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

Save this PDF as:

Size: px
Start display at page:

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

## Transcription

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

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

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

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

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

### Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis

Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, P-value Type I Error, Type II Error, Sigificace Level, Power Sectio 8-1: Overview Cofidece Itervals (Chapter 7) are

### 1. C. The formula for the confidence interval for a population mean is: x t, which was

s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-value

### Definition. Definition. 7-2 Estimating a Population Proportion. Definition. Definition

7- stimatig a Populatio Proportio I this sectio we preset methods for usig a sample proportio to estimate the value of a populatio proportio. The sample proportio is the best poit estimate of the populatio

### Inference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval

Chapter 8 Tests of Statistical Hypotheses 8. Tests about Proportios HT - Iferece o Proportio Parameter: Populatio Proportio p (or π) (Percetage of people has o health isurace) x Statistic: Sample Proportio

### Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.

Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).

### Hypothesis testing. Null and alternative hypotheses

Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate

### Section 7-3 Estimating a Population. Requirements

Sectio 7-3 Estimatig a Populatio Mea: σ Kow Key Cocept This sectio presets methods for usig sample data to fid a poit estimate ad cofidece iterval estimate of a populatio mea. A key requiremet i this sectio

### Unit 20 Hypotheses Testing

Uit 2 Hypotheses Testig Objectives: To uderstad how to formulate a ull hypothesis ad a alterative hypothesis about a populatio proportio, ad how to choose a sigificace level To uderstad how to collect

### Review for 1 sample CI Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Review for 1 sample CI Name MULTIPLE CHOICE. Choose the oe alterative that best completes the statemet or aswers the questio. Fid the margi of error for the give cofidece iterval. 1) A survey foud that

### Determining the sample size

Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors

### Hypothesis Tests Applied to Means

The Samplig Distributio of the Mea Hypothesis Tests Applied to Meas Recall that the samplig distributio of the mea is the distributio of sample meas that would be obtaied from a particular populatio (with

### 9.8: THE POWER OF A TEST

9.8: The Power of a Test CD9-1 9.8: THE POWER OF A TEST I the iitial discussio of statistical hypothesis testig, the two types of risks that are take whe decisios are made about populatio parameters based

### Practice Problems for Test 3

Practice Problems for Test 3 Note: these problems oly cover CIs ad hypothesis testig You are also resposible for kowig the samplig distributio of the sample meas, ad the Cetral Limit Theorem Review all

### Confidence Intervals and Sample Size

8/7/015 C H A P T E R S E V E N Cofidece Itervals ad Copyright 015 The McGraw-Hill Compaies, Ic. Permissio required for reproductio or display. 1 Cofidece Itervals ad Outlie 7-1 Cofidece Itervals for the

### Review for Test 3. b. Construct the 90% and 95% confidence intervals for the population mean. Interpret the CIs.

Review for Test 3 1 From a radom sample of 36 days i a recet year, the closig stock prices of Hasbro had a mea of \$1931 From past studies we kow that the populatio stadard deviatio is \$237 a Should you

### CHAPTER 7: Central Limit Theorem: CLT for Averages (Means)

CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:

### Center, Spread, and Shape in Inference: Claims, Caveats, and Insights

Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the

### Homework 7 Solutions Total Points

Homework 7 Solutios - 165 Total Poits STAT 201-502 Lecture 11, 12, & 13 Material 1. Studies that compare treatmets for chroic medical coditios such as headaches ca use the same subjects for each treatmet.

### Statistics Lecture 14. Introduction to Inference. Administrative Notes. Hypothesis Tests. Last Class: Confidence Intervals

Statistics 111 - Lecture 14 Itroductio to Iferece Hypothesis Tests Admiistrative Notes Sprig Break! No lectures o Tuesday, March 8 th ad Thursday March 10 th Exteded Sprig Break! There is o Stat 111 recitatio

### Overview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals

Overview Estimatig the Value of a Parameter Usig Cofidece Itervals We apply the results about the sample mea the problem of estimatio Estimatio is the process of usig sample data estimate the value of

### Confidence Intervals for the Population Mean

Cofidece Itervals Math 283 Cofidece Itervals for the Populatio Mea Recall that from the empirical rule that the iterval of the mea plus/mius 2 times the stadard deviatio will cotai about 95% of the observatios.

