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

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1 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 that is selected for study. Iferetial Statistics is the process of usig sample iformatio to draw ifereces or coclusios about the populatio. Cosider a populatio of 5 commuters who are all eighbors. Each commuter was asked how may miles he/she commutes to work each day. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles Fid the populatio mea of the set of data: Fid the populatio stadard deviatio: Look at a sample of two commuters ( 2 ), ad fid the mea to estimate µ. Like: C1 = 50 ad C2 = 84, the C1 ad C3, the C1 ad C4, the C1 ad C5 the C4 ad C5. How may should we have? List all possible samples of two commuters ad calculate the mea,, for each sample. Commuters Data Values Mea, The data set of all the sample meas i colum 3 is called a samplig distributio of the meas. The samplig error is the differece betwee the value of a sample mea,, ad the populatio mea, µ. Samplig error of the mea = = µ For the sample mea 67, calculate the sample error: For the sample mea 102, calculate the sample error: 1

2 Lookig back at colum 3 of our table, this data set of all the sample meas is called a samplig distributio of the meas. Calculate the mea of this data set we are calculatig the mea of the samplig distributio of the meas: The mea of all the sample meas of a samplig distributio has special otatio: mu sub -bar or Calculate the stadard deviatio of this data set we are calculatig the stadard deviatio of the samplig distributio of the meas: The stadard deviatio of the all the sample meas of a samplig distributio has special otatio: sigma sub -bar or The stadard deviatio is a measure of how spread out the sample meas are from µ. The mea of the samplig distributio of the meas is: = What is the populatio mea which we calculated before? = So, the mea of the samplig distributio of the meas is equal to the mea of the populatio from which the samples were selected: The stadard deviatio of the samplig distributio of the meas is: = What is the populatio stadard deviatio which we calculated before? = The stadard deviatio of the samplig distributio of the mea, also kow as the stadard error of the mea, will always be smaller tha the populatio stadard deviatio ad the formula is: The larger the stadard error of the mea is, the more dispersed the samples meas are from the populatio mea. The smaller the stadard error, the closer the sample meas are to the populatio mea. Fiite Correctio Factor: For a fiite populatio (havig a limit), the formula for the stadard deviatio of the samplig distributio of the mea, is: N N 1 2

3 8.1 The Samplig Distributio of the Mea Populatios are geerally either quite large, have a ulimited umber of data values (that is, ifiite), or partly uobtaiable. Sice a populatio ca rarely be studied completely, a sample radomly selected from the populatio serves as a coveiet ad ecoomical procedure to estimate the characteristics of a populatio. Sample iformatio is used to estimate the mea of a populatio. Whe the populatio mea is beig estimated usig sample data, the mea of the sample will probably ot be equal to the mea of the populatio. I fact, if a few samples were radomly selected from the same populatio, it is very likely that oe of these sample meas would be eactly equal to the mea of the populatio. Although there is oly oe value for the true mea of a populatio, there are may differet values for the mea whe differet samples are selected from the populatio. Therefore, the estimates of a true populatio mea will vary from sample to sample due to the data values radomly selected withi each sample, that is, by chace aloe. These variatios i the estimates of the populatio mea from sample to sample are due to chace ad are called samplig errors. Samplig error is the differece betwee the value of a sample statistic, such as the sample mea, ad the value of the correspodig populatio parameter, such as the populatio mea, μ. Thus, the samplig error for the mea is: samplig error = sample mea populatio mea assumig the sample is radom ad there are o osample errors. I symbols, samplig error of the mea = μ. Samplig error is ievitable because we are usig a chose umber of data values which are radomly selected, by chace, from the populatio. I practice, we will select oly oe sample from the populatio to estimate the mea of the populatio. A populatio ca have oly oe mea. Yet, depedig upo which sample is selected from a populatio, the mea of a sample ca vary from sample to sample as differet samples of the same size are radomly selected from the same populatio. Thus, the sample mea is a radom variable because it is depedet upo the particular data values which are radomly selected from the populatio. Eample 8.1 pg. 423 Suppose the populatio of seve college studets o the Studet Govermet Associatio (SGA) have the followig ages: 23, 19, 20, 21, 18, 19, 25 a. Compute the populatio mea age of all the studets o the SGA. b. If a radom sample of 3 studets was selected from this populatio havig ages: 19, 21, ad 18, the compute the sample mea age for this sample. c. Determie the samplig error if this sample (from part b) was used to estimate the populatio mea age. Eplai or iterpret the meaig of this samplig error. *Review parts d, e, ad f of this eample too! 3

