A point estimate is the value of a statistic that estimates the value of a parameter.

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1 Chapter 9 Estimatig the Value of a Parameter Chapter 9.1 Estimatig a Populatio Proportio Objective A : Poit Estimate A poit estimate is the value of a statistic that estimates the value of a parameter. x The best poit estimate of the populatio proportio is a sample proportio ( p ˆ = ). x The best poit estimate of the populatio mea is a sample mea ( x = ). Sice pˆ varies from sample to sample, we use a iterval based o pˆ to capture the ukow populatio proportio with a level of cofidece. Objective B : Cofidece Iterval A cofidece iterval for a ukow parameter cosists of a iterval of umbers based o a poit estimate. The level of cofidece represets the expected proportio of itervals that will cotai the parameter if a large umber of differet samples is obtaied. 1 α 100%. The level of cofidece cotrols the width The level of cofidece is deoted as ( ) of the iterval. Cofidece iterval estimates for a parameter are of the form: Poit estimate ± margi of error. Cofidece iterval for p : pˆ(1 pˆ) pˆ ± Z α / σ pˆ where σ pˆ = provided that pˆ (1 pˆ) 10. The value of Z α / is called the critical value of the distributio. The margi of error, E, i a ( α ) give by 1 100% cofidece iterval for a populatio proportio is pˆ(1 pˆ) E = Z α /. The width of the iterval is determied by the margi of error. 1

2 Example 1:Use StatCruch to determie the critical value Z α / that correspods to the give level of cofidece. (a) 90% (b) 95% (c) 98% (d) 9% Example : Determie the margi of error for p with x = 540 ad = 900 at a 99% level of cofidece. Example 3: A Rasmusse Reports atioal survey of 1000 adult Americas foud that 18% dreaded Valetie's Day. Costruct a 95% cofidece iterval for the populatio proportio of adult Americas who dread Valetie's Day. Explai what does the iterval mea.

3 Example 4: Costruct a cofidece iterval of the populatio proportio at the give level of cofidece. x= 80, = 00, 96% cofidece Example 5: I a study of 18 radomly selected medical malpractice lawsuits, it is foud that 856 if them were later dropped or dismissed. (a) What is the best poit of estimate of the proportio of medical malpractice lawsuits that are dropped or dismissed? (b) Use StatCruch to costruct a 99% cofidece iterval for the populatio proportio of medical malpractice lawsuits that are dropped or dismissed? (c) Iterpret the iterval. 3

4 Objective C: Sample Size Needed for Estimatig the Populatio Proportio p The sample size required to obtai a ( α ) margi of error E is give by Z = pˆ (1 pˆ) E α / 1 100% cofidece iterval for p with a Roud up to the ext iteger pˆ is a prior estimate of p If a prior estimate of p is uavailable, the sample size required is / 0.5 Z α = E Roud up to the ext iteger Example 1 : A urba ecoomist wishes to estimate the proportio of Americas who ow their homes. What size sample should be obtaied if he wishes the estimate to be withi 0.0 with 90% cofidece if (a) he uses a 010 estimate of obtaied from the U.S Cesus Bureau? (b) he does ot use ay prior estimates? 4

5 Example : I a Gallup poll coducted i October 010, 64% of the people polled aswered "more strict" to the followig questio: "Do you feel that the laws coverig the sale of firearms should be made more strict as they are ow?" Suppose the margi of error i the poll was 3.5% ad the estimate was made with 95% cofidece. At least how may people were surveyed? Example 3: A Gallup poll coducted i November 010 foud that 493 of 1050 adult Americas believe it is the resposibility of the federal govermet to make sure all Americas have healthcare coverage. (a) Obtai a poit estimate for the proportio of adult Americas who believe it is the resposibility of the federal govermet to make sure all Americas have healthcare coverage. (b) Verify the requiremets for costructig a cofidece iterval for p are satisfied. 5

6 (c) Use StatCruch to costruct a 95% cofidece iterval for the proportio of adult Americas who believe it is the resposibility of the federal govermet to make sure all Americas healthcare coverage. Iterpret the iterval. (d) You wish to coduct your ow study for the proportio of adult Americas who believe it is the resposibility of the federal govermet to make sure all Americas have healthcare coverage. What sample size would be eeded for the estimate to be withi 3 percetage poits with 90% cofidece if you use the estimate obtaied i part (a). (e) You wish to coduct your ow study for the proportio of adult Americas who believe it is the resposibility of the federal govermet to make sure all Americas have healthcare coverage. What sample size would be eeded for the estimate to be withi 3 percetage poits with 90% cofidece if you do ot have a prior estimate? 6

7 Chapter 9. Estimatig a Populatio Mea Objective A : Poit Estimate The best poit estimate of the populatio mea, µ, is the sample mea, x. Objective B :Studet's t - distributio Properties of the t - distributio 1. The t - distributio is differet for differet degrees of freedom ( df = 1).. The t - distributio has the same geeral symmetric bell shape as the stadard ormal distributio but its area i the tails is a little greater tha the area i the tails of the stadard ormal distributio due to the greater variability that is expected with small samples. 3. The t - distributio has a mea of t = 0 at the ceter of the distributio. 4. As the sample size gets larger, the t - distributio gets closer to the stadard ormal distributio. Example 1: Use StatCruch to determie the t -value. (a) Fid the t -value such that the area i the right tail is 0.05 with 19 degrees of freedom. (b) Fid the t -value such that the area left of the t -value is 0.0 with 6 degrees of freedom. (c) Fid the critical t -value that correspods to 90% cofidece. Assume 1degrees of freedom. 7

