Text&Tests5. Project Maths SUPPLEMENT. Frances O Regan O. D. Morris. Leaving Certificate Higher Level Maths

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1 Project Maths SUPPLEMENT Text&Tests5 Leavig Certificate Higher Level Maths Cotais all the Deferred Material ad Cetral Limit Theorem Fraces O Rega O. D. Morris

2 O.D. Morris, Fraces O Rega, 2014 All rights reserved. No part of this publicatio may be reproduced, stored i a retrieval system or trasmitted i ay form or by ay meas, electroic, mechaical, photocopyig, recordig or otherwise, without the prior writte coset of the copyright holders. First published Jue 2014 by The Celtic Press Groud Floor Block B Liffey Valley Office Campus Dubli 22 ISBN: Prited i Irelad by Turer Prit Group Earl Street Logford

3 Preface This booklet cotais all the Deferred Material from Strad 1 of the Leavig Certificate Higher Level Course. This material was itroduced i September 2013 for examiatio i Jue 2015 ad owards. This chapter is prited as a supplemet for those studets who bought Text & Tests 5 at the begiig of 5th Year i September All future editios of Text & Tests 5 will cotai this ew chapter. Chapter 5 also cotais a sectio o the Cetral Limit Theorem which was belatedly added to the origially published course. O. D. Morris Fraces O Rega Jue 2014

4 chapter 5 Iferetial statistics Key words samplig distributio Cetral Limit Theorem parameter statistic stadard error cofidece iterval proportio hypothesis ull hypothesis critical regio test statistic p-value Sectio 5.1 The samplig distributio of the mea The Cetral Limit Theorem I Sectio 4.4 it was stated that the purpose of samplig is to obtai iformatio about a whole populatio by surveyig a small part of the populatio. This small part is called a sample. Whe we select a sample from a populatio, ad study it, we hope that it is represetative of the populatio as a whole. To esure that it is represetative, it must be a radom sample. By a radom sample we mea that (i) every member of the populatio has a equal chace of beig selected (ii) the selectios are made idepedetly. A very importat part of the work of a statisticia ivolves drawig coclusios about a populatio based o evidece gathered from the sample. This process is kow as statistical iferece. Parameters Statistics It is kow that the mea height of me i Irelad is 176 cm. The mea height of a sample of Muster rugby players is 186 cm. The value 176 cm is called a parameter as it is a umerical property of a populatio. The value 186 cm is called a statistic as it is a umerical property of a sample. Parameter Statistic A parameter is a umerical property of a populatio. A statistic is a umerical property of a sample. The samplig distributio of the mea If we are iterested i the weights, for example, of all sixtee-year-olds i Irelad, we geerally require the mea ad stadard deviatios of these weights. We use the symbols (i) to deote the populatio mea (ii) to deote the populatio stadard deviatio. 4

5 I such a large populatio it would be impossible to obtai the weight of each perso ad so the values of ad will ot be kow. However, if we take a radom sample of this populatio, we ca get approximate values for ad. Obviously, the larger the sample, the more accurate we would expect the approximatios to be. If we take a large umber of differet radom samples of size, each sample will have its ow mea, _ x, ad stadard deviatio, _ x. Some of these samples are illustrated o the right. The differet meas of these samples are called the sample meas. If a large umber of samples of the same size are take, you get a correspodigly large umber of meas. These meas form their ow distributio givig us the distributio of sample mea. This distributio is also called the samplig distributio of the mea. The followig example illustrates the shape a distributio might take whe differet samples (of the same size) from a populatio are selected. Example 1 Sample 1 A populatio cosists of five digits 2, 4, 6, 8, 10. (i) Write dow all the possible samples of 2 differet digits that ca occur if radom samples are take. (ii) Fid the mea of each sample ad plot the distributio of the sample meas. (iii) Compare the value of the mea of the sample meas with the value of the populatio mea. x 1 x1 Sample 2 x 2 Sample 3 x 3 Sample 4 Sample 5 Sample 6 x 4 x 5 x 6 (i) The possible samples are: (2, 4), (2, 6), (2, 8), (2, 10), (4, 6), (4, 8), (4, 10), (6, 8), (6, 10), (8, 10) (ii) Their meas are: 3, 4, 5, 6, 5, 6, 7, 7, 8, 9 The distributio of the sample meas is plotted below. f f x Populatio Mea of populatio Sample of meas 60 Mea of sample meas 6 (iii) The mea of the populatio is 6. The mea of the sample meas is also 6. Thus the mea of the sample meas ad the populatio mea are equal. 10 x 5

6 If you examie the distributio of the sample meas plotted o the right i the worked example above, you will otice that it begis to approximate to a ormal distributio. I this case the sample size was oly 2. However, as the sample size icreases the closer the distributio will approximate to a ormal distributio. Also the mea of the samplig distributio will be the same as the mea of the populatio. The successive diagrams below illustrate the shape of the samplig distributio of meas resultig from differet-sized samples from a give populatio with a ormal distributio. Distributio whe 2, 5 ad Meas of samples of size Meas of samples of size Meas of samples of size 25 From the diagrams, you ca see that if samples are take from a ormal populatio, the samplig distributio of meas is ormal for ay sample size. As icreases, the curve represetig the samplig distributio of the mea gets taller ad arrower. These diagrams also show how the stadard deviatio decreases as icreases. The sample meas will be packed tightly aroud the populatio mea. The larger the samples become, the tighter the meas will be packed. From the worked example ad from the three diagrams show above, we ca see that whe samples are take from a populatio, the samplig distributio of the mea takes o the characteristics of a ormal curve as the sample size icreases. This observatio leads us to oe of the most importat theorems i statistics that is widely used i samplig. It is called the Cetral Limit Theorem ad is stated more formally below. The Cetral Limit Theorem If a radom sample of size with mea _ x is take from a populatio with mea ad stadard deviatio, the > If the sample size is large ( 30), the distributio of the sample meas will approximate to a ormal distributio regardless of what the populatio distributio is. > The mea of the distributio will be the same as the populatio mea. > The stadard deviatio of the samplig distributio (deoted by _ x ) is give by. [ is ofte referred to as the stadard error of the mea. ] As icreases, the stadard error gets smaller. > If the uderlyig populatio is ormal, the samplig distributio of the mea will always have a ormal distributio eve if the sample size is small ( 30). 6

