LP10 INFERENTIAL STATISTICS - Cofidece iterval. Objective: - how to determie the cofidece iterval for the mea of a ample - Determiig Sample Size for a Specified Width Cofidece Iterval Theoretical coideratio Mot tatitical tudie are ot performed o the etire tatitical populatio from oe or more icoveiece: - Populatio ize ca ometime be very high; - The actual tudy i proportioal to the umber of etitie urveyed; - Cot ad reource allocated proportioately more etitie tudied; - There are ituatio where you ca ot collect iformatio about all idividual i the populatio. Therefore, the tudy populatio characteritic tatitical parameter are determied a follow: a. extracted a repreetative ample. The ample ize i choe o a to allow a tudy of the characteritic exhautiv; b. Depedig o the ature characteritic (quatitative or qualitative) uig decriptive tatitic to determie the mai parameter; c. uig iferetial tatitic i tryig etimate parameter for the etire populatio tartig from the reult obtaied from the ample. POINT ESTIMATION A poit etimate of a populatio parameter i a igle value of a tatitic. For example, The ample mea x i a poit etimate of the populatio mea μ. INTERVAL ESTIMATION A iterval etimate i defied by two umber, betwee which a populatio parameter i aid to lie. For example a < x < b i a iterval etimate of the populatio mea μ. It idicate that the populatio mea i greater tha a but le tha b. I ay etimatio problem, we eed to obtai both a poit etimate ad a iterval etimate. The poit etimate i our bet gue of the true value of the parameter, while the iterval etimate give a meaure of accuracy of that poit etimate by providig a iterval that cotai plauible value. Cofidece Iterval Statiticia ue a cofidece iterval to expre the preciio ad ucertaity aociated with a particular amplig method. A cofidece iterval coit of three part. A cofidece level. A tatitic. A margi of error. The cofidece level decribe the ucertaity of a amplig method. The tatitic ad the margi of error defie a iterval etimate that decribe the preciio of the method. The iterval etimate of a cofidece iterval i defied by the ample tatitic + margi of error. For example, uppoe we compute a iterval etimate of a populatio parameter. We might decribe thi iterval etimate a a 95% cofidece iterval. Thi mea that if we ued the ame amplig method to elect differet ample ad compute differet iterval etimate, the true populatio parameter would fall withi a rage defied by the ample tatitic + margi of error 95% of the time.
Cofidece iterval are preferred to poit etimate, becaue cofidece iterval idicate (a) the preciio of the etimate ad (b) the ucertaity of the etimate. Cofidece Level The probability part of a cofidece iterval i called a cofidece level. The cofidece level decribe the likelihood that a particular amplig method will produce a cofidece iterval that iclude the true populatio parameter. Here i how to iterpret a cofidece level. Suppoe we collected all poible ample from a give populatio, ad computed cofidece iterval for each ample. Some cofidece iterval would iclude the true populatio parameter; other would ot. A 95% cofidece level mea that 95% of the iterval cotai the true populatio parameter; a 90% cofidece level mea that 90% of the iterval cotai the populatio parameter; ad o o. Margi of Error I a cofidece iterval, the rage of value above ad below the ample tatitic i called the margi of error. How to Compute the Margi of Error The margi of error ca be defied by either of the followig equatio. Margi of error = Critical value x Stadard deviatio of the tatitic Margi of error = Critical value x Stadard error of the tatitic If you kow the tadard deviatio of the tatitic, ue the firt equatio to compute the margi of