### Case Study. Normal and t Distributions. Density Plot. Normal Distributions

Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca

### 7.1 Inference for a Population Proportion

7.1 Iferece for a Populatio Proportio Defiitio. The statistic that estimates the parameter p is the sample proportio cout of successes i the sample ˆp = cout of observatios i the sample. Assumptios for

### Chapter 10. Hypothesis Tests Regarding a Parameter. 10.1 The Language of Hypothesis Testing

Chapter 10 Hypothesis Tests Regardig a Parameter A secod type of statistical iferece is hypothesis testig. Here, rather tha use either a poit (or iterval) estimate from a simple radom sample to approximate

### One-sample test of proportions

Oe-sample test of proportios The Settig: Idividuals i some populatio ca be classified ito oe of two categories. You wat to make iferece about the proportio i each category, so you draw a sample. Examples:

### ˆ p 2. ˆ p 1. ˆ p 3. p 4. ˆ p 8

Sectio 8 1C The Techiques of Hypothesis Testig A claim is made that 10% of the populatio is left haded. A alterate claim is made that less tha 10% of the populatio is left haded. We will use the techiques

### Lesson 17 Pearson s Correlation Coefficient

Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig

### I. Chi-squared Distributions

1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

### Statistical inference: example 1. Inferential Statistics

Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either

### Confidence Intervals for One Mean with Tolerance Probability

Chapter 421 Cofidece Itervals for Oe Mea with Tolerace Probability Itroductio This procedure calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) with

### PSYCHOLOGICAL STATISTICS

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics

### Confidence Intervals for the Mean of Non-normal Data Class 23, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Cofidece Itervals for the Mea of No-ormal Data Class 23, 8.05, Sprig 204 Jeremy Orloff ad Joatha Bloom Learig Goals. Be able to derive the formula for coservative ormal cofidece itervals for the proportio

### The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles

The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio

### Ch 7.1 pg. 364 #11, 13, 15, 17, 19, 21, 23, 25

Math 7 Elemetary Statistics: A Brief Versio, 5/e Bluma Ch 7.1 pg. 364 #11, 13, 15, 17, 19, 1, 3, 5 11. Readig Scores: A sample of the readig scores of 35 fifth-graders has a mea of 8. The stadard deviatio

### Confidence Intervals for One Mean

Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a

### Probability & Statistics Chapter 9 Hypothesis Testing

I Itroductio to Probability & Statistics A statisticia s most importat job is to draw ifereces about populatios based o samples take from the populatio Methods for drawig ifereces about parameters: ) Make

### Standard Errors and Confidence Intervals

Stadard Errors ad Cofidece Itervals Itroductio I the documet Data Descriptio, Populatios ad the Normal Distributio a sample had bee obtaied from the populatio of heights of 5-year-old boys. If we assume

### CHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Means and Proportions

CHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Meas ad Proportios Itroductio: We wat to kow the value of a parameter for a populatio. We do t kow the value of this parameter for the etire populatio because

### Section 7.2 Confidence Interval for a Proportion

Sectio 7.2 Cofidece Iterval for a Proportio Before ay ifereces ca be made about a proportio, certai coditios must be satisfied: 1. The sample must be a SRS from the populatio of iterest. 2. The populatio

### Hypothesis Testing. Definitions. H 0 : The Null Hypothesis This is the hypothesis or claim that is initially assumed to be true.

Hypothesis Testig Hypothesis testig allows us to use a sample to decide betwee two statemets made about a Populatio characteristic. These two statemets are called the Null Hypothesis ad the Alterative

### Elementary Statistics and Inference. Elementary Statistics and Inference. Chapter 26 Tests of Significance (cont.) 22S:025 or 7P:025.