7 The diagram o the right illustrates how the distributio of the sample mea approximates to a ormal distributio eve whe the uderlyig populatio is skewed. Distributio of sample mea Paret distributio Whe dealig with the samplig distributio of the mea, we covert the give uits to stadard uits usig the formula give o the right. z x x x Example 2 A radom sample of 250 is selected from a populatio havig mea 30 ad stadard deviatio 5. Fid the probability that the sample mea is greater tha Sice 250, the sample mea is ormally distributed sice 30. Chagig to stadard uits we get: _ x z z Now P(x 30.5) P(z 1.581) 1 P(z 1.581) The probability that the mea is greater tha 30.5 is Example 3 A ormal distributio has a mea of 40 ad a stadard deviatio of 4. If 25 items are draw at radom, fid the probability that their mea lies betwee 38 ad

8 Covertig the give uits to stadard uits we get: _ x z For x 38, z For x 40.5, z P(38 x 40.5) P( 2.5 z 0.625) P(z 0.625) P(z 2.5) P(z 0.625) [1 P(z 2.5)] [ ] [ ] P(mea lies betwee 38 ad 40.5) Example 4 A populatio is ormally distributed with mea 12 ad stadard deviatio 3. Fid the sample size such that P( _ x 12.5) 0.05, where _ x is the sample mea. P(z z 1 ) 0.05 P(z z 1 ) 0.95 z _ x z ( ) (1.645)3 0.5 The required sample size is i.e. 98 roud up 8

9 Example 5 A compay istalls ew machies for packig peauts. The compay claims that the machies fill packets with a mea mass of 500 g ad a stadard deviatio of 18 g. To test the compay s claim several samples of size 40 packets are take ad their mea masses, _ x grams, are recorded. (i) Describe the samplig distributio of _ x ad explai your aswer, referrig to the theorem you have used. (ii) Write dow the mea ad stadard deviatio of the distributio of _ x. (iii) Draw a rough sketch of the samplig distributio of _ x. (iv) Fid the probability that the mea of the distributio of _ x is less tha 496. (v) What sample size is required so that P( _ x ) ? (i) The samplig distributio of _ x is approximately ormal as the sample size of 40 is sufficietly large (i.e. 30) to apply The Cetral Limit Theorem. (ii) The mea of the distributio of the sample meas is 18 g, the same as the populatio mea. The stadard deviatio (or stadard error) is (iii) A sketch of the distributio of _ x is show below _ x (iv) Covertig the give uits to z-scores, we use z. For x 496, z P( _ x 496) P(z 1.405) 1 P(z 1.405) The probability that _ x or 7.35%

10 (v) P(z z 1 ) 0.06 P(z z 1 ) z _ x z (1.56) 9.36 (9.36) roud up The sample size required is 88. Exercise Fill i the correct word or symbol to complete the followig statemets: (i) Whe a large umber of samples of size are take from a populatio, the the distributio of _ x, the sample mea, is kow as the of the mea. (ii) As the sample size icreases, the stadard deviatio of the samplig distributio of the sample meas will. (iii) If the mea of the uderlyig populatio is, the mea of the samplig distributio of the meas is. (iv) If the stadard deviatio of a populatio is ad samples of size are take from it, the the stadard deviatio of the distributio of the sample meas is. 2. The diagram o the right shows two curves. Oe of these curves represets a distributio ad the other represets the distributio of the sample meas of size take from this distributio. Which curve represets the distributio of the sample meas? A B 3. Samples of size 36 are take from a populatio with mea 12 ad stadard deviatio 2. The samplig distributio of the meas are plotted i a curve. (i) Describe the shape of this curve amig the theorem you have used to support your descriptio. (ii) Explai why the theorem you have metioed ca be applied whe the shape of the uderlyig populatio is ukow. (iii) Write dow the mea ad stadard deviatio of the samplig distributio of the mea. 10

11 4. A populatio cosists of the elemets {4, 6, 8, 10}. (i) Write dow all possible samples of size 2 (chose with replacemet) from this populatio. (ii) Give the sample mea, _ x, for each pair. (iii) Are each of the values you have foud a statistic or a parameter? (iv) Show that the mea of all possible samples of size 2 equals the mea of the populatio. 5. Explai the differece betwee a parameter ad a statistic. 6. The diagram o the right shows two curves A ad B. Diagram A represets the distributio of a A populatio ad diagram B represets the distributio of the meas from a large umber of samples of size 40. (i) Is distributio A skewed positively or B egatively? (ii) Describe distributio B. (iii) Explai why the Cetral Limit Theorem ca be used to describe distributio B eve though the uderlyig populatio is ot ormally distributed. 7. A radom sample of size 36 is chose from a populatio with a mea of 12 ad a stadard deviatio of 3. Fid the probability that the sample mea is greater tha A radom sample of size 15 is take from a ormal distributio with mea 60 ad stadard deviatio 4. Fid the probability that the mea of the sample is less tha Me have a mea height of 176 cm with stadard deviatio 11 cm. Fid the probability that the mea of a radom sample of 80 me (i) exceeds 177 cm (ii) is less tha cm. 10. At a certai college, studets sped o average 4.2 hours per week at a computer termial, with a stadard deviatio of 1.8 hours. (i) Fid the stadard error for a radom sample of 36 studets. (ii) Fid the probability that the average time spet usig a computer termial is (a) at least 4.8 hours (b) betwee 4.1 ad 4.5 hours. 11. The sugar cotet per litre bottle of a soft drik is kow to be distributed with mea 5.8 ad stadard deviatio 1.2. A sample of 900 bottles is take at radom ad the sugar cotet of each bottle is measured. Estimate to 3 decimal places the probability that the mea sugar cotet of the 900 bottles will be less tha

