So we find a sample mean but what can we say about the General Education Statistics

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1 So we fid a ample mea but what ca we ay about the Geeral Educatio Statitic populatio? Cla Note Cofidece Iterval for Populatio Mea (Sectio 9.) We will be doig early the ame tuff a we did i the firt ectio of thi chapter. However, there are ome igificat differece. The mai oe i that we do ot ue the ormal ditributio. We will be uig a imilar, bell-haped curve called Studet t-ditributio. To make a log tory ot o log, uig the method dicued i the previou ectio to fid the σ cofidece iterval for x would be poit etimate ± margi of error or rather x ± zα. Thi i a problem becaue thi aume we kow the tadard deviatio of the populatio, which we would ot. We ca ubtitute the tadard deviatio of the ample, givig u x ± zα. However, by uig itead of σ, we itroduce too much variatio ad we ee that the x μx x μ z-core of x (ow zx = = ) i ot ormally ditributed. σ x At firt, thi eem like we hit a brick wall. Luckily, a ma by the ame of William Goett i the early 1900 olved thi problem by creatig what i kow a Studet t-ditributio. It i imilar to the ormal curve but ha a little more variatio. We ee the ormal curve here compared with two curve of the t-ditributio. The t-curve ha fatter tail but a le-high ceter poit. The t-curve deped o the value of the ample ize. 1

2 Recall: Defiitio: A poit etimate i the value of a (ample) tatitic that etimate the value of a (populatio) parameter. I thi ectio, the poit etimate we are playig with i x, the ample mea. It i aid to etimate μ, the populatio mea. Studet t-ditributio: Suppoe that a imple radom ample of ize i take from a populatio. If the populatio from which the ample i draw follow a ormal ditributio, the ditributio of x μx x μ t = = follow Studet t-ditributio with 1 degree of freedom, where x i σ x the ample mea, i the ample tadard deviatio, ad μ i the populatio mea. Propertie of the t-ditributio: There are imilaritie to the ormal ditributio a we will ee. However, there are igificat differece. 1. The t-ditributio i differet for differet degree of freedom. The degree of freedom will be calculated a 1.. The t-ditributio i cetered at 0 ad i ymmetric about The area uder the curve i 1. The area uder the curve to the right of 0 i equal to the area uder the curve to the left of 0. Thi area i ½. 4. A t icreae without boud (get larger ad larger), the graph approache, but ever reache, the horizotal axi. A t decreae without boud (get more ad more egative), the graph approache, but ever reache, the horizotal axi. 5. The area i the tail of the t-ditributio i a little greater tha the area i the tail of the tadard ormal ditributio, becaue we are uig a a etimate of σ, thereby itroducig further variability ito the t- tatitic. 6. A the ample ize icreae, the deity curve of t get cloer to the tadard ormal deity curve. Thi reult occur becaue, a the ample ize icreae, the value of get cloer to the value of σ, by the Law of Large Number.

3 Determiig t-value: Jut a we looked up z-core o a table, we will look upp t-value. We will ue the otatio to deotee the t-valuee whoe areaa uder the curve to the right of t α i α. We will eed to ue the ample ize,. A metioed earlier, the degree of f freedom will be calculated a 1. expl 1: Fid the t-value uch that the area uder the t-ditributio curve to the right i 0.05, aumig 19 degree of freedom. Ue the partial table give below. t α Label your t-value a a 0.05 t. Draw a quick t-curve with your t-value labeled ad the area 0.05 haded. To look upp a t-value, fid the row with the correct degree of freedom ad the colum with the correct value off α. Table VII i the book, o page A-14, will give you thee ame value. 3

4 Uig the TI calculator: There i a fuctio o the TI calculator that ca be ued for thi. However, you mut eter the area to the left of the deired t-value. Let redo example 1. expl : Fid the t-value uch that the area uder the t-ditributio curve to the right i 0.05, aumig 19 degree of freedom. Follow thee tep to do thi o the calculator. a.) If the area uder the curve to the right of the t-value i 0.05, what i the area to the left uder the curve? b.) Ue the ivt feature o the calculator to fid thi t-value. Older calculator may ot have thi fuctio. Pre d VARS to get to the DISTR meu. Select 4: ivt(. After the parethei, eter the area to the left ad the degree of freedom, eparated by a comma. Now, the reao we eed t-value i that we wat to create cofidece iterval a we did i the previou ectio. Cotructig a (1 α)100% Cofidece Iterval for μ: We mut verify that the followig i true before cotiuig. ample data come from a imple radom ample or radomized experimet, ample data ha o eriou outlier, ample ize i mall relative to the populatio ize ( 0.05N), ad the data come from a populatio that i ormally ditributed, or the ample ize i large. A (1 α) 100% cofidece iterval for μ i give by Lower boud: Upper boud: x t α x + t α Here we ee poit etimate ± margi of error. Sometime it called a t-iterval. The margi of error may be labeled E. where t α i the critical value with 1 degree of freedom. 4

5 expl 3: A ample of ixtee 011 Ford Focu car ha a mea ga mileage of 36.8 mile per gallo. The ample tadard deviatio i.9 mile per gallo. A ormal probability plot ha how that the data could come from a ormal populatio. I additio, a boxplot ha how that there are o outlier. Cotruct a 95% cofidece iterval for the mea ga mileage of all 011 Ford Focu car. Follow the tep below. a.) Fid the t-value eeded o the table. A complete table i give o the lat page of thee ote. b.) Calculate the margi of error a t α. c.) Form the 95% cofidece iterval ad tate it i etece form. Determiig the Sample Size : The ample ize required to etimate the populatio mea, µ, with a level of cofidece z α (1 α) 100% with a pecified margi of error, E, i give by =. E We roud up a before. expl 4: Coider agai the Ford Focue. How large a ample do we eed if we wat to form a 95% cofidece iterval to etimate the mea withi 0.5 mile per gallo? Notice we ue z α here. Recall that z 0.05 =

6 Whe ca we ot make a cofidece iterval?: Recall the populatio mut be ormal or the ample ize large to be jutified i uig the t- ditributio to make a cofidece iterval. Alo, there hould be o eriou outlier i the data. We ue the cocept of previou ectio to determie if the data come from a ormal populatio or ha outlier. expl 5: The data from a imple radom ample with 5 obervatio wa ued to cotruct the plot give below. The ormal probability plot that wa cotructed ha a correlatio coefficiet of Judge whether a t-iterval could be cotructed uig the data i the ample. Follow the tep below. a.) Compare the give correlatio coefficiet to the critical value i the partial table below. What i your cocluio? Table VI for Aeig Normality Sample Critical ize, Value b.) Coult the boxplot to determie if the data et ha outlier. If it doe, the cofidece iterval hould ot be performed. Are we jutified i makig a cofidece iterval uig the t-ditributio? Why? 6

7 Give the degree of freedom (firt colum, labeled df) ad area to the right uder the curve (title row, give a α ), look up the appropriate t-value. Be midful of decimal poit! (ource: If you do ot ee the correct degree of freedom (df), ue the cloet value. Ue the value of z give at the bottom if df i greater tha 1,000. 7