1. If six 6-sided dice are rolled,
a) What are the minimum, maximum and expected values for the total?
b) Find the probabilities that the total will equal the following values:
(i) 6 + b
(ii) 12 + c
(iii) 18 + d
(iv) 24 + e
c) What is the probability of matching or exceeding the values in (i), (ii) and (iii) if only five dice are used?
d) In terms of minimum, maximum, expected and probabilities of the total, what are differences in rolling four 8-sided dice compared with eight 4-sided dice?
2. Use your f value to choose a month according to the table below:
|
f
|
Month
|
|
0
|
January
|
|
1
|
February
|
|
2
|
March
|
|
3
|
April
|
|
4
|
May
|
|
5
|
June
|
|
6
|
July
|
|
7
|
August
|
|
8
|
September
|
|
9
|
October
|
a) Use the data for hours of sunshine per month for your particular month, to draw a graph with the years plotted on the horizontal axis, and the hours of sunlight on the vertical axis.
b) Calculate the mean and standard deviation for your month
c) Draw a second graph showing both your month and the following month.
d) Replot the graph from c) using a seven year moving average on both sets of data.
e) In no more than 100 words, compare the graphs in c) and d).
3. Using your f value, select 20 years of sunshine data from July and August, with the first year being 198f (i.e. 1980+f)
a) Use the Mann-Whitney test to investigate whether there is a difference between the medians of the hours of sunshine.
b) Use an appropriate statistical test to investigate whether there is a significant difference between the means of the hours of sunshine.
c) Investigate how your results for a) and b) change by artificially introducing errors to your data.
4. The table below shows minimum and maximum daily river levels in metres on the Mersey published by www.riverlevels.uk. Change these values using your ID number as indicated before proceeding.
|
Date
|
Minimum
|
Maximum
|
|
01/06/2016
|
2.05
|
8.32
|
|
02/06/2016
|
1.52
|
3.66
|
|
03/06/2016
|
1.88-0.1*a
|
2
|
|
04/06/2016
|
1.25
|
2.71
|
|
05/06/2016
|
1.5
|
3.83
|
|
06/06/2016
|
2.32
|
5.18+0.1*b
|
|
07/06/2016
|
3.42
|
6.6
|
|
08/06/2016
|
4.66
|
7.8
|
|
09/06/2016
|
5.89-0.1*c
|
8.49
|
|
10/06/2016
|
7.08
|
8.52
|
|
11/06/2016
|
7.89
|
7.91
|
|
12/06/2016
|
6.99
|
8.18+0.1*d
|
|
13/06/2016
|
5.62
|
7.78
|
|
14/06/2016
|
4.18
|
6.95
|
|
15/06/2016
|
3.28-0.1*e
|
5.81
|
|
16/06/2016
|
2.71
|
4.55
|
|
17/06/2016
|
2.59
|
3.3
|
|
18/06/2016
|
2.51
|
2.85+0.1*f
|
|
19/06/2016
|
2.15
|
3.39
|
|
20/06/2016
|
2.48
|
4.15
|
Draw a correctly labelled scatter diagram with the minimum depth on the horizontal axis, and the maximum depth on the vertical axis, find the correlation coefficient, and calculate the equation of the line of best fit.
Double your maximum depth values and then recalculate the correlation coefficient and the equation of the line of best fit. Discuss these results.
5. A survey has been carried out to investigate preferences on the following alternative uses of farmland.
• Biofuels for transport
• Photovoltaic arrays
• Wind turbines
Each respondent has been asked to pick which option they would prefer. The numbers for each option have been split into age ranges, as shown in the table.
|
Age range (years)
|
Biofuels for transport
|
Photovoltaic arrays
|
Wind turbines
|
|
< 40
|
22 + a
|
64
|
83
|
|
40 - 60
|
52
|
78 - b
|
46+c
|
|
> 60
|
31 + e
|
56 - f
|
19
|
You are to use the chi squared test to assess whether different age ranges responded differently to the questions.