12 12. A firm produces alterators for cars. The alterators are kow to have a mea lifetime of 8 years with stadard deviatio 6 moths. Forty samples of 144 alterators produced by the firm are tested. Estimate the umber of samples which would be expected to have a mea lifetime of more tha 8 years ad 1 moth. 13. A radom sample of size 10 is take from a ormal distributio with mea 200 ad stadard deviatio 10. Fid the probability that the sample mea lies outside the rage 198 to I the give diagram, curve A represets a B ormal distributio. Curve B represets the samplig distributio of meas take from samples of size 36. A The distributio represeted by A has mea 80 ad stadard deviatio 8. The poit C represets the mea of both distributios. E C D The poit D represets the value of the variable that is two stadard deviatios from C i distributio A. The poit E represets the value of the variable that is oe stadard error from C i distributio B. Write dow the values of C, D ad E. 15. A ormal distributio has mea 75 ad stadard deviatio 9. A sample of size is selected at radom ad the mea of this sample is _ x. Fid if P( _ x 73) A ormal distributio has a mea of 30 ad a stadard deviatio of 5. (i) Fid the probability that the mea of a radom sample of 40 exceeds (ii) Fid the value of such that the probability that the mea of a sample of size exceeds 30.4 is less tha Free-rage eggs supplied by a health food cooperative have a mea weight of 52 g with a stadard deviatio of 4 g. Assumig the weights are ormally distributed fid the probability that: (i) a radomly selected egg will weigh more tha 60 g (ii) the mea weight of five radomly selected eggs will be betwee 50 g ad 55 g (iii) the mea weight of 90 radomly selected eggs will be betwee 52.1 g ad 52.2 g. Which of your aswers would be uchaged if the weights are ot ormally distributed?

13 Sectio 5.2 Cofidece iterval for a mea I Sectio 5.1, the Cetral Limit Theorem was used to show that the samplig distributio of the mea approximates to a ormal distributio for large ( 30). I this sectio we itroduce a differet way of presetig iformatio provided by a sample mea to estimate the mea of the populatio from which the sample came. If samples of size are take from a populatio, the meas of the samples will vary. To accommodate this variety, we itroduce the cocept of a cofidece iterval. This iterval will produce a rage of values i which we are quite cofidet the populatio mea lies. The edpoits of this iterval are called cofidece limits. But how do we measure this cofidece? The degree of cofidece is geerally give as a percetage. These percetages are geerally 90%, 95% ad 99%. The most commoly used measure of cofidece is a 95% cofidece level. This meas that there is a 95% probability that the populatio mea lies i the give iterval. I the stadard ormal distributio, we require the values of z such that 95% of the populatio lies i the iterval z 1 z z 1. The work ivolved i fidig the value of z, is show below. We use the stadard ormal tables o pages 36 ad 37 of Formulae ad Tables. From the give diagram, P(z z 1 ) From the tables z z z 1 0 z Thus i the ormal distributio, 95% of the populatio lies withi 1.96 stadard deviatios of the mea. Sice the sample mea is ormally distributed, 95% of the populatio will lie i the iterval _ x 1.96 _ x, where _ x is the stadard error of the mea. _ x If is the populatio mea, the 95% of the sample meas lie i the iterval _ x 1.96 _ x _ x 1.96 _ x where _ x, beig the stadard deviatio of the populatio. This ca be writte as _ x 1.96, which are the ed-poits (or cofidece limits) of the mea. 13

14 Cofidece Iterval for Mea If _ x is the mea of a radom sample of size take from a populatio with a ormal distributio with kow stadard deviatio, the the ed-poits of the 95% cofidece iterval for, the populatio mea, are give by _ x 1.96 Note: If, the stadard deviatio of the populatio, is ot give, use the stadard deviatio of the sample as a approximatio. Example 1 A radom sample of 400 orages was take from a large cosigmet with ukow mea ad stadard deviatio 15 grams. The mea weight of the radom sample was 81.4 grams. Fid a 95% cofidece iterval for the mea weight of the orages i the cosigmet. _ The 95% cofidece iterval for is x _ x ( ) 15 ad (0.75) , The mea of the cosigmet lies betwee g ad g. Example 2 A certai type of teis ball is kow to have a height of bouce which is ormally distributed with stadard deviatio 2 cm. A sample of 60 such teis balls is tested ad the mea height of the bouce of the sample is 140 cm. (i) Fid a 95% cofidece iterval for the mea height of the bouce of this type of teis ball. (ii) Explai what is meat by a 95% cofidece iterval. (iii) If a teis ball is selected at radom, what is the probability that its bouce is outside the cofidece iterval foud i (i) above? 14

15 (i) The 95% cofidece iterval is give by _ x ( 2 60 ) (0.258) , The 95% cofidece iterval is (ii) A 95% cofidece iterval, meas that o 95 occasios out of 100 the iterval will cotai the true populatio mea. (iii) P(ball bouce lies outside 95% cofidece iterval) Example 3 The heights of people have a stadard deviatio of 11.5 cm. It is required to estimate the mea height of people, with 95% cofidece, to withi 0.4 cm. What sample size should be take i order to achieve this estimate? Let be the mea height of people. The 95% cofidece limits for are _ x stadard error is 0.4 cm 1.96 ( 11.5 ) (1.96) (56.35) Therefore, a sample of at least 3176 should be take. 15

