Iferetial Statistics ad Probability a Holistic Approach Chapter 8 Poit Estimatio ad Cofidece Itervals This Course Material by Maurice Geraghty is licesed uder a Creative Commos Attributio-ShareAlike 4.0 Iteratioal Licese. Coditios for use are show here: https://creativecommos.org/liceses/by-sa/4.0/ 1 Iferece Process Iferece Process 3 Maurice Geraghty, 018 1
Iferece Process 4 Iferece Process 5 Iferetial Statistics Populatio Parameters Mea = μ Proportio = p Stadard Deviatio = σ Sample Statistics Mea = X Proportio = ˆp Stadard Deviatio = s 6 Maurice Geraghty, 018
Iferetial Statistics Estimatio Usig sample data to estimate populatio parameters. Example: Public opiio polls Hypothesis Testig Usig sample data to make decisios or claims about populatio Example: A drug effectively treats a disease 7 Estimatio of μ X is a ubiased poit estimator of μ Example: The umber of defective items produced by a machie was recorded for five radomly selected hours durig a 40-hour work week. The observed umber of defectives were 1, 4, 7, 14, ad 10. So the sample mea is 9.4. Thus a poit estimate for μ, the hourly mea umber of defectives, is 9.4. 8 Cofidece Itervals A Iterval Estimate states the rage withi which a populatio parameter probably lies. The iterval withi which a populatio parameter is expected to occur is called a Cofidece Iterval. The distace from the ceter of the cofidece iterval to the edpoit is called the Margi of Error The three cofidece itervals that are used extesively are the 90%, 95% ad 99%. 9 Maurice Geraghty, 018 3
Cofidece Itervals A 95% cofidece iterval meas that about 95% of the similarly costructed itervals will cotai the parameter beig estimated, or 95% of the sample meas for a specified sample size will lie withi 1.96 stadard deviatios of the hypothesized populatio mea. For the 99% cofidece iterval, 99% of the sample meas for a specified sample size will lie withi.58 stadard deviatios of the hypothesized populatio mea. For the 90% cofidece iterval, 90% of the sample meas for a specified sample size will lie withi 1.645 stadard deviatios of the hypothesized populatio mea. 10 8-18 90%, 95% ad 99% Cofidece Itervals for µ The 90%, 95% ad 99% cofidece itervals for μ are costructed as follows whe 30 90% CI for the populatio mea is give by σ X ±1.645 95% CI for the populatio mea is give by σ X ± 1.96 99% CI for the populatio mea is give by σ X ±.58 11 8-19 Costructig Geeral Cofidece Itervals for µ I geeral, a cofidece iterval for the mea is computed by: X ± Z σ This ca also be thought of as: Poit Estimator ± Margi of Error 1 Maurice Geraghty, 018 4
8-19 The ature of Cofidece Itervals The Populatio mea μ is fixed. The cofidece iterval is cetered aroud the sample mea which is a Radom Variable. So the Cofidece Iterval (Radom Variable) is like a target tryig hit a fixed dart (μ). 13 8-0 EXAMPLE The Dea wats to estimate the mea umber of hours worked per week by studets. A sample of 49 studets showed a mea of 4 hours with a stadard deviatio of 4 hours. The poit estimate is 4 hours (sample mea). What is the 95% cofidece iterval for the average umber of hours worked per week by the studets? 14 8-1 EXAMPLE cotiued Usig the 95% CI for the populatio mea, we have 4 ± 1.96(4 / 7) =.88 to 5.1 The edpoits of the cofidece iterval are the cofidece limits. The lower cofidece limit is.88 ad the upper cofidece limit is 5.1 15 Maurice Geraghty, 018 5
8-1 EXAMPLE cotiued Usig the 99% CI for the populatio mea, we have 4 ±.58(4 / 7) =.53 to 5.47 Compare to the 95% cofidece iterval. A higher level of cofidece meas the cofidece iterval must be wider. 16 8-7 Selectig a Sample Size There are 3 factors that determie the size of a sample, oe of which has ay direct relatioship to the size of the populatio. They are: The degree of cofidece selected. The maximum allowable error. The variatio of the populatio. 17 8-8 Sample Size for the Mea A coveiet computatioal formula for determiig is: Zσ = E where E is the allowable error (margi of error), Z is the z score associated with the degree of cofidece selected, ad σ is the sample deviatio of the pilot survey. σ ca be estimated by past data, target sample or rage of data. 18 Maurice Geraghty, 018 6
