Macintosh Users: For a more readable version download the original lesson in Clarisworks 4.0 or higher.

**Grade Level:** High School.

**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.

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

Biased

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

Biased

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

Biased

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

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

Boys 15 12 3 9 0.75

Boys 18 12 6 36 3

Boys 20 12 8 64 5.333

Experiment 1:

Modeled with

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

Modeled with

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

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)

1. Number of female students in class

2. Number of male students in class

3. Total number of students in class

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

Boys 13 18 -5 25 1.389

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

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

6. Make a Decision: If P(X2

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