Standard deviation The formula for the best estimate of the population standard deviation from a sample is:

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1 Geder differeces Are there sigificat differeces betwee body measuremets take from male ad female childre? Do differeces emerge at particular ages? I this activity you will use athropometric data to carry out sigificace tests to aswer such questios. Iformatio sheet A Athropometric data Maufacturers of childre s clothig eed to cosider the body measuremets of boys ad girls. Toy maufacturers may also eed to cosider their weights. These body measuremets will deped o age ad sometimes geder. I some cases boys ad girls of the same age may be similar i size, but i other cases they may be differet. The spreadsheet cotais athropometric data collected for a 1977 study carried out i the USA by the Cosumer Product Safety Commissio (CPSC). The data icludes: weight (ewtos) stature (mm) head, chest ad waist circumfereces (mm) shoulder breadth (mm) had legth ad breadth (mm) foot legth ad breadth (mm) age (moths) The 'Male' ad 'Female' worksheets give data for males ad females sorted by age ad stature. Mea The mea of a sample ca be foud usig = or usig the spreadsheet formula AVERAGE. Stadard deviatio The formula for the best estimate of the populatio stadard deviatio from a sample is: σ - 1 = 1 The correspodig spreadsheet formula is STDEV. Nuffield Free-Stadig Mathematics Activity Geder differeces Studet sheets Copiable page 1 of 6 Nuffield Foudatio 01 dowloaded from

2 Iformatio sheet B Testig a mea Distributio of a sample mea Whe samples of size are take from a ormal populatio with mea μ ad stadard deviatio σ, the the sample mea, X, follows a ormal distributio with mea μ ad stadard deviatio. Mea μ stadard deviatio σ Distributio of Sample Mea Thik about Why is the stadard deviatio of the sample mea smaller? Summary of method for testig a mea To test whether the mea of a populatio has a value μ: Null hypothesis Alterative hypothesis H 0 : populatio mea, μ = value suggested Mea μ stadard deviatio H 1 : μ value suggested (two-tail test) or μ < value suggested or μ > value suggested (oe-tail test) σ Test statistic z = Thik about Eplai the formula for the test statistic. where is the sample mea, σ is the stadard deviatio ad is the size of the sample. If the stadard deviatio of the populatio is ot kow, the the best estimate from the sample, σ 1 is used istead (see page 1). Compare the test statistic with critical values of z. Critical values Test type Sigificace level Critical values 5% 95% oe-tail test 5% 1.65 or z 1%.33 or.33 two-tail test 5% %.58 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..5% % % z Nuffield Free-Stadig Mathematics Activity Geder differeces Studet sheets Copiable page of 6 Nuffield Foudatio 01 dowloaded from

3 Testig a mea: T-shirt eample A clothig maufacturer desigs boys t-shirts for a chest circumferece of 540 mm. Is the mea for 4-year-old boys larger tha this? Usig data from the spreadsheet to test this mea: H 0 : populatio mea, μ = 540 mm H 1 : μ >540 mm (oe-tail test) Thik about Why is a oe-tail test used here rather tha a two-tail test? Test statistic z = with μ = From the data, = ad = 14 So z = =.75 For a oe-tail 1% sigificace test, the critical value is.33. The test statistic is i the critical regio (greater tha the critical value). 99% 1% The result is sigificat at the 1% level z So reject the ull hypothesis ad accept the alterative. The test has provided strog evidece that the mea chest circumferece of 4-year-old boys is more tha 540 mm. Thik about Eplai the reasoig behid this coclusio. Nuffield Free-Stadig Mathematics Activity Geder differeces Studet sheets Copiable page 3 of 6 Nuffield Foudatio 01 dowloaded from

4 Iformatio sheet C Testig the differece betwee meas Summary of method for testig the differece betwee meas To test the differece betwee meas: Null hypothesis, H 0 : μ A = μ B (μ A μ B = 0) Alterative hypothesis, H 1 : μ A μ B (two-tail test) μ A < μ B (oe-tail test) μ A > μ B (oe-tail test) The test statistic is z = A A B A B B where A ad B are the meas from samples of size A ad B. σ A ad σ B are stadard deviatios of the populatios. If these are ot kow, use the best estimates that ca be foud from the samples. Thik about Why are the variaces added i the formula for the test statistic? Compare the test statistic with critical values of z. Critical values: Test type Sigificace level Critical values oe-tail test 5% 1.65 or %.33 or.33 two-tail test 5% %.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 Geder differeces Studet sheets Copiable page 4 of 6 Nuffield Foudatio 01 dowloaded from

5 Testig the differece betwee meas: Had legth eample Usig data from the spreadsheet to test whether the had legths of -year-old boys are sigificatly differet from those of -year-old girls: H 0 : μ M = μ F (μ M μ F = 0) H 1 : μ M μ F (two-tail test) The test statistic is z = M ( 1) M ( 1) F M F F The table below gives the values calculated from the data o the spreadsheet: Geder Sample mea, Stadard deviatio, σ 1 Sample size, Male mm mm 5 Female 101 mm mm 49 Usig these values, the test statistic is: z = = 1.70 For a two-tail test at the 5% level, the critical values of z are 1.96, so this value of z is ot sigificat at the 5% level. 95%.5%.5% z There is o sigificat differece betwee the had legths of -year-old boys ad girls. Thik about Eplai the reasoig behid this coclusio. Nuffield Free-Stadig Mathematics Activity Geder differeces Studet sheets Copiable page 5 of 6 Nuffield Foudatio 01 dowloaded from

6 Try these 1 Chest circumferece a A maufacturer assumes that the mea chest circumferece of 6-year-old boys ad the mea chest circumferece of 6-year-old girls are both 580 mm. Use the data i the spreadsheet to carry out hypothesis tests to decide whether the true meas are greater tha 580 mm. b Repeat part a to test whether the mea chest circumferece of 8-year-old boys ad the mea chest circumferece of 8-year-old girls are both 640 mm or greater tha this. Head circumferece a Use data from the spreadsheet to test whether there is ay differece betwee the mea head circumfereces of -year-old boys ad -year-old girls. b Repeat part a for oe other age group chose from: 4-year-olds, 8-year-olds, 10-year-olds, 16-year-olds 3 Differece betwee meas a Choose a body measuremet ad age group where you thik there will be a sigificat differece betwee boys ad girls. Use the data o the spreadsheet to test the differece betwee meas. b Choose a body measuremet ad age group where you thik there will ot be a sigificat differece betwee boys ad girls. Use the data o the spreadsheet to test the differece betwee meas. Reflect o your work What are the mea ad stadard deviatio of the distributio of a sample mea? Describe the steps i a sigificace test for a populatio mea. Describe the steps i a sigificace test for the differece betwee meas. 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? Eplai why. Nuffield Free-Stadig Mathematics Activity Geder differeces Studet sheets Copiable page 6 of 6 Nuffield Foudatio 01 dowloaded from