1. © Nuffield Foundation 2012 © Rudolf Stricker Nuffield Free-Standing Mathematics Activity Gender differences

2. © Nuffield Foundation 2010 Manufacturers of children’s clothing need to consider the body measurements of boys and girls, and find answers to questions such as: What are the mean values? Are there significant differences between the body measurements of boys and girls? Do differences emerge at different ages? Carrying out significance tests on mean values can help to answer such questions. In this activity you will use anthropometric data to carry out significance tests of this type.

3. © Nuffield Foundation 2010 Distribution of the sample mean When random variable X follows a normal distribution with mean μ and standard deviation σ and standard deviation Think about Why is the standard deviation of the sample mean smaller? μ x μ the sample mean follows a normal distribution with mean μ μ + 3 σ μ – 3 σ

4. © Nuffield Foundation 2010 Calculate the test statistic: Summary of method for testing a mean State the null hypothesis: H 0 : mean, μ = value suggested and the alternative hypothesis: H 1 : μ ≠ value suggested (2-tail test) or μ < value suggested or μ > value suggested (1-tail test) Think about Can you explain this formula? What do you do if σ is not known? where is the mean of a sample of size n and σ is the standard deviation of the population

5. © 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 mean 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 =  1.96 1% level, critical values =  2.58 –1.65 z 0 5% 95% z 0 1.96 2.5% –1.96 2.5%

6. © Nuffield Foundation 2010 Testing a mean: T-shirt example A clothing manufacturer designs t-shirts for a chest circumference of 540 mm. Is the mean for 4-year-old boys larger than this? Null hypothesisH 0 : μ = 540 mm Alternative hypothesis H 1 : μ > 540 mm Test statistic = 2.75 μ = 540 1-tail test Think about Why is a 1-tail test used? Using the data for 4-year-old males: n = 124 = 546.94355 σ n – 1 = 28.07432

7. © Nuffield Foundation 2010 Test statistic z = 2.75 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 mean is more than 540 mm. 2.33 z 0 1% 99% Think about Explain the reasoning behind this conclusion. 2.75

8. © Nuffield Foundation 2010 Calculate the test statistic: Compare with the critical value of z. State the null hypothesis: H 0 : μ A = μ B H 1 : μ A ≠ μ B Summary of method for testing the difference between means and alternative hypothesis: or μ A < μ B or μ A > μ B 2-tail test 1-tail test ( μ A – μ B = 0) Think about Why are the variances added? If the test statistic is in the critical region reject the null hypothesis and accept the alternative.

9. © Nuffield Foundation 2010 Using the data to test whether the hand lengths of 2-year-old boys are significantly different from those of 2-year-old girls. Test statistic: H 0 : μ M = μ F ( μ M – μ F = 0) H 1 : μ M ≠ μ F 2-tail test Testing the difference between means: Hand length example

10. © Nuffield Foundation 2010 Testing the difference between proportions: Example Using the hand length data for 2-year-olds on the spreadsheet gives: n M = 52= 103.25 mm = 1.70 Test statistic: n F = 49= 101 mm = 7.9332 mm = 5.1478 mm

11. © Nuffield Foundation 2010 z = 1.70 95% z 0 1.96 2.5% – 1.96 2.5% 1.7 Conclusion There is no significant difference between the hand lengths of 2-year-old boys and girls. The test statistic is not in the critical region. Think about Explain the reasoning behind this conclusion. For a 2-tail 5% testTest statistic: Testing the difference between means: Hand length example

12. At the end of the activity What are the mean and standard deviation of the distribution of a sample mean? Describe the steps in a significance test for a mean value. Describe the steps in a significance test for the difference between means. 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