Download
1 / 12
120 likes | 130 Views
Use a significance test to determine if certain weather hypotheses are likely true. Calculate the mean and standard deviation of monthly hours of sunshine and compare results to make conclusions.
E N D
Nuffield Free-Standing Mathematics ActivityRain or shine? Spreadsheet activity
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.
Monthly hours of sunshine England & Wales (2001 – 2010) 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?
To find the mean and standard deviation The sample mean is given by the formula: The sample standard deviation is: s= sn- 1 = The best estimator of the standard deviation of the population is: You can use a calculator or a spreadsheet to work out the mean and standard deviation. 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.
Monthly hours of sunshine in England & Wales 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 with mean m1 - m2 is normal and standard deviation Central Limit TheoremThis is also true for other underlying distributions if n1, n2 are large.
Summary of method for testing the difference between sample means H0:m1 = m2 (i.e.m1 – m2 = 0) State the null hypothesis: and alternative hypothesis: H1:m1¹m2 2-tailed test H1:m1 < m2 1-tailed test or H1:m1 > m2 Calculate the test statistic: Compare this value with the relevant critical value of z.
95% 2.5% 0 z 2.5% 1.96 – 1.96 Critical values of z for 2-tailed tests: For a 5% significance test use z = ± 1.96 Thinkabout Why is the critical value 1.96? 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.
95% 5% 0 z 1.65 Critical values of z for 1-tailed tests For a 5% significance test use z = 1.65 or – 1.65 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.
95% 5% 1.39 z 1.65 0 Testing whether July is significantly more sunny than August n1 = n2 = 50 H0:m1 = m2 Using the 5% level H1:m1 > m2 1-tailed test Test statistic z = 1.39 is not significant Conclusion: July is not significantly sunnier than August.
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 snand 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?
