At a college, two examiners are responsible for marking, independently, the students' projects. Each examiner awards a mark out of 100 to each project. There is some concern that the examiners' marks do not agree, on average. Consequently a random sample of 12 projects is selected and the marks awarded to them are compared.
Describe how a random sample of projects should be chosen.
The marks given for the projects in the sample are as follows.
Project
1
2
3
4
5
6
7
8
9
10
11
12
Examiner A
58
37
72
78
67
77
62
41
80
60
65
70
Examiner B
73
47
74
71
78
96
54
27
97
73
60
66
Carry out a test at the \(10 \%\) level of significance of the hypotheses \(\mathrm { H } _ { 0 } : m = 0 , \mathrm { H } _ { 1 } : m \neq 0\), where \(m\) is the population median difference.
A calculator has a built-in random number function which can be used to generate a list of random digits. If it functions correctly then each digit is equally likely to be generated. When it was used to generate 100 random digits, the frequencies of the digits were as follows.
Digit
0
1
2
3
4
5
6
7
8
9
Frequency
6
8
11
14
12
9
15
5
14
6
Use a goodness of fit test, with a significance level of \(10 \%\), to investigate whether the random number function is generating digits with equal probability.