Section 73 Estimating a Population. Requirements


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1 Sectio 73 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 is that we kow the stadard deviatio of the populatio. Requiremets 1. The sample is a simple radom sample. (All samples of the same size have a equal chace of beig selected.). The value of the populatio stadard deviatio σ is kow. 3. Either or both of these coditios is satisfied: The populatio is ormally distributed or > 30. Slide 1 Slide Poit Estimate of the Populatio Mea Sample Mea The sample mea x is the best poit estimate of the populatio mea µ. 1. For all populatios, the sample mea x is a ubiased estimator of the populatio mea µ, meaig that the distributio of sample meas teds to ceter about the value of the populatio mea µ.. For may populatios, the distributio of sample meas x teds to be more cosistet (with less variatio) tha the distributios of other sample statistics. Slide 3 Slide 4 Example: A study foud the body temperatures of 106 healthy adults. The sample mea was 98. degrees ad the sample stadard deviatio was 0.6 degrees. Fid the poit estimate of the populatio mea µ of all body temperatures. Because the sample mea x is the best poit estimate of the populatio mea µ, we coclude that the best poit estimate of the populatio mea µ of all body temperatures is 98.0 o F. Defiitio The margi of error is the maximum likely differece observed betwee sample mea x ad populatio mea µ, ad is deoted by E. Slide 5 Slide 6
2 E = z α/ Formula Margi of Error σ Formula 74 Margi of error for mea (based o kow σ) Cofidece Iterval estimate of the Populatio Mea µ (with σ Kow) or or x + E (x E, x + E) Slide 7 Slide 8 Defiitio The two values x E ad x + E are called cofidece iterval limits. Slide 9 Procedure for Costructig a Cofidece Iterval for µ (with Kow σ) 1. Verify that the requiremets are satisfied.. Refer to Table A ad fid the critical value z α/ that correspods to the desired degree of cofidece. 3. Evaluate the margi of error E = z α/ σ/. 4. Fid the values of x E ad x + E. Substitute those values i the geeral format of the cofidece iterval: 5. Roud usig the cofidece itervals roudoff rules. Slide 10 RoudOff Rule for Cofidece Itervals Used to Estimate µ 1. Whe usig the origial set of data, roud the cofidece iterval limits to oe more decimal place tha used i origial set of data.. Whe the origial set of data is ukow ad oly the summary statistics (, x, s) are used, roud the cofidece iterval limits to the same umber of decimal places used for the sample mea. Slide 11 Example: A study foud the body temperatures of 106 healthy adults. The sample mea was 98. degrees ad the sample stadard deviatio was 0.6 degrees. Fid the margi of error E ad the 95% cofidece iterval for µ. x = 98.0 o s = 0.6 o α / = 0.05 z α/ = 1.96 E = z α/ σ = = o < µ < 98.3 o 98.0 o 0.1 < µ < 98.0 o Slide 1
3 Example: A study foud the body temperatures of 106 healthy adults. The sample mea was 98. degrees ad the sample stadard deviatio was 0.6 degrees. Fid the margi of error E ad the 95% cofidece iterval for µ. x = 98.0 o s = 0.6 o α / = 0.05 z α/ = 1.96 E = z α/ σ = = o < µ < 98.3 o Based o the sample provided, the cofidece iterval for the populatio mea is o < µ < 98.3 o. If we were to select may differet samples of the same size, 95% of the cofidece itervals would actually cotai the populatio mea µ. Slide 13 Sample Size for Estimatig Mea µ Where = (z α/ ) σ E Formula 75 z α/ = critical z score based o the desired cofidece level E = desired margi of error σ = populatio stadard deviatio Slide 14 RoudOff Rule for Sample Size Whe fidig the sample size, if the use of Formula 75 does ot result i a whole umber, always icrease the value of to the ext larger whole umber. Fidig the Sample Size Whe σ is Ukow 1. Use the rage rule of thumb (see Sectio 33) to estimate the stadard deviatio as follows: σ rage/4.. Coduct a pilot study by startig the samplig process. Start the sample collectio process ad, usig the first several values, calculate the sample stadard deviatio s ad use it i place of σ. 3. Estimate the value of σ by usig the results of some other study that was doe earlier. Slide 15 Slide 16 Example: Assume that we wat to estimate the mea IQ score for the populatio of statistics professors. How may statistics professors must be radomly selected for IQ tests if we wat 95% cofidece that the sample mea is withi IQ poits of the populatio mea? Assume that σ = 15, as is foud i the geeral populatio. α / = 0.05 z α/ = 1.96 E = σ = 15 = = = 17 With a simple radom sample of oly 17 statistics professors, we will be 95% cofidet that the sample mea will be withi IQ poits of the true populatio mea µ. Slide 17 Sectio 74 Estimatig a Populatio Mea: σ Not Kow Key Cocept This sectio presets methods for fidig a cofidece iterval estimate of a populatio mea whe the populatio stadard deviatio is ot kow. With σ ukow, we will use the Studet t distributio assumig that certai requiremets are satisfied. Slide 18