16 Example 4 O the basis of the results obtaied from a radom sample of 100 me from a particular district, the 95% cofidece iterval for the mea height of the me i the district is foud to be ( cm, cm). Fid the value of _ x, the mea of the sample, ad, the stadard deviatio of the ormal populatio from which the sample is draw. The 95% cofidece iterval is give by _ x 1.96 (177.22,179.18) _ x _ ad x Addig 1 ad 2 : 2 _ x _ x Subtractig 1 ad 2 : 2(1.96) The sample mea _ x cm. The populatio stadard deviatio is 5 cm. Exercise A populatio has mea ad stadard deviatio 12. A radom sample of 800 from this populatio has mea 63. Fid a 95% cofidece iterval for. 2. The weights of dairy cows are kow to have a stadard deviatio of 42 kg. A radom sample of 280 dairy cows has a mea weight of 284 kg. Fid a 95% cofidece limit for the mea weight of all the cows. 3. Sevety packs of butter, selected at radom from a large batch delivered to a supermarket, are weighed. The mea weight is foud to be 227 g ad the stadard deviatio is foud to be 7.5 g. (i) Calculate a 95% cofidece iterval for the mea weight of all packs i the batch. (ii) If oe pack is selected at radom from the sevety packs, fid the probability that its weight is ot i the give iterval. 4. I a radom sample of 100 studets takig a state examiatio, it was foud that the mea mark was 62.7 with a stadard deviatio of 9.2 marks. Fid the 95% cofidece limits for the mea score of all the studets who took the examiatio. 16

17 5. The weight of vitami E i a capsule maufactured by a drug compay is ormally distributed with stadard deviatio 0.04 mg. A radom sample of 12 capsules was aalysed ad the mea weight of vitami E was foud to be 5.12 mg. (i) Calculate a 95% cofidece iterval for the populatio mea weight of vitami E per capsule. (ii) Give the values of the ed-poits of the iterval, correct to three sigificat figures. (iii) Explai what is meat by 95% cofidece? 6. A bak selected a radom sample of 400 customers ad foud that they had a mea credit of 280 with a stadard deviatio of 105 i their accouts. Calculate a 95% cofidece iterval for the mea credit of all the bak s customers. 7. Shoe shop staff routiely measure the legths of their customers feet. Measuremets of the legth of oe foot (without shoes) from each of 180 adult male customers yielded a mea legth of 29.2 cm ad a stadard deviatio of 1.47 cm. (i) Calculate a 95% cofidece iterval for the mea legth of male feet. (ii) Why was it ot ecessary to assume that the legths of feet are ormally distributed i order to calculate the cofidece iterval i (i) above? 8. A radom sample of 64 sweets is selected from a large batch. The sweets are foud to have a mea weight of grams ad a stadard deviatio of 0.1 grams. (i) Calculate the stadard error of the mea. (ii) What is the best estimate for, the mea of the large batch? (iii) Costruct a 95% cofidece iterval for. (iv) What would happe if a sample size of 100 was selected rather tha a sample of 64? (v) What coclusio ca you draw from your result i part (iv)? 9. A radom sample of 240 cars had a mea age of 4.6 years with a stadard deviatio of 2.5 years. (i) Give a 95% cofidece iterval for the mea age of all cars. (ii) What size of sample would be eeded to estimate the mea age, with 95% cofidece, to withi 0.2 years? boxes of cereal of a certai brad are weighed ad the mea weight is 748 grams with stadard deviatio 3.6 grams. (i) Fid a 95% cofidece iterval for the mea weight of all boxes of cereal of that brad. (ii) What size of sample would be eeded to estimate the mea weight, with 95% cofidece, to withi 1.5 grams? 11. Eighty people were asked to measure their pulse rates whe they woke up i the morig. The mea was 69 beats per miute ad the stadard deviatio 4 beats. (i) Fid a 95% cofidece iterval for the populatio mea. (ii) What size of sample would be eeded to estimate the mea umber of beats, with 95% cofidece, to withi 1.5 beats? 17

18 12. The weights of pebbles o a beach are distributed with mea 48.6 g ad stadard deviatio 8.5 g. A radom sample of 50 pebbles is chose. (i) Fid the probability that the mea weight will be less tha 49 g. (ii) Fid the limits withi which the cetral 95% of such sample meas would lie. (iii) How large a sample would be eeded i order that the cetral 95% of sample meas would lie i a iterval of width at most 4 g? 13. The 95% cofidece iterval for the mea mark of a group of studets is (54.09, 60.71). This iterval is based o the results from a radom sample of 80 studets. (i) Fid _ x, the mea of the sample. (ii) Fid, the stadard deviatio of the ormal populatio from which the sample is take. Sectio 5.3 Cofidece iterval for a proportio I Sectio 4.4 of Chapter 4 it was show how to fid the cofidece iterval for a populatio proportio usig the margi of error 1, where is the sample size. This cofidece iterval is show agai o the right. The 95% cofidece iterval for a proportio p is p^ 1 p p^ 1 Here is a remider of what a proportio is! If 150 televisio viewers are iterviewed i a sample survey ad 63 say they like a ew situatio comedy, the is the proportio of the sample who like the ew show. 150 This sample proportio, p^, is used as a estimate of the true populatio proportio p of televisio viewers who like the ew show. The otatio p^ is used to deote sample proportio. The otatio p is used to deote populatio proportio. Sice p is geerally ot kow, p^ is used as a estimator for the true populatio proportio, p. If may samples of the same size are take from a populatio, each sample will produce a differet (but similar) proportio. All these proportios form their ow distributio called the samplig distributio of the proportio. The stadard error, p^, of this distributio is give o page 34 of Formulae ad Tables ad is show o the right. p^ p(1 p) 18

19 I this sectio we will use the Stadard Normal Tables ( rather tha the margi of error, 1 ) to get a more accurate cofidece iterval for a populatio proportio. Sice the 95% level of cofidece will be used, the diagram o the right will remid us that 95% of a ormal distributio lies withi 1.96 stadard deviatios of the mea If p^ is the sample proportio ad p is the populatio proportio, the the 95% cofidece iterval for p is give by p^ 1.96 p(1 p) p p^ 1.96 p(1 p) This ca be writte more cocisely as p^ 1.96 p(1 p). The 95% cofidece iterval for a populatio proportio p^ 1.96 p(1 p) Note: A icrease i cofidece levels results i a icrease i the iterval width. Example 1 I a survey carried out i a large city, 170 households out of a radom sample of 250 owed at least oe pet. (i) Fid the stadard error of the samplig distributio of the proportio at the 95% cofidece level. (ii) Fid the 95% cofidece iterval for the proportio of households i the city who ow at least oe pet. (i) The sample proportio p^ Stadard error p^ p(1 p) 0.68(1 0.68) 250 p^

