Confidence Intervals for One Population Mean. Part I

Transcription

1 Cnfidence Intervals fr One Ppulatin Mean Part I

2 Quick Overview Objective: T estimate the value f ne ppulatin mean Inferential statistics using statistics (e.g. sample mean) in rder t estimate parameters (e.g. ppulatin mean) A knwledge f sampling distributins and the Central Limit Therem is necessary fr this unit We will be wrking with tw inference prcedures: (1) Z-Intervals (2) T-Intervals

3 Estimatin A statistic is a value that measures a sample A parameter is a value that measures a ppulatin Sme cmmn examples: Sample mean estimates a ppulatin mean Sample prprtin estimates a ppulatin prprtin Sample variance estimates a ppulatin variance

4 Mtivatin Presidential Campaign CNN reprts that candidate A had an apprval rating f 72% with a sampling errr f plus r minus 3%. A randm selectin f 1,000 individuals were cntacted by telephne. Interpret these results. Sme preliminary remarks: Here, we wish t estimate the prprtin f all individuals in the ppulatin wh favr candidate A. In rder t d s, we will need t use a cnfidence interval. Althugh the cveted parameter in this example is a ppulatin prprtin (and nt a ppulatin mean), the cncept will be the same fr ppulatin means.

5 Presidential Z-Interval (cntinued) Campaign (cntinued) We are 95% cnfident that candidate A s true apprval D hand calculatins..assumptins.. rating amng all registered vters is between 69% and 75% In the plls, 95% cnfidence intervals are the nrm. Nte: We are nt restricted t using a 95% cnfidence-level Using an interval (called cnfidence interval) in rder t make an inference abut the prprtin f all registered vters wh favr candidate A Sampling Errr, r margin f errr, measures the spread in ur cnfidence interval

6 Pint Estimates The fllwing is a naïve argument: since 72% f the sample favred the candidate, then 72% f all registered vters favr the candidate. A better argument invlves estimatin by using a cnfidence interval. 72% is called a pint estimate since it is ne pint that is being used in rder t estimate a parameter Cnfidence Intervals are frmed by a tw-step prcess: (1) Start with a pint estimate (e.g. x) (2) Create an interval arund the pint estimate, with the pint estimate always lying at the center.

7 Z-Interval Prcedure There are 3 assumptins fr using the z-interval: (1) Simple Randm Sample (2) Large sample (30 r greater) r nrmal ppulatin (3) Ppulatin standard deviatin is knwn If these 3 assumptins are satisfied, then the cnfidence interval fr estimating μ is σ x ± z n Pint Estimate (sample mean) Margin f errr (smaller is better)

8 Example: Z-Interval Fr a sample f 100 randmly selected adults, the mean she size is 10. What is a 95% Cnfidence Interval that estimates the average she size fr the entire ppulatin? Assume the ppulatin standard deviatin is 1.

9 Z-Interval Example: Pick (using (cntinued) a card Excel) D hand calculatins..assumptins.. Pick [1] T a card find the frm z value: a standard 52-card deck S = { A, 2,, 10, = J, nrmsinv(0.975) Q, K, A, 2,, 10, J, Q, K, A, 2,, 10, J, Q, K, A, 2,, Since 10, we J, have Q, 95% K area } in the middle f ur Nrmal curve, there is 2.5% area in the left-tail and 2.5% area in the right- Each tail. card S, t is btain equally the likely z value, we need t tell Excel hw much area is t the left f ur z value this is (alternatively, we culd als use 0.025, and this will yield a P( Ace negative r z-value). ) = P( Ace ) Yu shuld + P( ) btain P( Ace a z value and f ) rughly Inserting = 4/52 this int + 13/52 the z-interval 1/52 = frmula 16/52 shuld 0.31 yield a cnfidence interval: (9.8, 10.2)

10 Example: Z-Interval Frmula Fr a sample f 100 randmly selected adults, the mean she size is 10. What is a 95% Cnfidence Interval that estimates the average she size fr the entire ppulatin? Assume the ppulatin standard deviatin is 1.0. Slutin: (Inference frm sample f 100 t ppulatin) Margin f errr x ± z σn = 10 ± We are 95% cnfident that the mean she size amng all individuals in the ppulatin is between 9.8 & The margin f errr is plus r minus 0.20.

11 Z-Interval Example: Pick (using (cntinued) a card TI-83) D hand calculatins..assumptins.. Pick [1] Since a card the frm ppulatin a standard standard 52-card deviatin deck is given, we create a Z-Interval: S = { A, 2,, 10, J, Q, K, A, 2,, 10, J, Q, K, A, Stat 2, >, Tests 10, > Z-Interval J, Q, K, > Stats A, 2,, 10, J, Q, K } [2] After entering the statistics, highlight the fllwing: Each card is equally likely C-level:.95 P( Ace r ) = P( Ace ) + P( ) P( Ace and ) = 4/ /52 1/52 = 16/ Cnclusin: We are 95% cnfident that the mean she size fr all individuals in the ppulatin is between 9.80 and (runded ff).

12 Z-Interval (cntinued) Z-Interval (cntinued) D hand calculatins..assumptins.. Fr the previus example, calculate 80%, 95%, and 99% cnfidence intervals. What d yu ntice abut these 3 intervals? What is the margin f errr f each interval? Prvide interpretatins fr each interval. What is the 100% Cnfidence Interval?