error. Otherwie, ue the ecod equatio. The variability of a tatitic i meaured by it tadard deviatio. The formula for computig the tadard deviatio from imple radom ample i: σ (σ: Populatio tadard deviatio; : Number of obervatio i the ample) Thi formula are valid whe the populatio ize i much larger (at leat 20 time larger) tha the ample ize. Sadly, the value of populatio parameter are ofte ukow, makig it impoible to compute the tadard deviatio of a tatitic. Whe thi occur, ue the tadard error. The formula for computig the tadard error i: (: tadard deviatio of ample, : Number of obervatio i the ample) Ue thi formula whe aume the populatio ize i at leat 20 time larger tha the ample ize. Etimate cofidece iterval The cofidece iterval i a rage bouded by value (limit called cofidece limit), which iclude feature media tudied. The rage i wider, the more we are ure that tudied media feature will be foud i that rage. Size of cofidece, cofidece i give by probability value () tudied lie i that rage. Trut (cofidece) i frequetly ued i 95%, 99% or 99.9% Quatitative variable - average etimate P - populatio X - quatitative variable From P (populatio) we radomly extracted a repreetative ample. The ample ha m (average) ad (tadard deviatio) ( m ad are puctual etimatio). Tryig to determie a cofidece iterval for μ average theoretical The probability to fid ukow theoretical average μ i a iterval [a, b] i 1-α:
Pr (a μ b) = 1 - α α i called igificace threhold or rik 1 - α i called cofidece level a ad b are called cofidece limit Determiatio iterval, trut i baed o the calculatio formula how below: There are may ituatio: A. A cofidece iterval of the mea whe the populatio variace i kow ( 2 ) x u x u i the ample ize x i the ample mea 2 i give ad repreet variatio of ample =0,05 or =5% u=1,96 =0,01 or =1% u=2,58 =0,001 or =0,1% u=3,29 : B. A cofidece iterval of the mea whe the populatio variace i ot kow N>120 x u x u where repreet the tadard error of the mea N<120 Where x t x t, 1, 1 repreet the tadard deviatio of ample i the ample ize x i the ample mea t,-1 read off from the table of the ditributio of "t" i the α ad -1 degree of freedom Calculatig the cofidece iterval formula i made accordig to the ituatio A or B or with CONFIDENCE fuctio implemeted i the tatitical fuctio i Excel CONFIDENCE.T fuctio Retur the cofidece iterval for a populatio mea, uig a Studet' t ditributio. Sytax CONFIDENCE.T(alpha,tadard_dev,ize) The CONFIDENCE.T fuctio ytax ha the followig argumet: Alpha Required. The igificace level ued to compute the cofidece level. The cofidece level equal 100*(1 - alpha)%, or i other word, a alpha of 0.05 idicate a 95 percet cofidece level. Stadard_dev Required. The populatio tadard deviatio for the data rage ad i aumed to be kow. Size Required. The ample ize. How large a ample hould be? Statitically peakig, the larger the better!!!
Sample ize determiatio i cloely related to tatitical etimatio. Quite ofte, oe ak, How large a ample i eceary to make a accurate etimate? The awer i ot imple, ice it deped o three thig: the maximum error of etimate, the populatio tadard deviatio, ad the degree of cofidece. The formula for ample ize i derived from the maximum error of etimate formula Ad thi formula i olved for a follow: Where E i the maximum error of etimate. If there i ay fractio or decimal portio i the awer, ue the ext whole umber for ample ize,. Exemplu. How may obervatio hould be made to etimate the average ize adult male populatio if it i kow that S 2 = 9 cm2 ad the required accuracy =1 cm, to be eured with probability 1 - α = 0.95 =0.05 S=3 cm z=1,96 for =34 z0,05;34=2,033 3 1,96 1 3 2.033 1 2 35,574 35 2 37,197 37 37 obervatio are eeded to etimate the medium to withi 1 cm.