20 (ii) The 95% cofidece iterval is give by p^ 1.96 p(1 p) (0.029) from (i) above (0.6232, ) or about (62%, 74%) Example 2 A radom sample of 250 cars were surveyed passig a certai juctio ad 36 were foud to have K registratios. (i) Determie a 95% cofidece iterval for the proportio of cars i that area that have a K registratio. (ii) What sample size would have to be take i order to estimate the percetage to withi 2%? (i) The sample proportio p^ The 95% cofidece iterval is give by p^ 1.96 p(1 p) ( ) (0.0222) , The 95% cofidece iterval is (0.100, ). (ii) Let be the sample size. We require such that p^ p(1 p) p^ % ( ) 0.02 (0.144)(0.856) (0.02) (0.02) So a sample size of 309 would have to be take. 20

21 Exercise A maufacturer wats to assess the proportio of defective items i a large batch produced by a particular machie. He tests a radom sample of 300 items ad fids that 45 are defective. Calculate a 95% cofidece iterval for the proportio of defective items i the complete batch. 2. I order to assess the probability of a successful outcome, a experimet is performed 200 times ad the umber of successful outcomes is foud to be 72. Fid a 95% cofidece iterval for p, the probability of a successful outcome. 3. A market researcher carries out a survey i order to determie the popularity of SUDZ washig powder i the Cork area. He visits every house i a large housig estate i Cork ad asks the questio: Do you use SUDZ washig powder? Of 235 people questioed, 75 aswered YES. Treatig the sample as beig radom, calculate a 95% cofidece iterval for the proportio of households i the Cork area which use SUDZ. 4. A importer has ordered a large cosigmet of tomatoes. Whe it arrives, he examies a radomly chose sample of 50 boxes ad fids that 12 cotai at least oe bad tomato. Assumig that these boxes may be regarded as beig a radom sample from the boxes i the cosigmet, obtai a approximate 95% cofidece iterval for the proportio of boxes cotaiig at least oe bad tomato, givig your cofidece limits correct to three decimal places. 5. If 400 persos, costitutig a radom sample, are give a flu vaccie ad 136 of them experieced some discomfort, costruct a 95% large-sample cofidece iterval for the correspodig true proportio. 6. A radom sample of 120 library books is take as they are borrowed. They are classified as fictio or o-fictio, ad hardback or paperback. 88 books are foud to be fictio, ad of these, 74 are paperback. Fid a 95% cofidece limit for: (i) the proportio of books borrowed that are fictio (ii) the proportio of fictio books borrowed that are paperback. 7. I a sample of 400 shops take i 2012, it was discovered that 136 of them sold carpets at below the list prices which had bee recommeded by maufacturers. (i) Estimate the percetage of all carpet-sellig shops sellig below list price. (ii) Calculate the 95% cofidece limits for this estimate, ad explai briefly what these mea. (iii) What size sample would have to be take i order to estimate the percetage to withi 2%? 21

22 22 8. I a radom sample of 1,200 voters iterviewed atiowide, oly 324 felt that the salaries of certai govermet officials should be raised. Costruct a 95% cofidece iterval of the correspodig true proportio. 9. I a market research survey, 15 people out of a radom sample of 100 from a certai area said that they used a particular brad of soap. (i) Calculate a 95% cofidece iterval for the proportio of people who use this brad of soap. (ii) What size sample would eed to be take i order to estimate the percetage to withi 1 1_ %? Give your aswer correct to the earest Sectio 5.4 Hypothesis testig for a populatio mea Hypothesis testig I Sectios 5.2 ad 5.3 we dealt with cofidece itervals, oe of the two most commo types of statistical iferece. The secod type of statistical iferece has a differet objective. It is called hypothesis testig. Its purpose is to test the truth or otherwise of a claim, statemet or hypothesis made about a populatio parameter. A hypothesis is a statemet or cojecture made about some characteristic or parameter of a populatio. Here is a example of a hypothesis: The mea age of me o their weddig day is 32 years. A hypothesis test is a statistical method of provig the truth or otherwise of this statemet. It has already bee show that i ay ormal distributio 95% of the populatio lies withi 1.96 stadard deviatios of the mea, that is, 95% of the populatio will be i the iterval If we are dealig with a ormal distributio ad a experimet produces a result which is outside the iterval 1.96, we would be iclied to suspect that factors other tha chace are ivolved i the result. For example, some form of bias may be preset. If we toss a coi 100 times we would expect heads to occur 50 times. Do we coclude that the coi is biased if heads occur 60 times? Is the uexpected result more tha mere chace? To aswer this questio we start with the assumptio, or hypothesis, that the coi is ot biased. This assumptio is called the ull hypothesis, deoted by H 0. Usually the ull hypothesis is a statemet of o differece, o effect or o chage. A hypothesis test is the carried out to accept or reject the ull hypothesis. I this test, we speak of rejectig the ull hypothesis at a certai level. This certai level is called the level of sigificace. The 5% level of sigificace is by far the most commoly-used oe. It is the oly oe that we deal with i our course.

