The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Itroductory Statistics Chapter 6: Itroductio to Iferece Lecture 18: Estimatio with Cofidece 11/14/06 Lecture 18 1
Itroductio Statistical Iferece Sample Populatio Statistical iferece is a class of procedures by which we acquire iformatio about populatios from samples. Three procedures for makig ifereces: Poit estimatio Cofidece iterval Hypotheses testig 11/14/06 Lecture 18 2
TV Watchig Time The umber ad the types of televisio programs ad commercials targeted at childre are affected by the amout of time childre watch TV. Average time childre watch TV A survey was coducted amog 100 childre, i which they were asked to record the umber of hours they watched TV per week. The sample mea is x = 27.191. 11/14/06 Lecture 18 3
Poit Estimatio Use sample mea to estimate populatio mea. A poit estimator makes iferece about a populatio by estimatig the value of a ukow parameter usig a sigle umerical value (a poit). Drawbacks: How differet is the estimate from the true parameter? How reliable is your estimate? How cofidet are you with your estimate? Ways to improve? 11/14/06 Lecture 18 4
Cofidece Iterval A cofidece iterval has the form: poit estimate ± margi of error The poit estimate is our guess for the value of a ukow parameter. The margi of error shows how accurate we believe our guess is, based o the samplig distributio of the estimate. C: cofidece level, which shows how cofidet we are for the cofidece iterval to cover the true populatio mea. 11/14/06 Lecture 18 5
Cofidece Iterval for µ We are iterested i estimatig the populatio mea µ. The populatio SD is assumed to be kow. To estimate µ, a sample of sie is draw from the populatio, ad its mea x is calculated. We kow that x has (approximately) a ormal distributio, ad Z = x µ ~ N(0,1). 11/14/06 Lecture 18 6
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11/14/06 Lecture 18 8 The,. ) ( * * C x P = + µ µ This leads to. ) ( * * C x x P = + µ. ) / ( * * C x P = µ Thus, a level C cofidece iterval for µ is ], [ * * x x +
TV Watchig Time (cotiued) The umber ad the types of televisio programs ad commercials targeted at childre are affected by the amout of time childre watch TV. A survey was coducted amog 100 childre, i which they were asked to record the umber of hours they watched TV per week. The sample mea is x = 27.191 The populatio stadard deviatio of TV watch was kow to be = 8.0. Estimate the average watchig time at 95% cofidece level. 11/14/06 Lecture 18 9.
TV Watchig Time (cotiued) The parameter to be estimated is µ, the average time that a child watches TV per week. We eed to compute the 95% cofidece iterval for µ. * x ± * = 27.191 ± 8.0 100 = 8.0 27.191 ± 1.96 = 27.191 ± 1.57 = 100 [ 25.621, 28.761 ] 11/14/06 Lecture 18 10
Ivetory Cost To lower ivetory costs, a computer compay wats to employ a ivetory model. Demad durig lead time is ormally distributed with a s.d. of 50 computers. It is required to kow the mea i order to calculate optimum ivetory levels. Demad durig 60 lead times has x = 499.75. Estimate the mea demad durig lead time with 95% cofidece. The 95% cofidece iterval is: x ± * = 499.75± 1.96 50 60 = 499.75± 12.65 = [ 487.1, 512.4] 11/14/06 Lecture 18 11
How should we uderstad ad iterpret CI? A 95% cofidece iterval (CI) meas that the cofidece iterval is calculated by a method that will cover the true value i 95% of all possible samples. For a give sample, whether the CI covers the true value is kow, i.e. o ucertaity. Imagie there are 100 repeated samples. Based o each sample, a 95% CI ca be costructed. There will be approximately 95 CI s that will cover the true mea. A commo wrog statemet: The CI will cover the true value with probability 95%. 11/14/06 Lecture 18 12
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Four commoly used cofidece levels Cofidece level 0.90 Z 1.645 0.95 1.96 0.98 2.33 0.99 2.575 11/14/06 Lecture 18 14
Margi of Error The legth of a CI is give by: 2 * The margi of error is half of the legth: Margi of error is a measure of precisio or accuracy. The smaller, the more accurate. * 11/14/06 Lecture 18 15
Precisio The margi of error is a fuctio of: the populatio stadard deviatio the cofidece level the sample sie * If everythig else remais the same, the The larger the sample sie, the arrower the CI. The higher the cofidece level, the wider the CI; The larger the populatio SD, the wider the CI. 11/14/06 Lecture 18 16
A commo strategy is to first specify both desired cofidece level (reliability) margi of error (accuracy) The determie the ecessary sample sie as follows: The phrase estimate the mea to withi W uits, traslates to a cofidece iterval of the form The required sample sie to estimate the mea is = * W Selectig the Sample Sie 2. x ± W. 11/14/06 Lecture 18 17
Lumber Productio To estimate the amout of lumber that ca be harvested i a area, the mea diameter of trees i the area must be estimated to withi oe ich with 99% cofidece. What sample sie should be take? (Assume diameters are ormally distributed with = 6 iches.) The margi of error is +/- 1 ich. i.e., W = 1. The cofidece level 99% leads to the -score 2.575. = * W 2 = 2.575 1 (6) 2 = 239. 11/14/06 Lecture 18 18
Respose Time Suppose that the respose time to a particular editig commad is ormally distributed with stadard deviatio 25 millisecods. What sample sie is ecessary to esure that the 95% CI for µ has margi of error of at most 5? Note: W = 5 The sample sie satisfies Solvig for, we have 5= 1.96 25/. = [1.96 25 / 5] 2 = (9.80) 2 = 96.04. Sice must be a iteger, a sample sie of 97 is required. 11/14/06 Lecture 18 19
Take Home Message Poit estimatio Cofidece iterval Defiitio Iterpretatio Cofidece iterval for a populatio mea Margi of error ad sample sie determiatio 11/14/06 Lecture 18 20