Section 8.2 Confidence Intervals for One Population Mean When is known and Margin of Error

Transcription

1 Sectio 8.2 Cofidece Itervals for Oe Populatio Mea Whe is kow ad Margi of Error There is a procedure that we use to costruct a cofidece iterval for oe populatio mea whe is kow. Oe Mea z-iterval Procedure Assumptios 1. Simple radom sample 2. Normal Populatio or large sample 3. σ is kow STEP 1: For a cofidece level of 1-α, Use Table 2 to fid z α/2. STEP 2: The cofidece iterval for µ is from x z / 2 to x z / 2 Where z α/2 is foud i STEP 1. x is computed from the sample data, is the sample size ad is give i the problem. STEP 3: Iterpret the cofidece iterval. Note that cofidece itervals are exact for ormal populatios ad are approximately correct for large samples from o-ormal populatios.

2 Example Sakes deposit chemical trails as they travel through their habitats. These trails are ofte detected ad recogized by lizards which are potetial prey. The ability to recogize their predators via togue flicks ca ofte mea life or death for lizards. Scietist were iterested i quatifyig the resposes of the commo lizard to atural predator cues to determie whether the behavior is leared or cogeital. Sevetee juveile commo lizards were exposed to the chemical cues of he viper sake. Their resposes i umber of togue flicks per twety miutes are preseted below: Fid ad iterpret a 90% cofidece iterval of the mea umber of togue flicks per 20 miutes for all juveile commo lizards. Whe to Use the Oe-Mea z-iterval Procedure For small samples- less tha 15- the z-iterval procedure should be used oly whe the variable uder cosideratio is ormally distributed or very close to beig so. For samples of moderate size-betwee 15 ad 30-the z-iterval procedure ca be used uless the data cotai outliers or the variable uder cosideratio is far from beig ormally distributed. For large samples-of size 30 or more-the z-iterval procedure ca be used essetially without restrictio. However, if outliers are preset ad their removal is ot justified, you should compare the cofidece itervals obtaied with ad without the outliers to see what effect the outliers have if the effect is substatial, use a differet procedure or take aother sample, if possible. If outliers are preset, but their removal is justified ad results i a data set for which the z-iterval procedure is appropriate (as previously stated), the procedure ca be used.

3 Also Note: The assumptios are the rules that must be met i order for the cofidece iterval to be costructed. If the assumptios are ot met (the data) is ot ormal ad the sample is ot large, the a trasformatio may eed to be performed (see Sectio 6.4) If the sample is ot radom, the aother sample (oe that is radom) should be take. If is ukow the the procedure i Sectio 8.4 should be followed. Cofidece ad Precisio How does the cofidece level effect the legth of the cofidece iterval ad the the precisio of the iterval? Example Costruct a 95% cofidece iterval for the umber of togue flicks from the lizard ad compare to the 90% cofidece iterval. Decreasig the cofidece level decrease the legth of the cofidece iterval. For a fixed sample size, decreasig the cofidece level improves the precisio ad vice versa. We will be quatifyig how sample sizes ca affect the accuracy of our estimates i this sectio as we study what is called the margi of error.

4 The Margi of Error for the Estimate of µ The margi of error for the estimate of µ is E z / 2 We ca illustrate the margi of error. E E x z 2 / x x z / 2 Example 1. If a cofidece iterval legth is from 2 to 12, the what is the margi of error? Margi of Error, Precisio, ad Sample Size The legth of a cofidece iterval for a populatio mea, µ, ad therefore the precisio with which x estimates µ is determied by the margi of error E. For a fixed cofidece level, icreasig the sample size improves the precisio, ad vice versa. Example A cofidece iterval for a populatio mea has a margi of error of Determie the legth of the cofidece iterval 2. If the sample mea is 0.205, obtai the cofidece iterval 3. Costruct a graph similar to the Fig 8.6 o pg. 320 (show above)

5 Example Determiig the Required Sample Size If a cofidece level ad margi of error is give, the the appropriate sample size eeded to meet those specificatios must be determied. The formula for the margi of error E z / 2 is used ad the sample size required is solved.

6 Sample Size for Estimatig µ The sample size required for a (1-α) level cofidece iterval for µ with a specified margi of error, E, is give by the formula z 2 / E Rouded up to the earest whole umber. 2