4 Requiremets with σ Ukow Studet t Distributio 1) The sample is a simple radom sample. ) Either the sample is from a ormally distributed populatio, or > 30. Use Studet t distributio Slide 19 If the distributio of a populatio is essetially ormal, the the distributio of t = x  µ s is a Studet t Distributio for all samples of size. It is ofte referred to a a t distributio ad is used to fid critical values deoted by t. α/ Slide 0 Defiitio The umber of degrees of freedom for a collectio of sample data is the umber of sample values that ca vary after certai restrictios have bee imposed o all data values. degrees of freedom = 1 i this sectio. Margi of Error E for Estimate of µ (With σ Not Kow) Formula 76 E = t α/ s where t α/ has 1 degrees of freedom. Table A3 lists values for t α/ Slide 1 Slide Cofidece Iterval for the Estimate of µ (With σ Not Kow) where E = t α/ s t α/ foud i Table A3 Slide 3 Procedure for Costructig a Cofidece Iterval for µ (With σ Ukow) 1. Verify that the requiremets are satisfied.. Usig  1 degrees of freedom, refer to Table A3 ad fid the critical value t α/ that correspods to the desired cofidece level. 3. Evaluate the margi of error E = t α/ s /. 4. Fid the values of x  E ad x + E. Substitute those values i the geeral format for the cofidece iterval: 5. Roud the resultig cofidece iterval limits. Slide 4
5 Example: A study foud the body temperatures of 106 healthy adults. The sample mea was 98. degrees ad the sample stadard deviatio was 0.6 degrees. Fid the margi of error E ad the 95% cofidece iterval for µ. Example: A study foud the body temperatures of 106 healthy adults. The sample mea was 98. degrees ad the sample stadard deviatio was 0.6 degrees. Fid the margi of error E ad the 95% cofidece iterval for µ. x = 98.0 o s = 0.6 o α / = 0.05 t α/ = 1.96 E = t α/ s = = o < µ < 98.3 o Slide 5 Slide 6 Example: A study foud the body temperatures of 106 healthy adults. The sample mea was 98. degrees ad the sample stadard deviatio was 0.6 degrees. Fid the margi of error E ad the 95% cofidece iterval for µ. x = 98.0 o s = 0.6 o α / = 0.05 t α/ = 1.96 E = t α/ s = = o < µ < 98.3 o Based o the sample provided, the cofidece iterval for the populatio mea is o < µ < 98.3 o. The iterval is the same here as i Sectio 7, but i some other cases, the differece would be much greater. Slide 7 Importat Properties of the Studet t Distributio 1. The Studet t distributio is differet for differet sample sizes (see Figure 75, followig, for the cases = 3 ad = 1).. The Studet t distributio has the same geeral symmetric bell shape as the stadard ormal distributio but it reflects the greater variability (with wider distributios) that is expected with small samples. 3. The Studet t distributio has a mea of t = 0 (just as the stadard ormal distributio has a mea of z = 0). 4. The stadard deviatio of the Studet t distributio varies with the sample size ad is greater tha 1 (ulike the stadard ormal distributio, which has a σ = 1). 5. As the sample size gets larger, the Studet t distributio gets closer to the ormal distributio. Slide 8 Studet t Distributios for = 3 ad = 1 Choosig the Appropriate Distributio Figure 75 Slide 9 Figure 76 Slide 30
6 Example: Flesch ease of readig scores for 1 differet pages radomly selected from J.K. Rowlig s Harry Potter ad the Sorcerer s Stoe. Fid the 95% iterval estimate of µ, the mea Flesch ease of readig score. (The 1 pages distributio appears to be bellshaped with x = ad s = 4.68.) Slide 31 Example: Flesch ease of readig scores for 1 differet pages radomly selected from J.K. Rowlig s Harry Potter ad the Sorcerer s Stoe. Fid the 95% iterval estimate of µ, the mea Flesch ease of readig score. (The 1 pages distributio appears to be bellshaped with x = ad s = 4.68.) x = s = 4.68 α/ = 0.05 t α/ =.01 E = t α / s = (.01)(4.68) = < µ < < µ < < µ < 83.7 We are 95% cofidet that this iterval cotais the mea Flesch ease of readig score for all pages. Slide 3 Fidig the Poit Estimate ad E from a Cofidece Iterval Poit estimate of µ: x = (upper cofidece limit) + (lower cofidece limit) Cofidece Itervals for Comparig Data As before i Sectios 7 ad 73, do ot use the overlappig of cofidece itervals as the basis for makig fial coclusios about the equality of meas. Margi of Error: E = (upper cofidece limit) (lower cofidece limit) Slide 33 Slide 34
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