8-9 EXAMPLE A cosumer group would like to estimate the mea mothly electric bill for a sigle family house i July. Based o similar studies the stadard deviatio is estimated to be $0.00. A 99% ± level of cofidece is desired, with a accuracy of $5.00. How large a sample is required? = [(. 58)( 0) / 5] = 106. 504 107 19 Normal Family of Distributios: Z, t, χ, F 0 10-3 Characteristics of Studet s t- Distributio The t-distributio has the followig properties: It is cotiuous, bell-shaped, ad symmetrical about zero like the z-distributio. There is a family of t-distributios sharig a mea of zero but havig differet stadard deviatios based o degrees of freedom. The t-distributio is more spread out ad flatter at the ceter tha the z-distributio, but approaches the z-distributio as the sample size gets larger. 1 Maurice Geraghty, 018 7
9-3 9-3 The degrees of freedom for the t-distributio is df = - 1. z-distributio t-distributio Cofidece Iterval for μ (σ ukow) Formula to fid a cofidece iterval usig the t-distributio for the appropriate level of cofidece: s X ± t df = 1 3 Example Cofidece Iterval I a radom sample of 13 America adults, the mea waste recycled per perso per day was 5.3 pouds ad the stadard deviatio was.0 pouds. Assume the variable is ormally distributed ad costruct a 95% cofidece iterval for μ. 4 Maurice Geraghty, 018 8
Example- Cofidece Iterval level of cofidece = 95% df=13-1=1 t=.18.0 5.3 ±.18 13 5.3 ± 1. = (4.1, 6.5) 5 Cofidece Itervals, Populatio Proportios Poit estimate for proportio of successes i populatio is: X is the umber of successes i a sample of size. Stadard deviatio of pˆ is Cofidece Iterval for p: X p ˆ = ( p )(1 p) p(1 p) pˆ(1 pˆ) pˆ ± Z pˆ ± Z 6 Populatio Proportio Example I a May 006 AP/ISPOS Poll, 1000 adults were asked if "Over the ext six moths, do you expect that icreases i the price of gasolie will cause fiacial hardship for you or your family, or ot? 700 of those sampled respoded yes! Fid the sample proportio ad margi of error for this poll. (This meas fid a 95% cofidece iterval.) 7 Maurice Geraghty, 018 9
Populatio Proportio Example Sample proportio 700 p ˆ = =.70 = 70% 1000 Margi of Error.70(1.70) MOE = 1.96 =.08 =.8% 1000 8 8-8 Sample Size for the Proportio A coveiet computatioal formula for determiig is: Z = ( p( 1 p) ) E where E is the allowable margi of error, Z is the z-score associated with the degree of cofidece selected, ad p is the populatio proportio. If p is completely ukow, p ca be set equal to ½ which maximizes the value of (p)(1-p) ad guaratees the cofidece iterval will fall withi the margi of error. 9 30 Maurice Geraghty, 018 10
Example I pollig, determie the miimum sample size eeded to have a margi of error of 3% whe p is ukow. = 1.96.03 (.5)( 1.5) = 1068 31 Example I pollig, determie the miimum sample size eeded to have a margi of error of 3% whe p is kow to be close to 1/4. = 1.96.03 (.5)( 1.5) = 801 3 14- Characteristics of the Chi-Square Distributio The major characteristics of the chisquare distributio are: It is positively skewed It is o-egative It is based o degrees of freedom Whe the degrees of freedom chage, a ew distributio is created 33 Maurice Geraghty, 018 11
- CHI-SQUARE DISTRIBUTION df = 3 df = 5 df = 10 χ 34 Iferece about Populatio Variace ad Stadard Deviatio s is a ubiased poit estimator for σ s is a poit estimator for σ Iterval estimates ad hypothesis testig for both σ ad σ require a ew distributio the χ (Chi-square) 35 Distributio of s ( 1) s has a chi-square distributio σ -1 is degrees of freedom s is sample variace σ is populatio variace 36 Maurice Geraghty, 018 1
Cofidece iterval for σ Cofidece is NOT symmetric sice chi-square distributio is ot symmetric. You must fid separate left ad right bouds. We ca costruct a cofidece iterval for σ ( 1) ( 1) s s, χ χ L R Take square root of both edpoits to get cofidece iterval for σ, the populatio stadard deviatio. 37 Example I performace measuremet of ivestmets, stadard deviatio is a measure of volatility or risk. Twety mothly returs from a mutual fud show a average mothly retur of 1% ad a sample stadard deviatio of 5% Fid a 95% cofidece iterval for the mothly stadard deviatio of the mutual fud. 38 Example (cot) df = -1 =19 95% CI for σ ( 19) 5 ( 19), 3.853 5 = 8.90655 ( 3.8,7.3) 39 Maurice Geraghty, 018 13