Can You = Pass The AP Statistics Exam By Guessing?

By:

Krissy Holl, Chris Kirchner, & Clay Pear&= nbsp;

May 18, 2006

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 th= is in mind if you use or report the results of this study.

Methodology

This study= was meant as a test to determine if, as a test taker of the multiple choice por= tion of the AP Statistics Exam, you would be better off simply leaving the probl= em blank or guessing on one of the five choices when you did not know the answ= er to a problem. Students are st= rongly encouraged to leave the problem blank if an answer is not know but, this experiment is to test if it is always the best option. Some may see the logic in that if = you guess and get it right, you get an entire point. However, if you get the question w= rong you only lose one-quarter of a point off of your entire score. A whole point, if guessed right, c= ould significantly boost your score in some cases.

To test the effects of guessing without first eliminating any answers answers, a calculator simulation was conducted.&nbs= p; We chose the 2002 AP Statistics Exam at random, and used the multiple choice portion key as the key to our exam.= Using the random number generator on the calculator, we then generat= ed a simple random sample of 50 responses, chosen from A through E (labeled 1-5)= at random for the 40 questions on this portion of the exam. These responses were then “graded” using the 2002 AP Statistics multiple choice portion k= ey for the exam.

The actual= AP Statistic multiple choice portion of the exam is weighted to make the scores out of an even 50 points, and thus the “grading” of our exams w= as a little more involved. A formu= la was used to then get this weighted score: (number correct – .25 incorrect= ) * 1.25. The proportions were la= beled according to the weighted score, p₁ through p₅ with weighted scores 0-10, 11-20, 21-30, 31-40, and 41-50, respectively. To compare the sample proportions = to the actual population proportion, we then conducted a chi-square goodness of fit test and were able to draw conclusions upon our results to answer the quest= ion.

It may be considered that the exam was not exactly an exam, and therefore the conditions of the test were not entirely realistic. Subjects were not used, and it was= not given to them in a typical, controlled AP testing environment. However, the simulation was exactly that, to simulate these subjects, taking the test, with no idea of which options to guess. It is almost better a simulation was used because if subjects were brought into the test, there may have been some new biases introduced, such as nonresponse bias, or other such confounding and lurking variables. Because of the random nature of the tests, there was little or no bias present.

Exploratory Data Analysis

2002 AP Statistics Multiple C= hoice Portion

Weighted Score

% of Total Test Takers

Who Scored That Amount

Simulated % of Students With Weighted Score based on Guessing With 5 Choices

0 to 10

10.5

100

11 to 20

28.5

21 to 30

31 to 40

20.6

Goodness of Fit

Weighted Score

Observed Count

Expected

Count=

0 to 10

50

5.25<= /p>

11 to 20

0

14.25=

21 to 30

0

17.5<= /p>

31 to 40

0

10.3<= /p>

41 to 50

0

2.7

41 to 50

5.4

Chi-Square Distribution (Density Curve)

Analysis and Inference/Conclusion

α =3D .05

H₀ : p₁= =3D.054, p₂=3D.206, p₃=3D.350, p₄=3D.285, p₅=3D.105_<=
/o:p>

H_a=: At least one of the proportions is different

Assumptions for Chi- Square= Tests:

ü All expected counts are at least (≥) 1=

ü No more than 20% of the expected counts are = < 5

Expected Counts:

p₁=3D(.054*50)= =3D2.7, p₂=3D(.206*50)=3D10.3, p₃=3D(.350*50)=3D17.5, p₄=3D(.285*50)=3D14.25, p_=
5=3D(.105*50)=3D5.25

3D"Text

Chi-Square Test Statistic =3D = (Observed Count-Expected Count)² = &nb= sp; = &nb= sp; = = &nb= sp; = &nb= sp; = &nb= sp; &= nbsp; &nbs= p; (Expected Count)

χ² =3D (50-= 5.25)²/5.25 =3D 381.4405 + (0-14.25)²/14.25 =3D= 14.25 + (0-17.5)²/17.5 =3D 17.5 + (0-10.3)²/10.3 =3D 10.3 + (0-2.7)²/2.7 =3D <= span class=3DGramE>2.7 =3D<= span style=3D'mso-spacerun:yes'> 426.190476

The larges= t χ² component came from the first statistic =3D 381.4404761. This statistic is a result of the greatest difference between the observed and expected counts of this propor= tion and therefore guessing produces a significantly different distribution. This difference can clearly be see= n on the graphs, that it is more likely for students, when guessing on the AP Statistics multiple choice portion of the exam, to receive a weighted score between 0 and 10, and less likely, or nearly impossible for them to pass (receive a 3) the entire exam.

df =3D (n-1) =3D (5-1) =3D 4

The P-value is P(χ²≥ Χ²) =3D 6.08864963E-91, p≈0

With a p-v= alue of 6.08864963E-91 the data is statistically significant at the generally accep= ted α =3D .05 level. Therefo= re, we may reject the null hypothesis, the claim that the proportions of students = within each weighted score are as follows: 0-10 correct p₁=3D.054, 11-20 correct p₂=3D.206, 21-30 correct p₃=3D.350, 31-40 cor= rect p₄=3D.285, and 41-50 correct p₅=3D.105, in favor = of the alternative, the claim that at least one of those proportions is different.=

&n= bsp; This means that if students were to guess from one of the five options on each problem throughout the entire AP Statistics multiple choice portion of the exam, there would be very little chance that the proportions of their scores would result in the same proportions as those found from the students who actually took the test and attempted the questions with at least some knowledge. Therefore, the stu= dy has shown that guessing without first eliminating any of the options throughout= the entire exam can be very harmful to your score.

Using the formula stated = before, which is used to weight the scores of the AP multiple choice exams, when answering every question you must get 34 out of the 40 questions correct ju= st to pass (get a 3) the multiple choice portion. Not one of the simulated responses received a weighted score of 10 or above and some even received negative sc= ores (shown as 0’s on the data tables).&n= bsp; Therefore, it is highly suggested that you have some type of knowledge when entering the AP Statistics exams, if you plan to pass even just the multiple choice portion. Guessing will not get you far.

<= /span>Some weaknesses in our study include a somewhat small sample size, and also the = fact that it was a simulation. If = real subjects had been used in a real testing environment, the data may have had= a little more variability. Howe= ver, it would have been difficult to do so because of lack of time, materials, a= nd also there is no guarantee that none of the students will have any type of = knowledge of the material on the exam.

Chi-Square Distribution (Density Curve)

Analysis and Inference/Conclusion

Chi-Square Test Statistic =3D = (Observed Count-Expected Count)2 = &nb= sp; = &nb= sp; = = &nb= sp; = &nb= sp; = &nb= sp; &= nbsp; &nbs= p; (Expected Count)

Chi-Square Test Statistic =3D = (Observed Count-Expected Count)² = &nb= sp; = &nb= sp; = = &nb= sp; = &nb= sp; = &nb= sp; &= nbsp; &nbs= p; (Expected Count)