23 The 5% level of sigificace meas that the result obtaied is likely to occur o oly 5 occasios out of 100. At the 5% level of sigificace, the set of values, z 1.96 or z 1.96, is kow as the critical regio ad the boudaries of the critical regio are called the critical values. If the values of z are i the critical regio (i.e. z 1.96 or z 1.96), we reject the ull hypothesis ad coclude that factors other tha chace are ivolved. The critical regios at the 5% level of sigificace are show below. Reject ull hypothesis Reject ull hypothesis 2.5% 2.5% Hypothesis Testig At the 5% level of sigificace, the ull hypothesis is rejected if z 1.96 or z 1.96 Hypothesis testig for a populatio mea Whe a populatio is very large, it is geerally ot practical to fid the true mea ad stadard deviatio of the total populatio. However, assumptios are ofte made about these values ad their validity is tested based o observatios made from radom samples take from the populatio. Take, for example, machies desiged to produce batteries which last for 120 hours with a stadard deviatio 4 hours. What coclusios ca we come to about oe of these machies if a radom sample of 50 batteries produced by it had a mea life of 121 hours? We ow begi the process of ivestigatig whether these machies are producig the type of battery they were desiged to produce. This process is called hypothesis testig. Here are the basic steps of a hypothesis test: 1. Write dow H 0, the ull hypothesis, ad H 1, the alterative hypothesis. H 0 : The mea life of a battery is 120 hours. H 1 : The mea life of a battery is ot 120 hours. 2. State the sigificace level,. The sigificace level o our course is 5% ( 0.05). This meas that if z 1.96 or z 1.96, we reject the ull hypothesis ad accept the alterative hypothesis. 3. Calculate the value of the test statistic. This ivolves covertig the give uits to z-uits. 23

24 To covert the give uits to stadard uits we use _ x the sample mea _ populatio mea z x, where populatio stadard deviatio size of sample For the machie metioed above, _ x z The test statistic is z Come to a coclusio. Sice z does ot lie outside the rage 1.96 z 1.96 it is ot i the critical regio. So we accept the ull hypothesis which states that the mea life of a battery is 120 hours. Note: If, the stadard deviatio of the populatio is ot give, use _ x the stadard deviatio of the sample istead. Example 1 Over the years, a market gardeer foud that the mea yield from his tomato plats was 1.83 kg per plat with a stadard deviatio of 0.35 kg per plat. Oe year he plated 600 of a ew variety ad these yielded 1.87 kg per plat. At the 5% level of sigificace, test whether the mea yield from the ew plats is differet from his ormal variety. 1. H 0 : The mea is H 1 : The mea is ot The level of sigificace is 5%. The critical regio is z 1.96 or z Calculate the test statistic by covertig to stadard uits. x z x ( 600 ) z Sice z ad , we reject the ull hypothesis ad coclude that the ew variety is differet from the ormal variety. 24

25 Usig p-values Suppose we carry out a hypothesis test ad fid the test statistic to be z Sice 2.16 is greater tha 1.96, we reject the ull hypothesis at the 5% level of sigificace ( 0.05). Istead of comparig z 2.16 with z 1.96 (ad z 1.96), we compare the total area of the two coloured regios below with the specific level of sigificace, We use pages 36 ad 37 of Formulae ad Tables to fid the probability that z 2.16 or z P(z 2.16) P(z 2.16) 2P(z 2.16) 2[1 P(z 2.16)] 2[ ] 2[0.0154] The shaded areas above are referred to as the p-value, or probability-value correspodig to the observed value of the test statistic. The value foud above is the p-value that correspods to the test statistic z The p-value is iterpreted as the lowest level of sigificace at which the ull hypothesis could have bee rejected. With a test statistic of z 2.16, we would certaily have rejected the ull hypothesis at the specified level of sigificace ( 0.5). The p-value of gives us a specific or more precise level of sigificace. The smaller the p-value is, the stroger is the evidece agaist H 0 provided by the data. The p-value of a Test Statistic p-value p-value z 1 0 z 1 The p-value is the sum of the two shaded areas. p-value 2 P(z z 1 ), where z 1 is the test statistic. 25

26 Example 2 Calculate the p-value for the sample statistic z Sample statistic is z The sum of the probabilities that z 2.08 ad z 2.08 is the p-value. p-value 2 P(z 2.08 ) [1 P(z 2.08)] 2 [ ] 2(0.0188) p-value Steps ivolved i a Test of Sigificace usig a p-value 1. Write dow the ull hypothesis H 0 ad the alterative hypothesis H State the sigificace level. (O our course 0.5.) 3. Calculate the test statistic. 4. Fid the p-value that correspods to the test statistic. 5. If the p-value 0.05, the result is ot sigificat ad we do ot reject the ull hypothesis H 0. If the p-value 0.05, we reject the ull hypothesis H 0 i favour of the alterative hypothesis H 1. Example 3 A radom sample of 36 observatios is to be take from a distributio with stadard deviatio 10. I the past, the distributio has had a mea of 83, but it is believed that the mea may have chaged. Whe the sample was take it was foud to have a mea of (i) State H 0 ad H 1. (ii) Calculate the value of the test statistic. (iii) Calculate the p-value for the test statistic. (iv) Use the p-value to state if the result is sigificat at the 5% level of sigificace. Explai your coclusio. 26

27 (i) H 0 : Mea 83 H 1 : Mea 83 _ x (ii) Test statistic z z The test statistic is z 1.92 (iii) The p-value 2 P(z 1.92) 2 [1 P(z 1.92)] 2 [ ] 2(0.0274) (iv) As the p-value is ot less tha or equal to 0.05, the result is ot sigificat; we do ot reject the ull hypothesis. Exercise A ormal distributio is thought to have a mea of 50. A radom sample of 100 gave a mea of 52.4 ad a stadard deviatio of Is there evidece to suggest that the true mea is differet from the assumed mea at the 5% level of sigificace? 2. Over a log period the scores obtaied i a particular itelligece test were ormally distributed with mea score 70 ad stadard deviatio 6. Whe a test was take by a radom sample of 64 studets, the mea score was 68. Is there sufficiet evidece, at the 5% level of sigificace, that these studets differ from the ormal studets? 3. The maagemet of a large hospital states that the mea age of its patiets is 45 years. The HSE statistics departmet decides to test this claim about the mea age of the patiets. It took a radom sample of 100 patiets ad foud that the mea age was 48.4 years with a stadard deviatio of 18 years. (i) What is the ull hypothesis? (ii) State the alterative hypothesis. (iii) Work out the test statistic for the sample mea. (iv) At the 5% level of sigificace, is there evidece to show that the mea age of the patiets is ot 45 years? Give a reaso for your coclusio. 27