Problem 1 (olved). The quetio of whether i the cae of treatmet with a certai type of drug that caue vaocotrictio it ifluece the ytolic blood preure of the patiet. It i kow that the tadard deviatio of ytolic blood preure i a populatio of healthy adult i 10 mm Hg. It require the calculatio of the cofidece iterval of the mea ytolic blood preure of adult populatio treated with medicie vaocotrictio. To reolve the requiremet ha tudied a ample of 40 radomly choe idividual from the populatio of healthy adult who coumed a particular drug previouly producig vaocotrictio. TAS value for patiet i the ample are: Nr_id TAS TAS Nr_id (mm/hg) (mm/hg) 1 126 21 126 2 130 22 125 3 135 23 115 4 116 24 117 5 122 25 121 6 126 26 125 7 128 27 129 8 130 28 124 9 123 29 131 10 124 30 127 11 140 31 127 12 125 32 125 13 120 33 140 14 121 34 115 15 140 35 110 16 135 36 112 17 115 37 125 18 135 38 143 19 132 39 122 20 128 40 135 Populatio variace i kow. Apply formula i cae A. The ext tep: 1. Calculate the ample mea (AVERAGE fuctio i Excel) 2. Iert formula (Excel) to calculate upper ad lower limit of the rage. 3. Shall be calculated 3 iterval for igificace threhold =0,05; 0,01; 0,001 Cofidece iterval ca be cotructed ad uig the CONFIDENCE.T (= CONFIDENCE.T (0,05, 10, 40)) The iterpretatio iterval of cofidece: the cofidece iterval of the mea etimate rage that iclude ukow media TAS for the populatio that coume the MPV with a certai level of trut. Whe alpha i 0.05, the we have a cofidece iterval of 95%. I our cae the cofidece iterval for the mea SBP of
populatio who coume MPV i [123,03, 129,22]. We ca affirm that the arithmetic mea of the TAS that coume drug that caue vaocotrictio (the etire populatio) i i the rage [123,03; 129,22] with a error of 0.05. Problem 2. A total choleterol level of 50 broiler feed uder itake alumium ratio i how i the followig table: 107 138 134 90 138 105 105 129 128 105 109 102 135 115 104 111 99 126 114 110 120 109 128 111 111 127 111 104 112 121 98 128 98 110 127 115 125 88 99 128 114 123 95 89 130 118 102 105 105 135 Required: Calculate the cofidece iterval for the average choleterol level raied Broiler populatio with alumium itake i feed itake. Fidig of the cofidece iterval for average Cofidece iterval i calculated accordig to the formula: Where: x i arithmetic mea of the ample, tcrit i t critic two tailed, i tadard deviatio of the ample. [x t critic, x + t critic ] I. Calculate Cofidece iterval uig the formula baed o the defiitio a. The lower limit uig the formula: : x t critic II. III. IV. b. The upper limit uig the formula:: x + t critic Calculate CI with CONFIDENCE fuctio. Calculate CI with Decriptive Statitic of Data Aalye Iterpret the cofidece iterval: Problema 3 Suppoe that the etimated average icome of a family i a city with a error up to 50 RON ad 90% cofidece. Sice i ukow, we aume that we choe a ample of 80 familie ad calculated = 650. How may familie mut be teted to obtai a accurate etimate? =50 Zcritic for =80 și =90% i 1,66 S=650
Problem 1 (problem olved) Nie hudred (900) cow were radomly elected for a urvey. The mea fat of milk wa 2.7%, ad the tadard deviatio wa 0.4. What i the margi of error, aumig a 95% cofidece level? (A) 0.013 (B) 0.025 (C) 0.500 (D) 1.960 (E) Noe of the above. Solutio The correct awer i (B). To compute the margi of error, we eed to fid: 1. the critical value ad 2. the tadard error of the mea. To fid the critical value, we take the followig tep. Compute alpha (α): α = 1 - (cofidece level / 100) = 1-0.95 = 0.05 Fid the critical probability (p*): p* = 1 - α/2 = 1-0.05/2 = 0.975 Fid the degree of freedom (df): df = - 1 = 900-1 = 899 Fid the critical value. Sice we do't kow the populatio tadard deviatio, we'll expre the critical value a a t tatitic. For thi problem, it will be the t tatitic havig 899 degree of freedom ad a cumulative probability equal to 0.975. Uig the t Ditributio Calculator, we fid that the critical value i 1.96. Next, we fid the tadard error of the mea, uig the followig equatio: SEx = / qrt( ) = 0.4 / qrt( 900 ) = 0.4 / 30 = 0.013 Ad fially, we compute the margi of error (ME). ME = Critical value x Stadard error = 1.96 * 0.013 = 0.025 Thi mea we ca be 95% cofidet that the mea fat of milk i the cow populatio i 2.7 plu or miu 0.025, ice the margi of error i 0.025. Note: The larger the ample ize, the more cloely the t ditributio look like the ormal ditributio. For thi problem, ice the ample ize i very large, we would have foud the ame reult with a z-core a we foud with a t tatitic. That i, the critical value would till have bee 1.96. The choice of t tatitic veru z-core doe ot make much practical differece whe the ample ize i very large.