Elemetary Statistics ad Iferece 22S:25 or 7P:25 Lecture 35 Elemetary Statistics ad Iferece 22S:25 or 7P:25 Chapter 26 (cot.) 2 A) Zero-Oe Boxes (Populatios) Recall we ca use the Normal Curve table to study

### Statistical Inference: Hypothesis Testing for Single Populations

Chapter 9 Statistical Iferece: Hypothesis Testig for Sigle Populatios A foremost statistical mechaism for decisio makig is the hypothesis test. The cocept of hypothesis testig lies at the heart of iferetial

### Inference for Proportions Inference for a Single Proportion

Iferece for Proportios Iferece for a Sigle Proportio IPS Chapter 8. 009 W.H. Freema ad Compay Objectives (IPS Chapter 8.) Iferece for a sigle proportio Large-sample cofidece iterval for p Plus four cofidece

### Math C067 Sampling Distributions

Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters

### Using Excel to Construct Confidence Intervals

OPIM 303 Statistics Ja Stallaert Usig Excel to Costruct Cofidece Itervals This hadout explais how to costruct cofidece itervals i Excel for the followig cases: 1. Cofidece Itervals for the mea of a populatio

### STA 2023 Practice Questions Exam 2 Chapter 7- sec 9.2. Case parameter estimator standard error Estimate of standard error

STA 2023 Practice Questios Exam 2 Chapter 7- sec 9.2 Formulas Give o the test: Case parameter estimator stadard error Estimate of stadard error Samplig Distributio oe mea x s t (-1) oe p ( 1 p) CI: prop.

### Chapter 7: Confidence Interval and Sample Size

Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum

### 1 Hypothesis testing for a single mean

BST 140.65 Hypothesis Testig Review otes 1 Hypothesis testig for a sigle mea 1. The ull, or status quo, hypothesis is labeled H 0, the alterative H a or H 1 or H.... A type I error occurs whe we falsely

### Hypergeometric Distributions

7.4 Hypergeometric Distributios Whe choosig the startig lie-up for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you

### University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution

Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.

### 5: Introduction to Estimation

5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample

### Measures of Spread and Boxplots Discrete Math, Section 9.4

Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,

### Example Consider the following set of data, showing the number of times a sample of 5 students check their per day:

Sectio 82: Measures of cetral tedecy Whe thikig about questios such as: how may calories do I eat per day? or how much time do I sped talkig per day?, we quickly realize that the aswer will vary from day

### 15.075 Exam 3. Instructor: Cynthia Rudin TA: Dimitrios Bisias. November 22, 2011

15.075 Exam 3 Istructor: Cythia Rudi TA: Dimitrios Bisias November 22, 2011 Gradig is based o demostratio of coceptual uderstadig, so you eed to show all of your work. Problem 1 A compay makes high-defiitio

### Z-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown

Z-TEST / Z-STATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large T-TEST / T-STATISTIC: used to test hypotheses about

### Statistical Methods. Chapter 1: Overview and Descriptive Statistics

Geeral Itroductio Statistical Methods Chapter 1: Overview ad Descriptive Statistics Statistics studies data, populatio, ad samples. Descriptive Statistics vs Iferetial Statistics. Descriptive Statistics

### 4.1 Sigma Notation and Riemann Sums

0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

### 1 Computing the Standard Deviation of Sample Means

Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

### AQA STATISTICS 1 REVISION NOTES

AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if

### Hypothesis testing in a Nutshell

Hypothesis testig i a Nutshell Summary by Pamela Peterso Drake Itroductio The purpose of this readig is to discuss aother aspect of statistical iferece, testig. A is a statemet about the value of a populatio

### 23.3 Sampling Distributions

COMMON CORE Locker LESSON Commo Core Math Stadards The studet is expected to: COMMON CORE S-IC.B.4 Use data from a sample survey to estimate a populatio mea or proportio; develop a margi of error through

### Economics 140A Confidence Intervals and Hypothesis Testing

Ecoomics 140A Cofidece Itervals ad Hypothesis Testig Obtaiig a estimate of a parameter is ot the al purpose of statistical iferece because it is highly ulikely that the populatio value of a parameter is

### ME 101 Measurement Demonstration (MD 1) DEFINITIONS Precision - A measure of agreement between repeated measurements (repeatability).

