Three classes of students take an exam in statistics on the same day. In the first class there are 15 students ...

  

1) Assume that a sample is observed from a population such that X ≈ N(10; 16).
a) derive the sampling distribution of (n-1)s^2/(standard deviation^2) for a sample of dimension n = 20, s^2 being the unbiased estimator of the variance.
b) what is the probability that a sample variance exceeds 15 in this experiment?

2) Three classes of students take an exam in statistics on the same day. In the first class there are 15 students, 16 in the second and 14 in the third. The Professor distributes three different exam texts in the three classes. The grades obtained by the students in the midterm exam are reported here below (the maximum grade is 50).
 

Student ID. Class 1. Class 2. Class 3.
1    34 42 50
2    36 45 37
3    32 36 48
4    37 28 44
5    36 40 25
6    24 37 33
7    31 39 39
8    28 41 43
9    22 44 41
10  22 42 37
11  34 45 43
12  23 29 28
13  33 35 34
14  30 21 45
15  34 28 --
16  -- 44  --
 

i) Calculate the mean grade in the three different classes.
ii) Derive the 95% confidence interval for the mean grade in the three different classes.
iii) Compare the three intervals. What can you say ? Can you use this result to establish whether the three exam texts had the same level of difficulty?