Katie, Sarah

Is there a relationship between a student’= s GPA and whether or not they own a graphing calculator?

= &nb= sp; <= o:p>

B-1

AP Statistics

Spring 2006

Sarah Chandler

Katie Officer

Jay Embree

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 if you us= e or report the results of this study.”

Abstract: <= /u> &nbs= p;

The resear= ch question for this survey was whether the ownership of a graphing calculator= is related with student GPAs. Th= e expected outcome for this survey was that ownership of a graphing calculator does af= fect student GPAs. This outcome was expected because it was assumed that a graphing calculator would be require= d by students in upper level courses such as Advanced and Advanced Placement (AP). In these courses, stude= nts typically earn better grades and have higher GPAs.

The population included freshmen (class of 2009), sophomores (class of 2008), a= nd juniors (class of 2007) at Tates<= /st1:PlaceName> Creek High School. The survey was originally stratifi= ed by grade. Students were randomly selected from each class roster.

The resear= chers gave all selected students a survey that asked for their cumulative unweighted G= PA and whether they owned a graphing calculator.

The data w= as categorized using a two-way table and inference was conducted using a chi-square test for independence. The chi-square test was performed for each grade level and after loo= king at the expected counts, the researchers realized that both assumptions had = been violated. Since chi square te= sts are not robust, a change was made in how the data was grouped. Instead of being grouped by grade,= the researchers changed the parameters of the GPA ranges and grouped all classes together.

After the necessary changes were made, the chi square test passed both assumptions and showed a p value of 3.77 e-4 or essentially 0. This shows that there is a relatio= nship between student GPA and the ownership of a graphing calculator.

Methodology:<= span style=3D'mso-spacerun:yes'>

Class rost= ers were obtained for the classes of 2007 (Juniors), 2008 (Sophomores) and 2009 (Freshmen) at Tates Creek High School in May 2006 and each student was assig= ned a number. A random integer generator function on a graphing calculator was used to select 10% of the s= tudents. The entire population was used for selection. This test was conducted to pr= ove there is a relationship between student GPAs and ownership of a graphing calculator. The population fo= r the class of 2007 was 342 students while the populations for the classes of 2008 and 2009 were 438 and 442 respectively.&nb= sp; As a result, the freshman class had 44 students in the sample while = the sophomores sample had 43 students and the junior sample contained 34 students.

A survey w= as constructed which asked students for their cumulative unweighted GPAs, grade level and whether or not they owned a graphing calculator. Students were located by their sch= edules and the researchers took the surveys to the students in their classes. Students answered the surveys, and= the surveys were collected. The d= ata was sorted using a two- way table. However, due to non-response from absent students, not all of the students selected to be sampled were able to be reached.

Data Analysis Section

ORIGINAL

The data was originally grouped in the following manne= r:

Two Way Tables (Observed Counts):

<= o:p>	<= o:p>	Freshman	<= o:p>	<= o:p>
<= o:p>	<= o:p>	<= o:p>	<= o:p>	<= o:p>
&= nbsp;	0.0-1.0	1.1-2.0	2.1-3.0	3.1-4.0
O= wns a Graphing Calculator	0	0	8	13
D= oesn't Own a Graphing Calculator	1	5	8	5
<= o:p>	<= o:p>	<= o:p>	<= o:p>	<= o:p>
<= o:p>	<= o:p>	<= o:p>	<= o:p>	<= o:p>
<= o:p>	<= o:p>	<= o:p>	<= o:p>	<= o:p>
<= o:p>	<= o:p>	Sophomores=	<= o:p>	<= o:p>
<= o:p>	<= o:p>	<= o:p>	<= o:p>	<= o:p>
&= nbsp;	0.0-1.0	1.1-2.0	2.1-3.0	3.1-4.0
O= wns a Graphing Calculator	0	0	13	14
D= oesn't Own a Graphing Calculator	1	2	7	3
<= o:p>	<= o:p>	<= o:p>	<= o:p>	<= o:p>
<= o:p>	<= o:p>	<= o:p>	<= o:p>	<= o:p>
<= o:p>	<= o:p>	<= o:p>	<= o:p>	<= o:p>
<= o:p>	<= o:p>	<= o:p>	<= o:p>	<= o:p>
<= o:p>	<= o:p>	Juniors	<= o:p>	<= o:p>
<= o:p>	<= o:p>	<= o:p>	<= o:p>	<= o:p>
&= nbsp;	0.0-1.0	1.1-2.0	2.1-3.0	3.1-4.0
O= wns a Graphing Calculator	0	1	10	12
D= oesn't Own a Graphing Calculator	0	1	4	2

For this survey, a chi-squa= re test for independence will be conducted to determine whether or not an associati= on exists between student GPAs and ownership of a graphing calculator.