4 The samplig distributio of the mea is a probability distributio which lists the sample meas from all possible samples of the same sample size selected from the same populatio alog with the probability associated with each sample mea. Notatio for the Mea of the Samplig Distributio of the Mea The mea of the samplig distributio of the mea is deoted by, read mu sub bar. Thus, = mea of all the sample meas of the samplig distributio. Notatio for the Stadard Deviatio of the Samplig Distributio of the Mea The stadard deviatio of the samplig distributio of the mea is deoted by, read sigma sub bar. Thus, = stadard deviatio of all the sample meas of the samplig distributio. 8.2 The Mea ad Stadard Deviatio of the Samplig Distributio of the Mea Mea of the Samplig Distributio of the Mea, The mea of the sample meas of all possible samples of size is called the mea of the samplig distributio of the mea, deoted by. It is equal to the mea of the populatio from which the samples were selected. I symbols, this is epressed as: Stadard Deviatio of the Samplig Distributio of the Mea or Stadard Error of the Mea, deoted by The stadard error of the mea is the stadard deviatio of the sample meas of all possible samples of size of the samplig distributio, deoted by. The stadard error of the mea is equal to the stadard deviatio of the populatio, σ, divided by the square root of the sample size. That is: stadard error of the mea populatio stadard deviatio = sample size Iterpretatio of the Stadard Error of the Mea The stadard deviatio of the samplig distributio of the mea is referred to as the stadard error of the mea because it is a measure of how much a sample mea is likely to deviate from the populatio mea, that is, a measure of the average samplig error. If the stadard error of the mea,, is a small umber, the the samplig distributio of the mea has relatively little dispersio ad the sample meas will be relatively close to the populatio mea. O the other had, if the stadard error of the mea,, is a large umber, the the samplig distributio of the mea has a relatively large dispersio ad the sample meas will be relatively far from the populatio mea. 4

5 Eample 8.3 pg. 430 Accordig to a study of TV viewig habits, the average umber of hours a teeager watches MTV per week is 17.9 hours with a stadard deviatio of 3.8 hours. If a sample of 64 teeagers is radomly selected from the populatio, the determie the mea ad stadard error of the mea of the samplig distributio of the mea. Eample 8.4 pg The registrar at a large Uiversity states that the mea grade poit average of all the studets is 2.95 with a populatio stadard deviatio of a. Determie the mea ad stadard error of the samplig distributio if the samplig distributio of the mea cosists of all possible sample meas from samples of size 25. b. Determie the mea ad stadard error of the samplig distributio if the samplig distributio of the mea cosists of all possible sample meas from samples of size 100. c. What effect did icreasig the sample size have o the mea ad stadard error of the samplig distributio? d. I which samplig distributio of the mea ( = 25 or = 100) would you have a better chace of selectig a sample mea which is closer to the populatio mea grade poit average? Review Eample 8.5 o pg

