Successful HE applicants. Information sheet A Number of applicants. Gender Applicants Accepts Applicants Accepts. Age. Domicile

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1 Successful HE applicats Sigificace tests use data from samples to test hypotheses. You will use data o successful applicatios for courses i higher educatio to aswer questios about proportios, for example, to test whether equal proportios of male ad female applicats are accepted. Iformatio sheet A Number of applicats The simulated data give below give the total umber of applicats for courses of higher educatio at a sample of uiversities ad colleges. Actual data ca be accessed from the UCAS website Geder Geder Applicats Accepts Applicats Accepts Male 35,117 5,273 45,455 7,062 Female 37,785 5,521 54,030 8,353 Age Number accepted Age ad uder 5,821 7, ,727 4, , ad over Total 10,794 15,415 Domicile Domicile Applicatios Accepts Applicatios Accepts UK 67,699 10,220 90,540 14,053 EU(ot UK) 1, , No EU 3, , Total 72,902 10,794 99,485 15,415 Nuffield Free-Stadig Mathematics Activity Successful HE applicats Studet sheets Copiable page 1 of 6

2 Iformatio sheet B Testig a proportio Distributio of a sample proportio If large samples of sie are take from a populatio i which there are a proportio p with a certai attribute, the the distributio of sample proportios, p s, is approximately ormal with mea p ad stadard deviatio where q = 1 p. Why is it importat that samples are large? stadard deviatio Mea p p s Summary of method for testig a proportio To test whether the proportio of a populatio has a value p: Null Hypothesis H 0 : populatio proportio, p = value suggested Alterative Hypothesis H 1 : p value suggested (two-tail test) or p < value suggested or p > value suggested (oe-tail test) Test statistic = p s p where p s is the proportio i a sample of sie ad q = 1 p Compare the test statistic with critical values of. Explai the formula for the test statistic. Critical values: Test type Sigificace level Critical values oe-tail test 5% 1.65 or % 95% 1% 2.33 or % two-tail test 5% % If the test statistic is i the critical regio (that is, a tail of the distributio), reject the ull hypothesis i favour of the alterative. If the test statistic is ot i the critical regio, accept the ull hypothesis. Nuffield Free-Stadig Mathematics Activity Successful HE applicats Studet sheets Copiable page 2 of 6

3 Testig a proportio: Example I 2010, a ewspaper article said that the proportio of people accepted o higher educatio courses who were over 20 years old had reached 16%. Usig 2010 data to test this percetage: H 0 : populatio proportio, p = 0.16 H 1 : p <0.16 (oe-tail test) Why is a oe-tail test used here rather tha a two-tail test? Test statistic = p s p where p = 0.16 ad q = = 0.84 From the 2010 data, p s = 2340 = ad = So = = 2.77 For a oe-tail 1% sigificace test, the critical value is The test statistic is i the critical regio (less tha the critical value). 1% 99% The result is sigificat at the 1% level So reject the ull hypothesis ad accept the alterative. The test has provided strog evidece that the proportio of people accepted o higher educatio courses who were over 20 years old had ot reached 16%. Explai the reasoig behid this coclusio. Nuffield Free-Stadig Mathematics Activity Successful HE applicats Studet sheets Copiable page 3 of 6

4 Iformatio sheet C Testig the differece betwee proportios Summary of method for testig the differece betwee proportios To test the differece betwee proportios: Null hypothesis, H 0 : p A = p B ( p A p B = 0) Alterative hypothesis, H 1 : p A p B (two-tail test) p A < p B (oe-tail test) p A > p B (oe-tail test) The test statistic is = p SA p SB 1 1 A B where p SA ad p SB are the proportios from samples of sie A ad B. p, the best estimate of the populatio proportio, is calculated from: p = Totalumberof itemswithattribute Totalumberof itemsisamples Explai the formula for the test statistic. ad q = 1 p Compare the test statistic with critical values of. Critical values: Test type Sigificace level Critical values oe-tail test 5% 1.65 or % 2.33 or 2.33 two-tail test 5% % 2.58 If the test statistic is i the critical regio, reject the ull hypothesis i favour of the alterative. If the test statistic is ot i the critical regio, accept the ull hypothesis. Nuffield Free-Stadig Mathematics Activity Successful HE applicats Studet sheets Copiable page 4 of 6

5 Testig the differece betwee proportios: Example Usig 2010 data to test whether the proportio of males that were accepted is equal to the proportio of females that were accepted: H 0 : p M = p F (p M p F = 0) H 1 : p M p F (two-tail test) The test statistic is = p SM 1 p M SF 1 F I the 2010 sample, 7062 out of applicatios from males were accepted, ad 8353 out of applicatios from females were accepted So p SM = = , p SF = M = ad F = = , p, the best estimate of the populatio proportio, is calculated from p = = ad q = = Usig these values, the test statistic is give by: = = For a two-tail test at the 5% level, the critical values of are 1.96, so this value of is ot sigificat at the 5% level. 95% There is o sigificat differece betwee the proportio of males that were accepted ad the proportio of females that were accepted. Explai the reasoig behid this coclusio. Nuffield Free-Stadig Mathematics Activity Successful HE applicats Studet sheets Copiable page 5 of 6

6 Try this 1 Cosider the data o Iformatio sheet A. Write a list of hypotheses you thik could be tested usig these data. 2 Choose some of the hypotheses you have listed i questio 1. Carry out sigificace tests o these hypotheses. At least oe of your tests should be of a proportio. At least oe of your tests should be of the differece betwee proportios. Reflect o your work What are the mea ad stadard deviatio of the distributio of a sample proportio? Describe the steps i a sigificace test for a proportio. Describe the steps i a sigificace test for the differece betwee proportios. Whe should you use a oe-tail test ad whe a two-tail test? Would you be more cofidet i a sigificat result from a 5% sigificace test or a 1% sigificace test? Explai why. Nuffield Free-Stadig Mathematics Activity Successful HE applicats Studet sheets Copiable page 6 of 6