A personal trainer believes that drinking a glass of beetroot juice an hour before exercising enables endurance tests to be completed more quickly. To test his belief he takes a random sample of 12 of his trainees and, on two occasions, asks them to carry out 100 repetitions of a particular exercise as quickly as possible. Each trainee drinks a glass of water on one occasion and a glass of beetroot juice on the other occasion.
The times in seconds taken by the trainees are given in the table.
Trainee
Water
Beetroot juice
A
75.1
72.9
B
86.2
79.9
C
77.3
71.6
D
89.1
90.2
E
67.9
68.2
F
101.5
95.2
G
82.5
76.5
H
83.3
80.2
I
102.5
99.1
J
91.3
82.2
K
92.5
90.1
L
77.2
77.9
The trainer wishes to test his belief using a paired \(t\) test at the \(1 \%\) level of significance. Assuming any necessary assumptions are valid, carry out a test of the hypotheses \(\mathrm { H } _ { 0 } : \mu _ { D } = 0 , \mathrm { H } _ { 1 } : \mu _ { D } < 0\), where \(\mu _ { D }\) is the population mean difference in times (time with beetroot juice minus time with water).
An ornithologist believes that the number of birds landing on the bird feeding station in her garden in a given interval of time during the morning should follow a Poisson distribution. In order to test her belief, she makes the following observations in 60 randomly chosen minutes one morning.
Number of birds
0
1
2
3
4
5
6
\(\geqslant 7\)
Frequency
2
5
10
17
14
7
4
1
Given that the data in the table have a mean value of 3.3, use a goodness of fit test, with a significance level of \(5 \%\), to investigate whether the ornithologist is justified in her belief.