6 8.4 The Shape of the Samplig Distributio of the Mea Samplig from a Normal Populatio THEOREM The Shape of the Samplig Distributio whe Samplig from a Normal Populatio If the populatio beig sampled is a ormal distributio, the the samplig distributio of the mea is a ormal distributio regardless of the sample size,. Characteristics of the Samplig Distributio of the Mea Whe Samplig from a Normal Populatio Whose Mea is μ ad Stadard Deviatio is σ If all possible samples of size are selected from a ormal populatio, the the samplig distributio of the mea has the followig three characteristics: 1. The samplig distributio of the mea is a ormal distributio, regardless of sample size,. 2. The mea of the samplig distributio of the mea,, is equal to the mea of the populatio, μ:. 3. The stadard error of the samplig distributio of the mea,, is equal to the stadard deviatio of the populatio, σ, divided by the square root of the sample size, :. Eample 8.7 pg. 438 At a large New Eglad college, the grade poit average (GPA) of all attedig studets is ormally distributed with a mea of 2.95 ad a populatio stadard deviatio of A sample is radomly selected from this populatio ad its sample mea,, is calculated. Determie the mea,, the stadard error, idetify the shape of the samplig distributio of the mea of the samples of size: a. = 9 b. = 49 c. = 100 6

7 Samplig from a No-Normal Populatio THEOREM 8.2 The Cetral Limit Theorem For ay populatio, the samplig distributio of the mea approaches a ormal distributio as the sample size becomes large. This is true regardless of the shape of the populatio beig radomly sampled. Geeral Rule for Applyig the Cetral Limit Theorem: The Greater tha 30 Rule For most applicatios, a sample size greater tha 30 is cosidered large eough to apply the Cetral Limit Theorem. Thus, the samplig distributio of the mea ca be reasoably approimated by a ormal distributio wheever the sample size is greater tha 30. THEOREM 8.3 Characteristics of the Samplig Distributio of the Mea whe Samplig from a No-Normal Populatio If, the followig three coditios are satisfied: give ay ifiite populatio with mea, μ, ad stadard deviatio, σ, ad all possible radom samples of size are selected from the populatio to form a samplig distributio of the mea, ad the sample size,, is large (greater tha 30). the: 1. the samplig distributio of the mea is approimately ormal. 2. the mea of the samplig distributio of the mea is equal to the mea of the populatio. This is epressed as: 3. the stadard error of the samplig distributio of the mea is equal to the stadard deviatio of the populatio divided by the square root of the sample size. This is writte as: Eample 8.8 pg. 442 GRAM-HAM Bell, a telephoe compay, states that the average legth of time of log-distace telephoe calls is 21.3 miutes with a stadard deviatio of 3.8 miutes. Determie the mea ad the stadard error of the samplig distributio of the mea ad describe the shape of the samplig distributio of the mea whe the sample size is: a. = 36 b. = 100 c. Compare the samplig distributio of the mea for the sample size of = 36 ad = 100. d. If you had to estimate the mea of the populatio by either radomly selectig a sample of size = 36 or of = 100 from the populatio, the which sample size would give you a better chace of obtaiig a smaller samplig error? Eplai. 7

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9 8.5 Calculatig Probabilities Usig the Samplig Distributio of the Mea Now i this chapter we are workig with the samplig distributio of the mea so the data values are sample meas,, thus, the z score formula becomes: or sample mea mea of the samplig distributio z score of a samplig mea stadard error of the samplig distributio z of OR we ca Use the TI-83/84 ormalcdf fuctio: 2 d DISTR:2 ( lower sample mea value, higher sample mea value,, Remember: ad ) ENTER Eample 8.9 pg At a large public state college i Virgiia, the mea Verbal SAT score of all attedig studets was 600 with a populatio stadard deviatio of 65. If a radom sample of 100 studets is selected from a populatio of studets to determie: a. The probability that the mea Verbal SAT score of the selected sample will be less tha 615. b. The probability that the mea Verbal SAT score of the selected sample will be withi 10 poits of the populatio mea. Eample 8.10 pg. 447 The populatio mea weight of ewbor babies for a wester suburb is 7.4 lbs. with a stadard deviatio of 0.8 lbs. What is the probability that a sample of 64 ewbors selected at radom will have a mea weight greater tha 7.5 lbs.? Eample 8.11 pg. 449 The populatio of the ages of all U. S. college studets is skewed to the right with a mea age of 27.4 years ad a stadard deviatio of 5.8 years. Determie the probability that a radom sample of 49 studets selected from the populatio will have a sample mea age withi oe year of the populatio mea age? 9