### Gender Issues through Chi-square Experiments

by: Chris Povich

Special Note: For this lesson you may need to download the program Resampling Statistics.

NCTM Standards: 1-1 Use, with increasing confidence, problem-solving approaches to investigate and understand mathematical content.
2-3 Express mathematical ideas orally and in writing. 11-3 Use simulations to estimate probabilities.
11-6 Apply the concept of a random variable to generate and interpret probability distributions including Chi-square.

Day One Committee Investigation.

Part 1: Your group has been created as an investigating committee for the local school board. Your school board has been discussing gender issues within mathematics and society. Your group has been appointed to observe three teachers teaching in the district to see if there exists any bias towards calling on one gender more than another. Three teachers each teach the same twenty students at different time periods. The class consists of twelve girls and eight boys. Your observations are to count the first thirty times the teacher calls on students and mark whether the teacher called on a girl, or a boy. Suppose the following three situations were the results of your observation. Within your group, discuss if the teacher was being biased towards calling on one gender over the other.

Sample:

Situation 1. The teacher calls on girls fifteen times and on boys fifteen times.

Biased Unbiased x

Situation 2. The teacher calls on girls twelve times and on boys eighteen times.

Biased x Unbiased

Situation 3. The teacher calls on girls ten times and on boys twenty times.

Biased x Unbiased

Now see if the previous situations were due to the teacher being biased, or possibly due to chance.

Part 2: If the teacher was being "fair," what would be the expected number of times she/he would call on girls and on boys?

Sample:

1. Expected number of times calling on girls (out of thirty questions) = 18

(12/20)30=18.

2. Expected number of times calling on boys (out of thirty questions) = 12

(8/20)30=12.

Part 3: Use a spreadsheet to calculate the Chi-square for each of the three situations in Part 1.

Sample:

Situation 1.

###### Gender Called on f (o) Expected f (e) o-e (o-e)^2 ((o-e)^2)/e
Girls 15 18 -3 9 0.5
Boys 15 12 3 9 0.75

30 30 0 18 1.25

Situation 2.
###### Gender Called on f (o) Expected f (e) o-e (o-e)^2 ((o-e)^2)/e
Girls 12 18 -6 36 2
Boys 18 12 6 36 3

30 30 0 72 5

Situation 3.
###### Gender Called on f (o) Expected f (e) o-e (o-e)^2 ((o-e)^2)/e
Girls 10 18 -8 64 3.556
Boys 20 12 8 64 5.333

###### Total
30 30 0 128 8.889

###### X^2 = 8.889

Part 4: Each person in the group needs to perform two random experiments using the equipment (dice, spinners, bags of blocks, TI-82 Random Functions, etc.) provided. You need to model the probabilities of calling on twelve girls versus calling on eight boys with thirty questions total. Record the results in the following Chi-square tables.

Experiment 1:

Modeled with . Why is this model appropriate?

Chi-square from Experiment
Gender Observed Expected (O-E) (O-E)^2 (O-E)^2/E

Girls

Boys

Chi Square = Sum (O-E)^2/E =

Experiment 2:

Modeled with . Why is this model appropriate?

Chi-square from Experiment
Gender Observed Expected (O-E) (O-E)^2 (O-E)^2/E

Girls

Boys

Chi Square = Sum (O-E)^2/E =

Part 5: Enter the class results in the stem and leaf diagram down below. Then determine the probabilities of generating the Chi-squares in Part 1 through random chance.

Sample:

Part 6: Now calculate the probability of getting a Chi-square as large, or larger, than the Chi-square in each situation.

Sample:

Calculate the following probabilities:
(the # of Chi-squares 1.25, 11)
P(Chi-square is 1.25) = ----------------------------------------- = .37.
(the total # of trials, 30)

(the # of Chi-squares 5.00, 5)
P(Chi-square is 5.00) = ----------------------------------------- = .17.
(the total # of trials, 30)

(the # of Chi-squares 8.889, 1)
P(Ch-square is 8.889) = ----------------------------------------- = .03.
(the total # of trials, 30)

Assessment: Explain in your journal your results and if there exist persuasive evidence that a/the teacher(s) was/were being biased towards calling on one gender over another. (Make sure to write a convincing argument that explains your evidence for each situation.)

Day Two Committee Field Work.

Part 1: Groups will observe different teachers throughout the school. Each group will be responsible for gathering the following information from their observation of one teacher.

