(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: = 26 / 39 = 0.667
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 = 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 < , 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 < (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 s_{1}/s_{2} = 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)
The Hypothesis:
This is a Two tailed test.
The Test Statistic:We use the students t test as population standard deviations are unknown.
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 = 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 < , 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 > (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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