1997 was the first year for Advanced Placement Statistics nationwide. Wachusett
ID: 3228893 • Letter: 1
Question
1997 was the first year for Advanced Placement Statistics nationwide. Wachusett Regional High School is considered a leading public school in central Massachusetts. How well did Wachusett's students do on the AP* exam? Here are the summary statistics reported by College Board for 2013's AP Statistics and AB Calculus exams, as well as a breakdown of the AP* Stat scores by gender. Some questions to consider: 1. How did WRHS AP* Stat scores compare to the national results? 2. How did AP* Stat students do compared to the AB Calc students? 3. Do these results indicate that WRHS males and females may perform differently on the AP* Statistics exam? Write a report assessing the success the 2012 group of AP* Statistics students with respect to the questions above. Indicate hypotheses you test and the types of tests you do (goodness of fit, homogeneity, independence), show appropriate calculations and results, and clearly state your conclusions.Explanation / Answer
1)
H0: The group scores are identically distributed as the national results.
H1: The group scores are not identically distributed as the national results.
We have the total number of candidates and also the % of candidates in each grade. Hence, we can calculate the observed and expected as under:-
National (Expected)
Group (Observed)
Total
5
5.544
14
12.89753535
4
8.888
15
4.203031503
3
11
12
0.090909091
2
8.272
1
6.392889749
1
10.296
2
6.684500389
Total
X^2STAT
30.26886608
Please note that in this case, the observed will be of group and expected will be of national. When calculated, we get the above values. The formula for X^2STAT is (fo-fe)^2/fe. Considering that, we get the X^2STAT value as 30.27. The critical value at 0.05 would be 9.4877. Here critical value is considered with (r-1)(c-1) DF. Hence, DF is (5-1)(2-1)=4. Since, X^2STAT is greater than critical value, we reject the null hypothesis.
2)
H0: The scores of Stats students are identically distributed as the scores of Calc results.
H1: The scores of Stats students are not identically distributed as the scores of Calc results.
Observed
Calculus AB
Statistics
Total
5
41
14
55
4
17
15
32
3
19
12
31
2,1
18
3
21
Total
95
44
139
Expected
Calculus AB
Statistics
5
37.58992806
17.41007194
0.309353895
4
21.8705036
10.1294964
1.084648334
3
21.18705036
9.81294964
0.225760037
2,1
14.35251799
6.647482014
0.926954076
X^2STAT
2.546716342
Expected value is calculated as (95*55)/139, (95*32)/139 and so on.
Here, we are combining 2 and 1 because the expected value if separated is less than 5 and hence, goodness of fit cannot be tested if that is the case. Critical value with DF=3 at 0.05 is 7.8147. Since, 2.5467 is less than 7.8147, we cannot reject the null hypothesis.
3)
H0: The scores of male students are identically distributed as the scores of female results.
H1: The scores of male students are not identically distributed as the scores of female results.
Observed
Male
Female
5
5
9
14
4
10
5
15
3,2,1
10
5
15
25
19
44
Expected
Male
Female
5
7.954545455
6.045454545
1.097402597
4
8.522727273
6.477272727
0.256060606
3,2,1
8.522727273
6.477272727
0.256060606
X^2STAT
1.60952381
Expected value is calculated as (25*14)/44, (25*15)/44 and so on.
Here, we are combining 3, 2 and 1 because the expected value if separated is less than 5 and hence, goodness of fit cannot be tested if that is the case. Critical value with DF=2 at 0.05 is 5.99. Since, 1.609 is less than 5.99, we cannot reject the null hypothesis.
Here, based on the above results, we can see that the group scores are not identically distributed as the national results. Looking at the numbers of observed and expected, we can say that group performed better than the national average. Whereas, when comparing stats students to calc and male and female students, we can say that they are similar or identically distributed.
National (Expected)
Group (Observed)
Total
5
5.544
14
12.89753535
4
8.888
15
4.203031503
3
11
12
0.090909091
2
8.272
1
6.392889749
1
10.296
2
6.684500389
Total
X^2STAT
30.26886608
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