Sample:

1. Number of female students in class 14 .

2. Number of male students in class 21 .

3. Total number of students in class 35 .

4. Record the gender the teacher calls on for the first thirty times:

Sample:

Sample:

###### Gender Called on f (o) Expected f (e) o-e (o-e)^2 ((o-e)^2)/e
Girls 17 12 5 25 2.083
Boys 13 18 -5 25 1.389

30 30 0 50 3.472

###### X^2 = 3.472

Part 3: Write a program using Resampling Stats to test if the teacher preferred to call on one gender over the other. (Make sure to run 100 trials in your program.) Use the six-step method to explain your program.

Sample:

1. Model: Generate 30 numbers 1-2 using, "copy (1 1 2 2 2) gender." Why? 1=girl, 2=boy, and p(1)=p(girl)=2/5. Also, the p(2)=p(boy)=3/5. The expected frequency for girls = (2/5)(30)=12 and the expected frequency for boys = (3/5)(30)=18. Now we will determine the P(X2> 3.472).

2. Trial: Generate 30 numbers 1-2. Why? There are thirty questions, therefore, one number for each question.

3. Outcomes: Record the frequency for the numbers 1 and 2 out of thirty generated numbers ranging from 1-2. Then calculate the Chi-square for the data of thirty generated numbers.

4. Repeat Trial: 100 times steps 2-3. Record 100 random Chi-squares.

5. Find Probability: Find P(X2> 3.472) = (# X2> 3.472)/(# of trials, 100).

6. Make a Decision: If P(X2> 3.472) > .05, then there is significant evidence that the obtained Chi-square, 3.472, is most likely due to chance. The Resampling Stats Program will then let "signif = 0." If P(X2> 3.472) < .05, then there is significant evidence that the obtained Chi-square, 3.472, is most likely due to the teacher preferring to call on one gender over another.
The Resampling Stats Program will then let "signif = 1."

Resampling Stats Program:
copy (1 1 2 2 2) gender 'Set probabilities, 1=g, 2=b
repeat 100 'Repeat 100 trials
shuffle gender class 'Shuffle order
generate 30 class x 'Generate 30 #'s 1-2
count x=1 girl 'Count the # girls
count x=2 boy 'Count the # boys
subtract girl 12 dgirl 'o-e for girls
subtract boy 18 dboy 'o-e for boys
square dgirl sdgirl 'square(o-e) girls
square dboy sdboy 'square(o-e) boys
divide sdgirl 12 e1 'square(o-e)/e girls
divide sdboy 18 e2 'square(o-e)/e boys
add e1 e2 total 'Calculate Chi-square
score total Chi 'Score Chi values
end
count Chi>=3.472 highChi 'Count Chi values>= 3.472
divide highChi 100 phighChi 'Probability of Chi-square
if phighChi >0.05 'If not significant,
set 1 0 signif 'give signif 0 value.
end
if phighChi <= 0.05 'If significant,
set 1 1 signif 'give signif 1 value.
end
print highChi phighChi signif 'Print #Chi>=3.472, prob(highChi), signif
histogram Chi 'Distribution of Chi-square values

Results of Resampling Stats Program:
Start execution.
HIGHCHI = 7
PHIGHCHI = 0.07
SIGNIF = 0
Vector no. 1: CHI
Bin Cum
Center Freq Pct Pct
--------------------------------------------
0 40 40.0 40.0
0.5 26 26.0 66.0
1.5 13 13.0 79.0
2 14 14.0 93.0
3.5 3 3.0 96.0
5 3 3.0 99.0
9 1 1.0 100.0
Note: Each bin covers all values within 0.25 of
its center.
Successful execution. (3.3 seconds)

Outcome of Resampling Stats Program: Out of 100 trials, there were seven times that the X2> 3.472. Therefore, the P(X2> 3.472)=.07, and was not significant, "SIGNIF = 0," according to .05 test. The teacher was found , in this test, to be biased towards calling on one gender over another. Notice this would have been different if we used the .10 test, though.

Assessment: Each group (committee) is responsible for creating a paper which will be presented to the teacher and school board of their findings. Students will be responsible for explaining their Chi-square test, calculations, and conclusion of results. Each group will also present their material to the class.

Extension: The class can pull their group information together and repeat Day 2 project with the combined data.