Seth Broster

Alex Frost

Jordan Woolum

 

 

Disclaimer: This study was done in an AP statistics course with relatively small sample sizes.  The validity of such studies must always be questioned.  Please keep this in mind is you use or report the results of this study.

Abstract: This study was done in order to test the hypothesis that there is a difference in basic algebra skills between freshman and seniors at Tates Creek High School (that the mean score of seniors would be higher than the mean score of freshman).  This hypothesis was tested with a two sample t-test, in which we used the mean scores on a five question algebra test that was administered to a sample from both populations (freshman and seniors).

     This particular experiment was conducted was because of our long standing assumption that seniors are more generally adept in all subject areas (including math) than freshman because they have more experience with the subject matter and have had more chances to use and apply what they have learned.  

     As all students in both populations will have completed a basic Algebra 1 course, because the completion of this course during freshman year is required by the Fayette County School Board, the test that was administered should be within comprehension for all of those tested.  

Methodology:

Sampling: In order to get a truly random sample for each population for testing, we assigned every freshman and every senior a number and used a random number generator to generate a set of forty numbers for each population.  We then tested those students whose numbers came up in the sets that we generated.   If there was a repeated number we simply added one more random number to the list and used the student with the corresponding number in out sample.  If the student no longer attended the school, we flipped a coin and if it was heads we took the person whose name was before the absent student (the names were in alphabetical order), if it was tails we took the person whose name was after the absent student.

Testing: The test consisted of five questions over material that should have been covered in freshman algebra classes.  They were as follows:

1.      Find “y”, given x =2:  y = x+3

2.      Find the slope of the line containing these 2 points:  (0,3) & (4,7)

3.      Find the X-intercepts of  y = x2 – 4

4.      Simplify: (x2 )

5.      Rationalize:

All students were given two minutes to complete the test.  If the student was not finished by the end of the two minutes they were asked to stop and the test was scored “as is”.

 

Results:

 

The scores on the math test are represented in the table and on the graph below:

Seniors

Freshmen

5

2

4

2

1

1

1

0

2

2

1

3

2

3

3

1

3

2

4

2

4

4

5

1

5

1

5

1

2

1

2

2

5

2

4

1

5

2

5

1

3

3

5

2

5

3

5

1

4

2

5

2

2

2

3

1

5

1

 -

3

 -

2

 -

1

 -

1

 -

1

avg = 3.6207

avg = 1.7353

St. Dev = 1.4495

St. Dev = .8637

 

 

 

 

 

 

 

 

 

 

 

 

 

     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Mean Score’s of Freshman and Senior Samples

 

           

 

 

 

 

 

 

 

 

 

 

 

 

 

Seniors

n1 = sample size = 29

Sample Mean 1 (µ1) = Total of Scores / n = 3.620689655

Sx1 (Sample Standard Deviation) = 1.449477569

 
 

Freshman

n2 = sample size = 34

Sample Mean 2 (µ2) = Total of Scores / n = 1.735294118

Sx2 (Sample Standard Deviation) = 0.8637067238

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Analysis and Inference: With the information that we now have, we can perform a two sample t-test, based on the assumption:

 

Notes:

Significance Level α = 0.05

Population one: Seniors

Population two: Freshman

 

Assumptions:

·         Both samples are SRS’s from distinct populations.

·         Both samples are independent

·         Both Populations are normally distributed (n ≥ 40)

o   n1 = 29 < 40

            * t is robust, proceed with caution

o   n2 = 34 < 40

      * t is robust, proceed with caution

 

Hypotheses:

H0: The mean of the Senior scores will be the same as the mean of the Freshman scores

(µ1 = µ2)

Ha: The mean of the Senior scores will be greater than the mean of the Freshman scores (µ1 > µ2)

 

Test:

 

t =                           (sample mean 1 – sample mean 2) – (µ1 - µ2)                               .                        

            Square root (Sample standard deviation 1) + (Sample standard deviation 2)

                                         Sample Size 1                                   Sample Size 2

 

t =                           (3.620689655– 1.735294118) – (0)                               .                        

                      Square root[(1.449477569) + (0.8637067238)]

                                                    29                          34

 

t = 6.6136809439

p = 1.0533209 X 1010

 

 

 

 

 

Conclusion: Because our data gives a p-value of 1.0533209 X 1010 , our data is statistically significant at the α = 0.05 level and we can reject our null hypothesis (H0) in favor of the alternative hypothesis Ha; that the mean of the Senior scores will be greater than the mean of the Freshman scores (µ1 > µ2). 

 

Our conclusion, based on the test that we administered, is that seniors have generally better algebra one skills than freshman do.

 

Possible Sources of Error:

There was some non-response bias in our experiment.  When we were testing we were not able to find all of the students that were in our samples so they were not tested; this resulted in the non-response bias.  This bias may have lowered the averages of the samples (because students that don’t regularly attend school would likely score lower on the test than students that do regularly attend school)  leading to results that are slightly skewed one way or the other depending on which sample the non-responder was in.