© Nuffield Foundation 2012 Nuffield Free-Standing Mathematics Activity Successful HE applicants © Rudolf Stricker

© Nuffield Foundation 2010 Are 50% of the applications for courses in higher education from females? Are the same proportion of male and female applicants successful? Carrying out significance tests on proportions, and the difference between proportions, can help to answer such questions. In this activity you will use data on successful applications for courses in higher education to carry out significance tests of this type.

© Nuffield Foundation 2010 Distribution of a sample proportion p psps If the sample size, n, is large (>30) the sample proportion, p s, will follow a normal distribution and standard deviation with mean p where q = 1 – p Think about Why is it important that the sample is large?

© Nuffield Foundation 2010 Calculate the test statistic: Summary of method for testing a proportion State the null hypothesis: H 0 : population proportion, p = value suggested and the alternative hypothesis: H 1 : p ≠ value suggested (2-tail test) or p < value suggested or p > value suggested (1-tail test) Think about Can you explain this formula? where p s is the proportion in a sample of size n and q = 1 – p

© Nuffield Foundation 2010 If the test statistic is in the critical region (tail of the distribution) Compare the test statistic with the critical value of z : Summary of method for testing a proportion For a 1-tail test 1% level, critical value = 2.33 or –2.33 reject the null hypothesis and accept the alternative. 5% level, critical value = 1.65 or –1.65 For a 2-tail test 5% level, critical values =  % level, critical values =  z 0 5% 95% z % %

© Nuffield Foundation 2010 Testing a proportion: Example In 2010 a newspaper article said that the proportion of people accepted on higher education courses over 20 years old was 16%. Null hypothesis H 0 : p = 0.16 Alternative hypothesis H 1 : p < 0.16 Test statistic = –2.77 From 2010 data: q = 1 – p p = 0.16 = 0.84 n = tail test Using 2010 data to test this percentage: Think about Why is a 1-tail test used? = psps

© Nuffield Foundation 2010 Test statistic= –2.77 –2.77 The test statistic, z, is in the critical region. The result is significant at the 1% level, so reject the null hypothesis. For a 1-tail 1% significance test: Conclusion There is strong evidence that the proportion reported is too high. –2.33 z 0 1% 99% Think about Explain the reasoning behind this conclusion.

© Nuffield Foundation 2010 Calculate the test statistic: Compare with the critical value of z. State the null hypothesis: H 0 : p A = p B H1: pA ≠ pBH1: pA ≠ pB Summary of method for testing the difference between proportions and alternative hypothesis: or p A < p B or p A > p B 2-tail test 1-tail test where q = 1 – p ( p A – p B = 0) Think about Explain the formula for the test statistic. If the test statistic is in the critical region reject the null hypothesis and accept the alternative. p SA, p SB, n A and n B are values from the samples.

© Nuffield Foundation 2010 Using 2010 data to test whether the proportion of males that were accepted is equal to the proportion of females that were accepted. Test statistic: H 0 : p M = p F ( p M – p F = 0) H1: pM ≠ pFH1: pM ≠ pF 2-tail test Testing the difference between proportions: Example

© Nuffield Foundation 2010 Testing the difference between proportions: Example In the 2010 sample: 7062 out of males and 8353 out of females were successful. = q = 1 – = = = Test statistic: = 0.331

© Nuffield Foundation 2010 z = % z % % Conclusion There is no significant difference between the proportion of males and the proportion of females accepted. The test statistic is not in the critical region. Think about Explain the reasoning behind this conclusion. For a 5% testTest statistic: Testing the difference between proportions: Example

At the end of the activity What are the mean and standard deviation of the distribution of a sample proportion? Describe the steps in a significance test for a sample proportion. Describe the steps in a significance test for the difference between sample proportions. When should you use a one-tail test and when a two-tail test? Would you be more confident in a significant result from a 5% significance test or a 1% significance test? Explain why. © Nuffield Foundation 2012