In order to prevent and/or control the spread of infectious diseases, the Government has various vaccination programmes. One such programme requires people to receive a booster injection at the age of 18. It is felt that the proportion of people receiving this booster could be increased and a publicity campaign is undertaken for this purpose. In order to assess the effectiveness of this campaign, health authorities across the country are asked to report the percentage of 18-year-olds receiving the booster before and after the campaign. The results for a randomly chosen sample of 9 authorities are as follows.
Authority
A
B
C
D
E
F
G
H
I
Before
76
98
88
81
86
84
83
93
80
After
82
97
93
77
83
95
91
95
89
This sample is to be tested to see whether the campaign appears to have been successful in raising the percentage receiving the booster.
Explain why the use of paired data is appropriate in this context.
Carry out an appropriate Wilcoxon signed rank test using these data, at the \(5 \%\) significance level.
Benford's Law predicts the following probability distribution for the first significant digit in some large data sets.
Digit
1
2
3
4
5
6
7
8
9
Probability
0.301
0.176
0.125
0.097
0.079
0.067
0.058
0.051
0.046
On one particular day, the first significant digits of the stock market prices of the shares of a random sample of 200 companies gave the following results.
Digit
1
2
3
4
5
6
7
8
9
Frequency
55
34
27
16
15
17
12
15
9
Test at the \(10 \%\) level of significance whether Benford's Law provides a reasonable model in the context of share prices.