen

alty

?

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

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

?

Step 3: Find 2 critical

• df = (R - 1)(C - 1)

• df = (2 - 1)(2 - 1) = 1 = .01

2 critical = 6.64

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

?

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

Step 6: Decision

• Thus, if 2 > than 2critical

– Reject H0, and accept H1

• If 2 < or = to 2critical

– Fail to reject H0

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

Step 7: Put 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 (.01) related with their opinion about the legalization of marijuana

Sam Sleepresearcher hypothesizes that people who are allowed to sleep for only four hours will score significantly lower than people who are allowed to sleep for eight hours on a cognitive skills test. He brings sixteen participants into his sleep lab and randomly assigns them to one of two groups. In one group he has participants sleep for eight hours and in the other group he has them sleep for four. The next morning he administers the SCAT (Sam's Cognitive Ability Test) to all participants. (Scores on the SCAT range from 1-9 with high scores representing better performance). Was Sam correct?

CAT scores

8 hours sleep group (X)

5 7 5 3 5 3 3 9

4 hours sleep group (Y)

8 1 4 6 6 4 1 2

Indepdent t-test (2 sample with equal n)

t obs = .847t crit (one tailed) = 1.76

Sam's hypothesis was not confirmed. He did not find a significant difference between those who slept for four hours versus those who slept for eight hours on cognitive test performance.

Below grade level

At grade level Advanced

Pre-School 8 6 6

No Pre-School 6 15 9

Algebra Achievement Levels

Before making their recommendations, administrators in a large urban district take a random sample of 50 seventh graders and compare the achievement levels of those who attended pre-school and those who did not. If achievement is independent of attending pre-school then the proportions at each level should be equal. Determine if there is an association between attending preschool and algebra achievement.