UNIT 4 Section 8 Estimating Population Parameters using Confidence Intervals

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1 UNIT 4 Sectio 8 Estimatig Populatio Parameters usig Cofidece Itervals To make ifereces about a populatio that caot be surveyed etirely, sample statistics ca be take from a SRS of the populatio ad used to estimate populatio parameters. Recall that parameters are ukow fixed values about populatios, such as m ad p. The populatio mea μ is estimated usig x (x-bar), the sample mea. The populatio proportio p is estimated usig p (p-hat), the sample proportio. The populatio stadard deviatio is estimated usig s, the sample stadard deviatio. These poit estimators are the best estimators of parameters; they are ubiased estimators (because the mea of the samplig distributio is equal to the value of the parameter). Usig these statistics from sample data, we create Cofidece Itervals to provide a rage of values to capture the true parameter. The iterval is based o a Cofidece Level that determies how precise our estimate is ad how cofidet we are that our iterval captures the true parameter. Cofidece Iterval & Cofidece Level Because the poit estimate (the mea of the sample x or the proportio of the sample p ) will vary with each ew sample ad will ot ecessarily be the exact value of the populatio parameter, a rage of plausible values must be provided. This Cofidece Iterval is a iterval of values costructed aroud the poit estimate, at the ceter of the iterval, with a margi of error. The margi of error is added to ad subtracted from the poit estimate to provide a iterval ad accouts for chace variatio i the poit estimate that exists betwee samples (However, the margi of error does ot accout for biased samplig methods.). A cofidece iterval has the geeral form: poit estimate margi of error statistic (critical value) (stadard deviatio of statistic) z or t score stadard error The critical value is the z-score (or later t-score) obtaied based o the cofidece level (which provides a area).

2 If give the cofidece iterval limits, the poit estimate ca be foud, usig: (Upper cofidece level) + (Lower cofidece level) p = 2 If give the cofidece iterval limits, the margi of error ca be foud, usig: (Upper cofidece level) (Lower cofidece level) p = 2 PRACTICE: Use the give cofidece iterval to fid the poit estimate ad the margi of error. a. (.344,.528) b. (-.18,.24) The Cofidece Level is a predetermied percetage that represets how cofidet we are that the cofidece iterval costructed will actually capture the true populatio parameter. The margi of error decreases as the Cofidece Level decreases ad as the sample size icreases, arrowig the iterval. Our use of a 95% cofidece level ca be iterpreted as, We are usig a method that will provide correct results i 95% of all cofidece itervals costructed usig radomly obtaied data. To sketch a cofidece iterval, draw a ormal curve ad label the itervals critical values based o the cofidece level. For a 95% cofidece level, fid the z-score that correspods to the area to the left of the lower boud (ad to the right of the upper boud) usig a z-score sheet or a calculator: Usig the z-score sheet, look i the body of the table for area.0250 [ ] to obtai a z score critical values ±1.96 (as stadard deviatios below ad above the poit estimate). Usig the calculator, press 2 d, Distributio, ivnorm(.025), Eter to obtai a z score critical values ±1.96 (as stadard deviatios below ad above the poit estimate). 2

3 Iterpret the iterval you have costructed i cotext of the problem. Be sure to correctly iterpret the meaig of the cofidece iterval: We are 95% cofidet that the (true parameter) lies betwee (lower limit) ad (upper limit). The remaiig 5%, split equally betwee the highest 2.5% ad the lowest 2.5%, is where the true parameter lies if our iterval does ot capture the actual parameter. PRACTICE: Sketch ad label a ormal curve with the z-scores for the give cofidece itervals. a. 85% b. 90% Margi of Error & Required Sample Size Sample size depeds o the desired cofidece level ad margi of error. Use the formulas for margi of error, ad roud up to the ext larger iteger. To estimate p To estimate with To estimate without m Z pˆ (1 pˆ) m Z m t s PRACTICE: Fid the margi of error that correspods to the give statistic ad cofidece level. a. x = 70, = 100, 95% cofidece, s =12

4 b. sample size 500, 20% successes, 99% cofidece PRACTICE: How may doctors must be radomly selected for IQ tests if we wat to estimate the mea IQ score with 95% cofidece that the sample mea is withi two IQ poits of the populatio mea? PRACTICE: What is the miimum sample size eeded i order to be 99% cofidet that the margi of error is at most 6% whe the p value is estimated to be.76?