INTRODUCTION This laboratory ivestigatio ivolves makig both legth ad mass measuremets of a populatio, ad the assessig statistical parameters to describe that populatio. For example, oe may wat to determie

### Stat 104 Lecture 16. Statistics 104 Lecture 16 (IPS 6.1) Confidence intervals - the general concept

Statistics 104 Lecture 16 (IPS 6.1) Outlie for today Cofidece itervals Cofidece itervals for a mea, µ (kow σ) Cofidece itervals for a proportio, p Margi of error ad sample size Review of mai topics for

### TIEE Teaching Issues and Experiments in Ecology - Volume 1, January 2004

TIEE Teachig Issues ad Experimets i Ecology - Volume 1, Jauary 2004 EXPERIMENTS Evirometal Correlates of Leaf Stomata Desity Bruce W. Grat ad Itzick Vatick Biology, Wideer Uiversity, Chester PA, 19013

### GCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.

GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea - add up all

### Confidence Intervals

Cofidece Itervals Cofidece Itervals are a extesio of the cocept of Margi of Error which we met earlier i this course. Remember we saw: The sample proportio will differ from the populatio proportio by more

### Descriptive statistics deals with the description or simple analysis of population or sample data.

Descriptive statistics Some basic cocepts A populatio is a fiite or ifiite collectio of idividuals or objects. Ofte it is impossible or impractical to get data o all the members of the populatio ad a small

### In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

### Case Study. Contingency Tables. Graphing Tabled Counts. Stacked Bar Graph

Case Study Cotigecy Tables Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 4 6, 2011 Case Study Example 9.3 begiig o page 213 of the text describes a experimet i which

### 1 Correlation and Regression Analysis

1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio

### Hypothesis testing: one sample

Hypothesis testig: oe sample Describig iformatios Flow-chart for QMS 202 Drawig coclusios Forecastig Improve busiess processes Data Collectio Probability & Probability Distributio Regressio Aalysis Time-series

### CHAPTER 11 Financial mathematics

CHAPTER 11 Fiacial mathematics I this chapter you will: Calculate iterest usig the simple iterest formula ( ) Use the simple iterest formula to calculate the pricipal (P) Use the simple iterest formula

### The shaded region above represents the region in which z lies.

GCE A Level H Maths Solutio Paper SECTION A (PURE MATHEMATICS) (i) Im 3 Note: Uless required i the questio, it would be sufficiet to just idicate the cetre ad radius of the circle i such a locus drawig.

### Descriptive Statistics

Descriptive Statistics We leared to describe data sets graphically. We ca also describe a data set umerically. Measures of Locatio Defiitio The sample mea is the arithmetic average of values. We deote

Ca you aswer these questios? A savigs accout gives % iterest per aum.. If 000 is ivested i this accout, how much will be i the accout at the ed of years? A ew car costs 16 000 ad its value falls by 1%

### Explore Identifying Likely Population Proportions

COMMON CORE Locker LESSON Cofidece Itervals ad Margis of Error Commo Core Math Stadards The studet is expected to: COMMON CORE S-IC.B.4 Use data from a sample survey to estimate a populatio mea or proportio;

### AP * Statistics Review. Inference

AP * Statistics Review Iferece Teacher Packet AP* is a trademark of the College Etrace Examiatio Board. The College Etrace Examiatio Board was ot ivolved i the productio of this material. Copyright 009

### Riemann Sums y = f (x)

Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, o-egative fuctio o the closed iterval [a, b] Fid