In order to conduct the chi square test for independen= ce, the researchers must find the expected counts for each grade level. Expected counts can be found using= the following formula:

&nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p;

Expected = counts=3D(row total)(column total)<= /p>

= (table total)

Two-Way Table= s (Expected Count):

Using the original grouping system, the expected count= s were as follows:

<= o:p>	<= o:p>	Freshman	<= o:p>	<= o:p>
<= o:p>	<= o:p>	G= PAs	<= o:p>	<= o:p>
&= nbsp;	0.0-1.0	1.1-2.0	2.1-3.0	3.1-4.0
O= wns a Graphing Calculator	0.525	2.625	8.4	9.45
D= oesn't Own a Graphing Calculator	0.475	2.375	7.6	8.55
<= o:p>	<= o:p>	<= o:p>	<= o:p>	<= o:p>
<= o:p>	<= o:p>	<= o:p>	<= o:p>	<= o:p>
<= o:p>	<= o:p>	<= o:p>	<= o:p>	<= o:p>
<= o:p>	<= o:p>	Sophomores=	<= o:p>	<= o:p>
<= o:p>	<= o:p>	G= PAs	<= o:p>	<= o:p>
&= nbsp;	0.0-1.0	1.1-2.0	2.1-3.0	3.1-4.0
O= wns a Graphing Calculator	0.675	1.35	13.5	11.475
D= oesn't Own a Graphing Calculator	0.325	0.65	6.5	5.525

It was at this point that the researchers realized that there was a problem with the data. The junior data would not work because when the expected counts were being calculated, there was a zero in two way table. It violates the assumption that all expected counts must be greater than 1.

However, the graphs are as follows:

Graph on Freshman Data<= /span>

<= /p>

Graph on Sophomore Data=

The researchers also realized that all their tests fai= led both assumptions for the chi square test.&= nbsp; The assumptions are:

1. All expected counts are greater than 1.

2. No more than 20% of expected counts are less than 5.

In order to pass both assumptions, the researchers re categorized their data. Inste= ad of grouping by grades, all grades were combined. The GPA ranges were changed to 2.5= and under, 2.51-3.5, 3.51-4.0. Af= ter re tallying, the observed counts were as follows:

Observed Counts

&= nbsp;	2= .5 and under	2= .51-3.5	3= .51-4.0
O= wns a Graphing Calculator	11	43	16
D= oesn't Own a Graphing Calculator	20	17	3

In order to conduct the chi square test for independen= ce, the researchers must find the expected counts. Expected counts can be found using= the following formula:

&nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p;

Expected = counts=3D(row total)(column total)<= /p>

= (table total)

Expected Counts

	2.5 and under	2.51-3.5	3.51-4.0
Owns a Graphing Calculator	19.7	38.2	12.1
Doesn't Own a Graphing Calcu= lator	11.3	21.8	6.9

After the changes were made, the new graphs were as fo= llows:

<= /o:p>

<= /p>

Inference Section

“H_ois the null hypothesis, which = states that there is no effect or change in the population. If the null hypothesis is true= , the sample data is just chance at work” ¹

= ;

Significance = Test:

Hypotheses:

H_o= =3D There is no relationship between the ownership of a graphing calculator and stude= nt GPAs.

H_ais = the alternative hypothesis, which states the alternative to the null hypothesis. ²

H_a= =3D There is a relationship between the ownership of a graphing calculator and student GPAs.

Assumptions:<= o:p>

19.7, 38.2, 12.1, 11.3, 21= .8, 6.9 > 1

= 2.&n= bsp;    No more than 20% of the expected counts are = less than 5.

19.7, 38.2, 12.1, 11.3, 21= .8, 6.9 >5

¹ Definition taken from “The Practice of Statistics” by Yates, Moore and Starnes, published in 2002 by Texas Instruments.

²<= span style=3D'font-size:10.0pt'>Definition taken from“The Prac= tice of Statistics” by Yates, Moore and Starnes, published in 2002 by Texas Instruments.

Chi Square Te= st for Independence<= /o:p>

Χ^{2=3D Σ}  &nb= sp; (Observed-Expected)²

^{&=
nbsp;           =

Expected}

Example: = (11-19.7)²

^{&nbs=
p;            &=
nbsp;}19.= 7         =3D   3.84

^{Degrees of Freedom}^:

(row-1<= /span>)(column-1)

(2-1)(3-1)=3D 2 degrees of freedom

After completing the chi square test, the results were= as follows:

Χ²=3D 15.7651

P=3D 3.7727 x 10^-4

Since the = x²=3D 15.7651 and x^2*=3D 9.21 the researchers can conclude that their = data is statistically significant at the α=3D .01 level. Therefore the researchers can reje= ct H_oin favor of H_swhich means that there is a relationship between student GPAs and whether or not they own a graphing calculator.

Possible Conf= ounding:
- Weaknesses:
  
  The weaknesses that most influenced this survey included students being unknowledgeable about their GPAs, the sample size was small, students were = absent and could not be surveyed, and seniors were not included.