© Nuffield Foundation 2011

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.

This activity will show you how to use a significance test to decide whether or not such hypotheses are likely to be true.

Some people think Scotland is sunnier in spring than England and Wales.

© Nuffield Foundation 2011

Monthly hours of sunshine England & Wales (2001–2010)Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec2001 74.5 86.7 90.2 134.6 226.2 192.7 190.0 177.8 113.8 103.6 65.8 75.42002 41.2 75.3 112.4 189.4 180.2 163.5 163.9 158.3 154.6 93.4 55.3 34.32003 70.5 95.3 165.2 191.3 189.9 209.4 172.5 207.6 166.0 130.3 67.4 52.12004 49.5 88.3 108.2 133.3 204.6 198.2 166.0 173.8 155.7 95.9 47.9 52.12005 56.4 72.0 79.5 146.9 216.6 190.8 178.4 210.6 150.6 81.1 90.8 57.02006 51.0 68.3 97.4 160.4 173.1 229.7 287.6 151.4 157.5 95.6 93.1 45.82007 60.5 70.4 153.9 216.5 167.1 149.8 179.7 198.1 149.3 109.4 71.2 51.22008 47.8 118.2 120.0 153.9 193.0 199.3 184.9 112.8 114.6 122.5 56.6 66.42009 56.9 55.6 153.2 170.3 212.7 202.0 177.1 173.5 142.8 90.2 65.7 62.22010 60.9 60.1 125.8 201.8 204.8 238.8 152.1 148.8 138.8 120.3 71.5 53.8

Think aboutWhat 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?

To find the mean and standard deviation

The best estimator of the standard deviation of the population is:

The sample mean is given by the formula: nxx

You can use a calculator or a spreadsheet to work out the mean and standard deviation.

s = 22x

nx

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.

Try this

The sample standard deviation is:

n – 1 =s

nn1

Monthly hours of sunshine in England & Wales

July August

Mean192.1 hours 180.9 hours

Standard deviation38.37 hours 31.54 hours

Think aboutCompare 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

is normal with mean 1 – 2

and standard deviation

Central Limit TheoremThis is also true for other underlying distributions if n1, n2 are large.

When samples of size n1 and n2 are taken at random from normal distributions with means 1, 2 and standard deviations 1, 2

2

22

1

21 n

σnσ

Distribution of 21 XX

Calculate the test statistic:

Compare this value with the relevant critical value of z.

State the null hypothesis: H0: 1 = 2

H1: 1 2

Summary of method for testing the difference between sample means

and alternative hypothesis:

H1: 1 < 2

or H1: 1 > 2

2-tailed test

1-tailed test

(i.e. 1 - 2 = 0)

22

11

21

nσ

nσ

xxz22

Critical values of z for 2-tailed tests:

For a 1% significance test use z = ± 2.58

If the test statistic is in the critical regionreject the null hypothesis and accept the alternative.

For a 5% significance test use z = ± 1.96

95%

z0 1.96

2.5%

1.96

2.5%

State your conclusion clearly in terms of the real context.

Think aboutWhy is the critical value 1.96?

Critical values of z for 1-tailed tests

For a 1% significance test use z = 2.33 or –2.33

If the test statistic is in the critical regionreject the null hypothesis and accept the alternative.

95%

z0 1.65

5%

For a 5% significance test use z = 1.65 or –1.65

State your conclusion clearly in terms of the real context.

22

11

21

nσ

nσ

xxz22

Test statistic

3031.54

3038.37

180.9192.122

z = 1.24

H0: 1 = 2 H1: 1 > 2 1-tailed test

is not significant

n1 = n2 = 30

Testing whether July is significantly more sunny than August

July August

Mean 192.1 hours 180.9 hoursStandard deviation 38.37 hours 31.54 hours

Using the 5% level

z0 1.65

5%

95%

1.24

Conclusion: July is not significantly sunnier than August.

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.

Try this

Rain or shineReflect on your work

•What is measured by standard deviation?

•When is it better to use n and when n – 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?