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Nuffield Free-Standing Mathematics Activity
Rain or shine?
Calculator activity
© Nuffield Foundation 2011
Parents sometimes say that it rains more in August
when children are not at school than it does in other
summer months.
Some people think Scotland is sunnier in spring
than England and Wales.
This activity will show you how to use a significance
test to decide whether or not such hypotheses are
likely to be true.
© Nuffield Foundation 2011
Monthly hours of sunshine England & Wales (2001–2010)
Year
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
Jan Feb Mar Apr
74.5 86.7 90.2 134.6
41.2 75.3 112.4 189.4
70.5 95.3 165.2 191.3
49.5 88.3 108.2 133.3
56.4 72.0 79.5 146.9
51.0 68.3 97.4 160.4
60.5 70.4 153.9 216.5
47.8 118.2 120.0 153.9
56.9 55.6 153.2 170.3
60.9 60.1 125.8 201.8
May
226.2
180.2
189.9
204.6
216.6
173.1
167.1
193.0
212.7
204.8
Jun
192.7
163.5
209.4
198.2
190.8
229.7
149.8
199.3
202.0
238.8
Jul
190.0
163.9
172.5
166.0
178.4
287.6
179.7
184.9
177.1
152.1
Aug
177.8
158.3
207.6
173.8
210.6
151.4
198.1
112.8
173.5
148.8
Sep
113.8
154.6
166.0
155.7
150.6
157.5
149.3
114.6
142.8
138.8
Oct
103.6
93.4
130.3
95.9
81.1
95.6
109.4
122.5
90.2
120.3
Nov
65.8
55.3
67.4
47.9
90.8
93.1
71.2
56.6
65.7
71.5
Dec
75.4
34.3
52.1
52.1
57.0
45.8
51.2
66.4
62.2
53.8
Think about
What are the three averages that can be used to represent data?
What measures of spread do you know?
Which is the most appropriate average and measure of spread to
use in this context?
© Nuffield Foundation 2011
To find the mean and standard deviation
x  nx
The sample mean is given by the formula:
The sample standard deviation is:
s=
x 2  x 2
n
The best estimator of the standard
s n–1 =
deviation of the population is:
n s
n 1
You can use a calculator or a spreadsheet to work out
the mean and standard deviation.
Try this Use the data on the information sheet to work out,
for July then August, the mean and standard deviation of the
monthly hours of sunshine.
Monthly hours of sunshine in England & Wales
July
Mean
Standard deviation
August
192.1 hours 180.9 hours
38.37 hours 31.54 hours
Think about
Compare the results.
How can you decide whether July is significantly more sunny
than August?
This can be done by carrying out a significance test.
Testing the difference between sample means
When samples of size n1 and n2 are taken at random from
normal distributions with means m1, m2 and standard
deviations s1, s2
Distribution of X 1  X 2 is normal with mean m1 – m2
and standard deviation
σ12 σ22
n1  n2
Central Limit Theorem
This is also true for other underlying distributions if
n1, n2 are large.
Summary of method for testing
the difference between sample means
State the null hypothesis:
H0: m1 = m2
(i.e. m1 - m2 = 0)
and alternative hypothesis: H1: m1  m2
2-tailed test
H1: m1 < m2
1-tailed test
or H1: m1 > m2
Calculate the test statistic:
z
x1  x2
σ12  σ22
n1 n2
Compare this value with the relevant critical value of z.
Critical values of z for 2-tailed tests:
For a 5% significance test use z = ± 1.96
95%
Think about
Why is the critical
value 1.96?
2.5%
 1.96 0
2.5%
1.96
z
For a 1% significance test use z = ± 2.58
If the test statistic is in the critical region
reject the null hypothesis and accept the alternative.
State your conclusion clearly in terms of the real context.
Critical values of z for 1-tailed tests
For a 5% significance test use z = 1.65 or –1.65
95%
5%
0
1.65
z
For a 1% significance test use z = 2.33 or –2.33
If the test statistic is in the critical region
reject the null hypothesis and accept the alternative.
State your conclusion clearly in terms of the real context.
Testing whether July is significantly more sunny than August
Mean
Standard deviation
July
August
192.1 hours
38.37 hours
180.9 hours
31.54 hours
n1 = n2 = 30
H0: m1 = m2
H1: m1 > m2
1-tailed test
Test statistic
z
x1  x2
σ12  σ22
n1 n2
Using the 5% level
95%

192.1180.9
38.372  31.542
30
30
z = 1.24 is not significant
5%
1.24
0
1.65
Conclusion: July is not significantly sunnier than August.
z
Try this
Write down and test other hypotheses comparing:
• the amount of sunshine in two other months of the year
• the amount of rainfall in two months
• the amount of sunshine and/or rainfall in the countries
of the UK
Write a short report summarising your findings.
Rain or shine
Reflect on your work
• What is measured by standard deviation?
• When is it better to use sn and when sn – 1 ?
• Describe the steps in a hypothesis test.
• What do you do if the value of z is in the critical region?
• Can you use a hypothesis test to prove that a
theory is true or false?