5 Cofidece Iterval for Estimatig Populatio PROPORTION ˆp A four-step system is used to orgaize the iferece process whe estimatig proportios: I. Idetify the populatio ad the parameter of iterest i cotext. II. III. Verify the assumptios/coditios (a-c), ad idetify the procedure (d) i cotext. a. Data is radomly obtaied (SRS). (Otherwise, this coditio may be assumed, which may limit the ability to geeralize the results to the populatio.) b. Populatio is at least 10 times sample size. (for idepedece ad to fid stadard deviatio.) c. Both ˆp 10 ad (1- ˆp ) 10 must be met. (for ormal approximatio) d. State, We have verified the coditios for a (type of iterval) for (parameter). With coditios met, carry out the selected procedure. Fid cofidece iterval: estimate ± margi of error (Show work) p ± Z p (1 p ) IV. Iterpret results i cotext of the problem. We are (cofidece level)% cofidet that the true (parameter) of (the specific populatio) lies betwee (the iterval bouds). (Cofirm results with calculator: STAT, TESTS, 1-PropZIt, (eter values), Calculate) Example 1: The New York Times ad CBS News coducted a atiowide poll of a SRS of radomly selected 13- to 17-year olds. Of these teeagers, 692 had a televisio i their room. Costruct a 95% cofidece iterval to estimate p. I. We are iterested i estimatig p, the proportio of 13- to 17-year olds who have a televisio i their room. II. As stated i the problem, data are obtaied from a SRS of 13- to 17-year olds. It is safe to assume that the populatio is comprised of at least 10,480 (10X sample size) 13- to 17-year olds. pˆ 10 ( 1 pˆ) ( ) 10 ( ) It is safe to use the ormal approximatio. We have verified the coditios for a oe-proportio Z cofidece iterval for p.

6 III. pˆ Z pˆ(1 pˆ) (.3397) (.6316,.6890) IV. We are 95% cofidet that the true proportio of 13- to 17-year olds who have a televisio i their room lies betwee 63.16% ad 68.90%. Example 2: We are iterested i the proportio of seiors at CBHS who pla to atted college i the state of Florida. We radomly selected 50 seiors who pla to atted college ad 37 of them say they will be goig to school i state. Costruct a 99% cofidece iterval to estimate p.

7 Cofidece Iterval for Estimatig Populatio MEAN m WITH stadard deviatio s A four-step system is used to orgaize the iferece process whe estimatig meas: I. Idetify the populatio ad the parameter of iterest i cotext. II. III. Verify the assumptios/coditios (a-c), ad idetify the procedure (d) i cotext. e. Data is radomly obtaied (SRS). (Otherwise, this coditio may be assumed, which may limit the ability to geeralize the results to the populatio.) f. Populatio is at least 10 times sample size. (for idepedece ad to fid stadard deviatio.) g. Either 30 or the origial populatio is ormally distributed. (for ormal approximatio) h. State, We have verified the coditios for a (type of iterval) for (parameter). With coditios met, carry out the selected procedure. Fid cofidece iterval: estimate ± margi of error (Show work) x ± Z σ IV. Iterpret results i cotext of the problem. We are (cofidece level)% cofidet that the true (parameter) of (the specific populatio) lies betwee (the iterval bouds). EXAMPLE 1: A sample of 54 bears from Yellowstoe Natioal Park has a mea weight of lb. Assumig that s is kow to be lb., fid a 99% CI estimate of the mea of the populatio of all such bear weights.

8 Cofidece Iterval for Populatio Mea m WITHOUT Stadard Deviatio s Follow the same procedures as The Cofidece Iterval, CI, is foud usig: x t s s, stadard deviatio of sample m, margi of error for mea without s x, the sample mea, the best poit estimate of the populatio mea Use t-distributio to obtai larger critical values whe the stadard deviatio is ot kow. The t- distributio will provide a wider iterval of values as we must approximate usig the sample stadard deviatio s. As with kow populatio stadard deviatio, the greater the sample size, the arrower the iterval. Whe determiig degrees of freedom, choose oe fewer tha sample size o t score sheet (or i calculator), df = 1, ad always roud dow to the ext lowest degrees of freedom whe the value is t exact o sheet. PRACTICE: Fid critical t value for the followig: 1. = 15; SRS; ormally distributed populatio; 95% cofidece level; s = Costruct a CI (without 4-step process) for estimatig mea: = 20 (from a ormally distributed populatio) x = 4.4 i s = 4.2 i 99% cofidece level EXAMPLE 2: A sample of 54 bears from Yellowstoe Natioal Park has a mea weight of lb. Assumig that s is ot kow, but the sample stadard deviatio is lb., costruct a 99% CI estimate of the mea of the populatio of all such bear weights.