28 4. A particular machie produces metal rods which are ormally distributed with a mea legth of 210 cm ad with a stadard deviatio of 6 cm. The machie is serviced ad a sample is take to ivestigate if the mea legth has chaged. The sample of 100 rods gave a mea legth of cm. (i) What is the ull hypothesis? (ii) What is the alterative hypothesis? (iii) Work out the test statistic for the sample mea. (iv) At the 5% level of sigificace, is there evidece of a chage i the mea legth of rods produced by the machie? Explai your coclusio. 5. Mice kept uder laboratory coditios have a mea lifespa of 258 days ad a stadard deviatio of 45 days. 64 of these mice, selected at radom, were each give a measured dose of a certai drug each day, ad the mea lifespa for this group was 269 days. At the 5% level of sigificace, is there evidece to suggest that the drug has altered the mea lifespa of the mice? 6. I 1970 the average umber of childre per family i a certai tow was 3.8 with a stadard deviatio of 0.6. I 1980 a radom sample of 40 families had a total of 144 childre. At the 5% level of sigificace, is there evidece to coclude that the mea umber of childre per family had chaged? 7. The mea mark for all studets takig a certai Leavig Certificate subject was I a particular tow, 120 studets took this examiatio. The mea mark of these studets was 46.5 with a stadard deviatio of 9.5. At the 5% level of sigificace, is there evidece to suggest that the mea mark of the studets of this tow differs from the rest of the populatio? 8. I each of the followig, the z-score for a sample mea is give. Work out the correspodig p-value for each test statistic. (i) z 1.73 (ii) z 1.91 (iii) z 1.65 (iv) z Stadard batteries have a mea lifetime of 85 hours with a stadard deviatio 12 hours. A sample of 200 log-life batteries had a mea lifetime of 86.5 hours. (i) Calculate the sample statistic for this sample. (ii) Work out the correspodig p-value for this sample statistic. (iii) Is the result sigificat at the 5% level of sigificace? 10. Experiece has show that the scores obtaied i a particular test are ormally distributed with mea score 70 ad stadard deviatio 6. Whe the test is take by a radom sample of 36 studets, the mea is

29 (i) Calculate the sample statistic for this sample. (ii) Calculate the p-value for this sample statistic. (iii) Use the p-value you have foud to ivestigate if the mea score of the sample differs from the mea score of the populatio at the 5% level of sigificace. 11. The security departmet of a warehouse wats to kow whether the average time required by the ight watchma to walk his roud is 12.0 miutes. I a radom sample of 36 rouds, the ight watchma averaged 12.3 miutes with a stadard deviatio of 1.2 miutes. (i) Calculate the test statistic for this sample. (ii) Ca we reject the ull hypothesis that 12.0 miutes at the 5% level of sigificace? (iii) Work out the p-value that correspods to the test statistic foud i (i) above. (iv) If this p-value is used, do you reach the same coclusio with regard to sigificace at the 5% level? 12. The legths of metal bars produced by a particular machie are ormally distributed with mea legth 420 cm ad stadard deviatio 12 cm. The machie is serviced, after which a sample of 100 bars gives a mea legth of 423 cm. (i) Calculate the sample statistic for this sample. (ii) Work out the p-value for this sample statistic. (iii) Use this p-value to determie if there is evidece, at the 5% level, of a chage i the mea legth of the bars produced by the machie, assumig that the stadard deviatio remais the same. 13. A machie is desiged to produce screws with a stated mea legth of 5 mm. A radom sample of 400 screws produced by the machie is foud to have a mea legth of mm ad a stadard deviatio of mm. Estimate the stadard error of the mea, ad obtai a approximate 95% cofidece iterval for the mea of the whole output of this machie. Ivestigate if the mea of the sample differs sigificatly from the stated mea at the 5% level of sigificace. 29

30 Test yourself 5 A questios 1. The weights of a large collectio of bags of potatoes have a mea of 25 kg ad a stadard deviatio of 5 kg. Estimate, to 2 decimal places, the probability that a radom sample of 50 bags will have a mea weight of betwee 24.5 kg ad 25.5 kg. 2. A radom sample of size 20 is take from a populatio of size 80 (with replacemet). Fid the mea ad stadard error of the sample if the populatio is ormally distributed with mea 2.85 ad stadard deviatio The pulse-rate of a sample of 32 people was measured. The mea was foud to be 26.2 with stadard deviatio _ x Calculate the 95% cofidece iterval for the populatio mea. 4. A machie is regulated to dispese liquid ito cartos i such a way that the amout of liquid dispesed o each occasio is ormally distributed with a stadard deviatio of 20 ml. Fid the cofidece limits for the mea amout of liquid dispesed if a radom sample of 40 cartos had a average cotet of 266 ml. 5. Amog the first 150 customers at a ew sack bar, 90 order coffee. Assumig that this is a radom sample from the populatio of future customers, estimate a 95% cofidece iterval for the proportio of future customers who will order coffee. 6. A sample poll of 100 voters chose at radom from all voters i a give costituecy idicated that 55% of them were i favour of cadidate A. Fid the 95% cofidece iterval for the proportio of all the voters i the district i favour of this cadidate. 7. Irish third-level studets are kow to have a mea height of 176 cm with a stadard deviatio 11 cm. A radom sample of 60 equivalet Germa studets had a mea height of 179 cm. Does this suggest that the mea height of Germa studets differs from that of Irish studets at the 95% cofidece level? 8. Jars of hoey are filled by a machie. It has bee foud that the quatity of hoey i a jar has a mea of g with a stadard deviatio of 3.2 g. It is believed that the machie cotrols have bee altered i such a way that, although the stadard deviatio is ualtered, the mea quatity may have chaged. A radom sample of 60 jars is take ad the mea quatity of hoey per jar is foud to be g. 30