### Institute for the Advancement of University Learning & Department of Statistics

Istitute for the Advacemet of Uiversity Learig & Departmet of Statistics Descriptive Statistics for Research (Hilary Term, 00) Lecture 5: Cofidece Itervals (I.) Itroductio Cofidece itervals (or regios)

### x : X bar Mean (i.e. Average) of a sample

A quick referece for symbols ad formulas covered i COGS14: MEAN OF SAMPLE: x = x i x : X bar Mea (i.e. Average) of a sample x i : X sub i This stads for each idividual value you have i your sample. For

### Laboratory: Case-Control Studies. Hypothesis Testing

Laboratory: Case-Cotrol Studies How may do I eed? is oe of the most commo questios addressed to a epidemiologist. The epidemiologist aswers with What questio are you attemptig to aswer? Sample size depeds

### 3. Continuous Random Variables

Statistics ad probability: 3-1 3. Cotiuous Radom Variables A cotiuous radom variable is a radom variable which ca take values measured o a cotiuous scale e.g. weights, stregths, times or legths. For ay

### Notes on Hypothesis Testing

Probability & Statistics Grishpa Notes o Hypothesis Testig A radom sample X = X 1,..., X is observed, with joit pmf/pdf f θ x 1,..., x. The values x = x 1,..., x of X lie i some sample space X. The parameter

### OMG! Excessive Texting Tied to Risky Teen Behaviors

BUSIESS WEEK: EXECUTIVE EALT ovember 09, 2010 OMG! Excessive Textig Tied to Risky Tee Behaviors Kids who sed more tha 120 a day more likely to try drugs, alcohol ad sex, researchers fid TUESDAY, ov. 9

### Recursion and Recurrences

Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,

### 3.1 Measures of Central Tendency. Introduction 5/28/2013. Data Description. Outline. Objectives. Objectives. Traditional Statistics Average

5/8/013 C H 3A P T E R Outlie 3 1 Measures of Cetral Tedecy 3 Measures of Variatio 3 3 3 Measuresof Positio 3 4 Exploratory Data Aalysis Copyright 013 The McGraw Hill Compaies, Ic. C H 3A P T E R Objectives

### 0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5

Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.

### Statistics for Clinicians. 7: Sample size

J. Paediatr. Child Health (2002) 38, 300 304 Statistics for Cliicias 7: Sample size JB CARLIN 1,3 ad LW DOYLE 2,3,4 1 Cliical Epidemiology ad Biostatistics Uit, Murdoch Childre s Research Istitute, Departmets

### MEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book)

MEI Mathematics i Educatio ad Idustry MEI Structured Mathematics Module Summary Sheets Statistics (Versio B: referece to ew book) Topic : The Poisso Distributio Topic : The Normal Distributio Topic 3:

### 1 The Binomial Theorem: Another Approach

The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets

### Analyzing Longitudinal Data from Complex Surveys Using SUDAAN

Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical

### FM4 CREDIT AND BORROWING

FM4 CREDIT AND BORROWING Whe you purchase big ticket items such as cars, boats, televisios ad the like, retailers ad fiacial istitutios have various terms ad coditios that are implemeted for the cosumer

### The Poisson Distribution

Lecture 5 The Poisso Distributio 5.1 Itroductio Example 5.1: Drowigs i Malta The book [Mou98] cites data from the St. Luke s Hospital Gazette, o the mothly umber of drowigs o Malta, over a period of early

### Estimating the Mean and Variance of a Normal Distribution

Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbers

### Quadrat Sampling in Population Ecology

Quadrat Samplig i Populatio Ecology Backgroud Estimatig the abudace of orgaisms. Ecology is ofte referred to as the "study of distributio ad abudace". This beig true, we would ofte like to kow how may

### SAMPLING NTI Bulletin 2006,42/3&4, 55-62

SAMPLING NTI Bulleti 006,4/3&4, 55-6 Sample size determiatio i health studies VK Chadha * Summary Oe of the most importat factors to cosider i the desig of a itervetio trial is the choice of a appropriate