1. © Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Rain or shine? Spreadsheet activity

2. © 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.

3. © Nuffield Foundation 2011 Monthly hours of sunshine England & Wales (2001 – 2010) YearJanFebMarAprMayJunJulAugSepOctNovDec 200174.586.790.2134.6226.2192.7190.0177.8113.8103.665.875.4 200241.275.3112.4189.4180.2163.5163.9158.3154.693.455.334.3 200370.595.3165.2191.3189.9209.4172.5207.6166.0130.367.452.1 200449.588.3108.2133.3204.6198.2166.0173.8155.795.947.952.1 200556.472.079.5146.9216.6190.8178.4210.6150.681.190.857.0 200651.068.397.4160.4173.1229.7287.6151.4157.595.693.145.8 200760.570.4153.9216.5167.1149.8179.7198.1149.3109.471.251.2 200847.8118.2120.0153.9193.0199.3184.9112.8114.6122.556.666.4 200956.955.6153.2170.3212.7202.0177.1173.5142.890.265.762.2 201060.960.1125.8201.8204.8238.8152.1148.8138.8120.371.553.8 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?

4. 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: You can use a calculator or a spreadsheet to work out the mean and standard deviation. s=s= Try this Use the data on the EWSun worksheet to work out, for July then August, the mean and standard deviation of the monthly hours of sunshine. The sample standard deviation is:   n - 1 =

5. Monthly hours of sunshine in England & Wales JulyAugust Mean 184.4 hours174.8 hours Standard deviation 37.33 hours31.47 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.

6. Testing the difference between sample means is normal with mean  1 -  2 and standard deviation Central Limit Theorem This is also true for other underlying distributions if n 1, n 2 are large. When samples of size n 1 and n 2 are taken at random from normal distributions with means  1,  2 and standard deviations  1,  2 Distribution of

7. Calculate the test statistic: Compare this value with the relevant critical value of z. State the null hypothesis: H 0 :  1 =  2 H1: 1  2H1: 1  2 Summary of method for testing the difference between sample means and alternative hypothesis: H 1 :  1 <  2 or H 1 :  1 >  2 2-tailed test 1-tailed test (i.e.  1 –  2 = 0)

8. 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 region reject the null hypothesis and accept the alternative. For a 5% significance test use z = ± 1.96 95% z 0 1.96 2.5% – 1.96 2.5% State your conclusion clearly in terms of the real context. Think about Why is the critical value 1.96?

9. 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 region reject the null hypothesis and accept the alternative. 95% z 0 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.

10. Test statistic z = 1.39 H 0 :  1 =  2 H 1 :  1 >  2 1-tailed test is not significant n 1 = n 2 = 50 Testing whether July is significantly more sunny than August JulyAugust Mean 184.4 hours174.8 hours Standard deviation 37.33 hours31.47 hours Using the 5% level z 0 1.65 5% 95% 1.39 Conclusion: July is not significantly sunnier than August.

11. 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.

12. Rain or shine Reflect 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?