31 (i) State the ull ad alterative hypotheses. (ii) Calculate the sample statistic for the mea. (iii) Is there evidece, at the 5% level of sigificace, that the sample mea is differet from the populatio mea? 9. A firm produces batteries which are kow to have a mea lifetime of 96 hours. Forty samples of 36 batteries each are tested. (i) Describe the samplig distributio of the meas of these samples, metioig the theorem you have used to justify your aswer. (ii) Explai why the theorem you have metioed ca be applied whe the shape of the uderlyig populatio is ot kow. (iii) Estimate the umber of samples i which the average lifetime of the 36 batteries is greater tha 98 hours if the stadard deviatio of the batteries is 6 hours. 10. Draw a rough sketch of the ormal curve showig the critical regios, at the 5% level of sigificace, of a hypothesis test. (i) Clearly idicate the rejectio regios. (ii) What are the critical z-values for the limits of these rejectio regios? (iii) For a z-value of 1.6, estimate the correspodig p-value for this statistic. B questios 1. A large umber of radom samples of size are take from a ormal distributio with a mea of 74 ad a stadard deviatio of 6. The meas, _ x, of these samples are calculated. Fid the sample size required to esure that the probability of _ x 72 is The weights of bags of fertiliser may be modelled by a ormal distributio with mea 12.1 kg ad stadard deviatio 0.4 kg. Fid the probability that: (i) a radomly selected bag will weigh less tha 12.0 kg, (ii) the mea weight of four bags selected at radom will weigh more tha 12.0 kg, (iii) the mea weight of 100 bags will be betwee 12.0 ad 12.1 kg. How would your aswer to (iii) be affected if the ormal distributio was ot a good model for the weights of the bags? Explai your aswer. 3. A plat produces steel sheets whose weights are kow to be ormally distributed with a stadard deviatio of 2.4 kg. A radom sample of 36 sheets had a mea weight of 31.4 kg. Fid a 95% cofidece iterval for the mea weight of sheets produced by the plat. 31

32 4. The residets of a rural area are beig asked for their views o a pla to build a wid farm i their area. Evirometal campaigers claim that 20% of the residets are agaist the pla. (i) State oe reaso why surveyig a radom sample of 30 residets will allow reliable coclusios to be draw. (ii) Usig a 5% sigificace level, calculate a 95% cofidece iterval for the populatio proportio agaist the pla. 5. (i) Explai briefly what is meat by the term 95% cofidece iterval. (ii) A car maufacturig compay tested a radom sample of 150 cars of the same model to estimate the mea umber of kilometres travelled per litre of petrol cosumptio for all cars of that model. The sample mea of kilometres travelled per litre cosumed was ad the stadard deviatio was Form a 95% cofidece iterval for the mea umber of kilometres travelled per litre of petrol cosumed for all cars of that make. Give all calculatios correct to two places of decimal. 6. A eurologist wats to test the effect a ew drug has o respose times. 100 rats are ijected with a uit dose of this drug ad the respose times are recorded. The eurologist kows that the mea respose time for rats ot ijected with the drug is 1.2 secods. The mea respose time of the 100 rats ijected with the drug is 1.05 secods with a sample stadard deviatio of 0.5 secods. (i) State the ull ad alterative hypotheses for this test. (ii) Determie the critical regio at the 5% level of sigificace ad illustrate your aswer with a sketch. (iii) Calculate the test statistic ad aswer the questio Do you thik that the drug has a effect o the respose time at the 5% level of sigificace? (iv) Calculate the p-value for the test statistic ad iterpret this value. 7. A school of motorig claims that 80% of its cliets are successful i their first drivig test. A perso who did ot believe this claim took a radom sample of 72 cliets ad foud that 50 of these had bee successful i their first drivig test. (i) Usig 1, write dow the margi of error. (ii) Calculate the sample proportio as a decimal correct to two decimal places. (iii) Write dow the cofidece iterval, at the 95% level of cofidece, i terms of p^ ad. (iv) Ca the school s claim be upheld at the 95% level of cofidece? [Note: See Sectio 4.4, page 166.] 32

33 8. A market gardeer sells carrots i 25 kg sacks. The wholesaler suspects that the true mea weight is ot 25 kg. He weighs a radom sample of 50 sacks ad fids that the mea weight is 24.5 kg with a stadard deviatio of 1.5 kg. (i) State the ull ad alterative hypotheses. (ii) Calculate the sample statistic for the sample. (iii) Calculate the p-value for this statistic. (iv) Is the wholesaler s suspicio justified at the 5% level of sigificace? (v) Complete the followig setece: The p-value is the level of sigificace at which the ull hypothesis could have bee. 9. The weights of male studets at a large uiversity are ormally distributed with a mea of 68 kg ad a stadard deviatio of 3 kg. Eighty samples of 25 studets are picked at radom (with replacemet). (i) Fid the mea ad stadard error of the resultig samplig distributio. (ii) I how may of the samples would you expect the sample mea to be less tha 67.5 kg? C questios 1. A compay istals a ew machie i a factory. The compay claims that the machie will fill bags with wholemeal flour havig a mea weight of 500 g ad a stadard deviatio of 18 g. 36 bags are checked i a radom sample to test this claim. Their mea weight is 505 g. (i) State the ull ad the alterative hypotheses. (ii) Calculate the test statistic for the sample mea. (iii) Fid the p-value that correspods to the test statistic. (iv) Is the result sigificat at the 5% level of sigificace? Explai your aswer. 2. Amog 80 fish caught i a certai lake, 28 were iedible as a result of the chemical pollutio of their eviromet. (i) Work out the stadard error for this proportio. (ii) Costruct a 95% cofidece iterval for the true proportio of fish i this lake which are iedible as a result of chemical pollutio. 3. The 95% cofidece iterval for the mea weight, i grams, of a cosigmet of orages is (79.93, 82.87). This result is based o a radom sample of 400 orages. Usig this cofidece iterval, fid (i) _ x, the mea of the sample (ii), the stadard deviatio of the ormal populatio from which the sample is take. 33

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