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The given data is the grades for people in this class. The goal here is to determine the factors that effect students Grade
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Answer #1

(a) There are 26 F and 13 Males out of a total of 39 Students

The Mens Quizzes scores are as below

1 M 55
2 M 86
3 M 58
4 M 70
5 M 83
6 M 80
7 M 49
8 M 55
9 M 67
10 M 74
11 M 81
12 M 78
13 M 84

Average = Sum of observations / Total Observations = 920 / 13 = 70.77

The Median. Since n is odd, the median = (n + 1)/ 2 th number = (13 + 1) / 2 = 7th number = 49

For Females

1 F 61
2 F 59
3 F 54
4 F 62
5 F 49
6 F 73
7 F 66
8 F 66
9 F 73
10 F 75
11 F 69
12 F 66
13 F 87
14 F 74
15 F 78
16 F 91
17 F 67
18 F 66
19 F 89
20 F 77
21 F 63
22 F 87
23 F 82
24 F 81
25 F 81
26 F 92

Average = Sum of observations / Total Observations = 1888 / 26 = 72.62

The Median = Middle value

Since n is even, the median = Average of n/2th and the next number = Average of 26/2 = 13th and 14th numbers

= (87 +74) / 2 = 80.5

Men Women
Mean 70.8 72.62
Median 49 80.5

________________________________________

(a) Test For proportion

The Hypothesis:

p = 0.5

p > 0.5

This is a right tailed test

The Test Statistic: \hat{p} = 26 / 39 = 0.667

Z = \frac{\hat{p}-p}{\sqrt{\frac{p(1-p)}{n}}} = \frac{0.667-0.5}{\sqrt{\frac{0.5*0.5}{39}}} = 2.8

The p Value:    The p value (Right tail) for Z = 2.08, is; p value = 0.0188

The Critical Value:   The critical value (Right tail) at \alpha = 0.05 (default level) Z critical = +1.645

The Decision Rule: If Z observed is > Z critical Then Reject H0.

Also If the P value is < \alpha , Then Reject H0

The Decision:    Since Z observed (2.08) is > Z critical (1.645), We Reject H0.

Also since P value (0.0188) is < \alpha (0.05), We Reject H0.

The Conclusion:   There is sufficient evidence at the 95% significance level to conclude that the majority of students in the class are women

____________________________________________

For Grades the sample statistics are as below

Men Women
Total 937 1898
n 13 26
Mean 72.08 73
SD 8.391 8.841

Since s1/s2 = 8.391 / 8.841 = 0.949 (it lies between 0.5 and 2) we used the pooled variance.

The degrees of freedom used is n1 + n2 - 2 = 13 + 26 - 2 = 37 (since pooled variance is used)

S_p^2 = \frac{(n_1-1)*s_1^2+(n_2-1)*s_2^2}{n_1+n_2-2} = \frac{12*8.391^2+25*8.841^2}{26+13-2} = 75.648

The Hypothesis:

H0: \mu_1 = \mu_2

Ha: \mu_1 \neq \mu_2

This is a Two tailed test.

The Test Statistic:We use the students t test as population standard deviations are unknown.

t = \frac{\bar{x}_1-\bar{x}_2}{\sqrt{S_p^2*(\frac{1}{n1}+\frac{1}{n2})}} = \frac{72.08-73}{\sqrt{75.648*(\frac{1}{26}+\frac{1}{13})}} = -0.32

The p Value:   The p value (2 Tail) for t = -0.32, df = 37, is; p value = 0.7575

The Critical Value:   The critical value (2 tail) at \alpha = 0.05 (default level), df = 37, t critical = - 2.03 and + 2.03

The Decision Rule: If t observed is > t critical or If   t observed is < -t critical, Then Reject H0.

Also If the P value is < \alpha , Then Reject H0

The Decision: Since t lies in between -2.03 and +2.03, We Fail To Reject H0

Also since P value (0.7575) is > \alpha (0.05), We Fail to Reject H0.

The Conclusion: There isn’t sufficient evidence at the 95% significance level to warrant rejection of the claim that Men and Women have the same mean grade.

___________________________________________

Calculation for the mean and standard deviation:

Mean = Sum of observation / Total Observations

Standard deviation = SQRT(Variance)

Variance = Sum Of Squares (SS) / n - 1, where SS = SUM(X - Mean)2.

Men
# X X - Mean (X - Mean)2
1 55 72.08 291.7264
2 72 72.08 0.0064
3 73 72.08 0.8464
4 70 72.08 4.3264
5 64 72.08 65.2864
6 75 72.08 8.5264
7 62 72.08 101.6064
8 68 72.08 16.6464
9 72 72.08 0.0064
10 82 72.08 98.4064
11 82 72.08 98.4064
12 81 72.08 79.5664
13 81 72.08 79.5664
Total 937 844.923
Women
# X X - Mean (X - Mean)2
1 59 73 196
2 63 73 100
3 65 73 64
4 66 73 49
5 61 73 144
6 59 73 196
7 59 73 196
8 67 73 36
9 70 73 9
10 74 73 1
11 72 73 1
12 72 73 1
13 79 73 36
14 75 73 4
15 73 73 0
16 88 73 225
17 76 73 9
18 73 73 0
19 78 73 25
20 83 73 100
21 73 73 0
22 85 73 144
23 80 73 49
24 81 73 64
25 77 73 16
26 90 73 289
Total 1898 1954
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