Objective C :Cofidece Iterval for a Populatio Mea Costructig a ( α ) 1 100% Cofidece Iterval for µ Poit estimate ± margi of error s s x ± tα / where E = tα

8 I geeral, the populatio stadard deviatio is ukow for estimatig a populatio mea based o a sample mea. The t -distributio is used to off-set the additioal variability itroduced by usig s i place of σ. Objective C :Cofidece Iterval for a Populatio Mea Costructig a ( α ) 1 100% Cofidece Iterval for µ Poit estimate ± margi of error s s x ± tα / where E = tα /. provided the data come from a populatio that is ormally distributed, or the sample size is large. Example 1: A simple radom sample of size < 30 has bee obtaied. From the ormal probability plot ad boxplot, judge whether a t -iterval should be costructed. (a) (b) 8

9 Example : A simple radom sample of size is draw from a populatio that is ormally distributed. The sample mea, x, is foud to be 50, ad the sample stadard deviatio, s, is foud to be 8. (a) Use StatCruch to costruct a 98% cofidece iterval for µ if the sample size,, is 0. (b) Use StatCruch to costruct a 98% cofidece iterval for µ if the sample size,, is 15. How does decreasig the sample size affect the margi of error, E? (c) Costruct a 95% cofidece iterval for µ if the sample size,, is 0. Compare the results to those obtaied i part (a). How does decreasig the level of cofidece affect the margi of error, E? (d) Could we have computed the cofidece itervals i parts (a) to (c) if the populatio had ot bee ormally distributed? Why? 9

10 Example 3: Determie the poit estimate of the populatio mea ad margi of error for the followig cofidece iterval. Lower boud: 5 Upper boud: 3 Example 4 : How much time do Americas sped eatig or drikig? Suppose for a radom sample of 1001 Americas age 15 or older, the mea amout of time spet eatig or drikig per day is 1. hours with a stadard deviatio of 0.65 hour. (a) A histogram of time spet eatig ad drikig each day is skewed right. Use this result to explai why a large sample size is eeded to costruct a cofidece iterval for the mea time spet eatig ad drikig each day. (b) Use StatCruch to determie ad a 95% cofidece iterval for the mea amout of time Americas age 15 or older sped eatig ad drikig each day. Iterpret the iterval. (c) Could the iterval be used to estimate the mea amout of time a 9-year-old America speds eatig ad drikig each day? Explai. 10

11 Objective D : Determiig the Sample Size The sample size required to estimate the populatio mea, µ, with a level of cofidece ( α ) 1 100% withi a specified margi of error, E, is give by Zα / s = E where is rouded up to the earest whole umber. Note: The t -distributio approaches the stadard ormal z - distributio as the samplesize icreases. Example 1: A researcher wated to determie the mea umber of hours per week(suday through Saturday) the typical perso watches televisio. Results from the Sulliva Statistics Survey idicate that s = 7.5 hours. (a) How may people are eeded to estimate the umber of hours people watch televisio per week withi hours with 95% cofidece? (b) How may people are eeded to estimate the umber of hours people watch televisio per week withi 1 hour with 95% cofidece? (c) What effect does doublig the required accuracy have o the sample size? 11

12 Chapter 9 Estimatig a Populatio Stadard Deviatio (Supplemetary Materials) Objective A : Poit Estimate The best poit estimate of the populatio variace, σ, is the sample variace, s. Objective B : Chi-Square Distributio 1

13 Example 1: Use StatCruch to fid the critical values ad sample size. (a) 90% cofidece, = 3 χ1 α / ad χ α / for the give level of cofidece (b) 99% cofidece, = 15 Objective C : Cofidece Iterval for a Populatio Variace or Stadard Deviatio ( 1 α) 100% of the values of χ will lie betwee χ1 α / ad ( 1) χ α /. ( Recall: χ σ = ) σ To fid a ( 1 α) 100% cofidece iterval about σ, take the square root of the lower boud ad upper boud. 13

14 Example 1: A simple radom sample of size is draw from a populatio that is kow to be ormally distributed. The sample variace, s, is determied to be (a) Use StatCruch to costruct a 95% cofidece iterval for σ if the sample size,, is 10. (b) Use StatCruch to costruct a 95% cofidece iterval for σ if the sample size,, is 5. How does icreasig the sample size affect the width of the iterval? (c) Use StatCruch to costruct a 99% cofidece iterval for σ if the sample size,, is 10. Compare the results with those obtaied i part (a). How does icreasig the level of cofidece affect the width of the cofidece iterval? 14

15 Example : Travelers per taxes for flyig, car retals, ad hotels. The followig data represet the total travel tax for a 3-day busiess trip i eight radomly selected cities. It was verified that the data are ormally distributed. Use StatCruch to costruct a 90% cofidece iterval for the stadard deviatio travel tax for a 3-day busiess trip. Iterpret the iterval. 15

Inferential Statistics and Probability a Holistic Approach. Inference Process. Inference Process. Chapter 8 Slides. Maurice Geraghty,

Inferential Statistics and Probability a Holistic Approach. Inference Process. Inference Process. Chapter 8 Slides. Maurice Geraghty, 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