13 Z-Interval (cntinued) 80% CI is (9.87, 10.13) 95% CI is (9.8, 10.2) 99% CI is (9.74, 10.26) Interpretatin: Fr example, we are 80% cnfident that the ppulatin mean she size is between 9.87 and The margin f errr is ±0.13. S, we nw have an estimate fr the parameter μ 100% CI is (-infinity, infinity) r (0, 50) r (-100, 100), etc. Nte: 100% C.I. s are nt practical.

14 Margin f Errr The margin f errr is defined t be half the length f a cnfidence interval. The margin f errr is a measure f the spread in a cnfidence interval. The margin f errr des nt accmmdate errrs due t sampling bias. Key fact: As the cnfidence-level increases, s des the margin f errr (fr a fixed sample size).

15 Interpretatin f Cnfidence Intervals We are 90% cnfident that the mean she size fr all individuals in ur ppulatin ranges between 9.84 and We cannt say, hwever, that there is 0.90 prbability that this ne interval cntains the ppulatin mean. Rather, if we cnstruct many, many 90% Cnfidence Intervals in the same way, each with a sample size f 50, then abut 90% f these cnfidence intervals will cntain the ppulatin mean, and abut 10% f these cnfidence intervals will nt cntain the ppulatin mean.

16 Example: Z-Interval Market researchers seek t estimate the average # f sentences in televisin cmmercials. A sample f 49 cmmercials yields an average length f 20 sentences per cmmercial with a ppulatin standard deviatin f 5 sentences. Calculate the 95% cnfidence interval. What is yur interpretatin? What is the margin f errr? What are tw ways that the researcher can reduce the margin f errr?

17 Z-Interval: (cntinued) Slutin: (Inference frm sample f 49 t ppulatin) x ± z σn = 20 ± We are 95% cnfident that the average number f sentences fr all tv cmmercials is between 18.6 sentences and 21.4 sentences The margin f errr is ±1.4 Increase the sample size; decrease the cnfidence-level

18 When Example: Z-Interval t use Pick (cntinued) Z-Interval a card Prcedure D hand calculatins..assumptins.. Pick a card frm a standard 52-card deck Fr small samples say, less than 15 as lng as the ppulatin is nrmal r clse t being nrmal. S = { A, 2,, 10, J, Q, K, A, 2,, 10, J, Q, K, A, 2,, 10, J, Q, K, A, 2,, Fr mderate samples say, frm 15 t 29 as lng as the 10, data J, d nt Q, cntain K } utliers, and as lng as the ppulatin is nt far frm being nrmally distributed Each (meaning card is equally that there likely is n strng skewness). P( Ace Fr large r ) samples = P( Ace ) 30 + r P( ) greater P( Ace the prcedure and ) can be used withut = 4/52 restrictin. + 13/52 1/52 = 16/ Nte: The same restrictins apply t the T-Interval.

19 Example: Z-Interval A cllege admissins directr at sme university wishes t estimate the mean age f all students enrlled in In a randm sample f 20 students, the mean age is 22.9 years. Frm past studies, the standard deviatin is 1.5 years. Assume that the ppulatin is nt that far ff frm being nrmally distributed, and that there are n utliers in ur sample. Cnstruct a 90% cnfidence interval f the ppulatin mean age.

20 Z-Interval (cntinued) Using nrmsinv(0.95), we get z = The calculatin f the z-interval is 22.9 ± (1.645)(1.5/ 20) = 22.9 ± The margin f errr is rughly ±0.552 Interpretatin: We are 90% cnfident that the mean age f all students attending this university is between and years.

21 Estimating Sample Size Nte that in the frmula fr calculating the margin f errr, we can slve fr n algebraically. This frmula will give us the minimum sample size required t attain a certain margin f errr at a specified cnfidence-level (E = margin f errr): E = z σ n E 2 = n E 2 = z σ 2 n z σ 2 n = z σ E 2 Sample Size Frmula

22 Example: Sample Size A researcher wants t estimate the mean age f all students enrlled at sme cllege. The estimate must be within 1 year f the ppulatin mean. Assume that the distributin f all ages is nrmally distributed. What is the minimum required sample size t cnstruct a 90% cnfidence interval fr the ppulatin mean? Assume the ppulatin standard deviatin is 1.2 years.

23 Sample Size (cntinued) E = 1 (the margin f errr) z = σ = 1. 2 n = = = 4 Always rund up! Cnclusin: Thus, n = 4 is the minimum sample size necessary in rder t ensure a margin f errr f at-mst ±1 with a 90% cnfidence-level.

24 T-Intervals It is nt practical t knw the ppulatin standard deviatin The t-distributin utilizes the sample standard deviatin, s, instead f σ (this is much mre practical) Develped by William Gssett in 1908

25 Prperties f the standard t-curve The ttal area under the standard t-curve is 1 The standard t-curve extends indefinitely in bth directins, but never tuches the hrizntal axis Standard t-curves are centered abut 0 As the number f degrees f freedm increase, the standard t-curve lks mre and mre like the standard nrmal curve Picture taken frm Pearsn Educatin: 2008 Pearsn Educatin, Inc.

26 Example: t-values Fr a t-curve with 13 degrees f freedm, determine the t- value that has an area f 0.05 t its right, as shwn in the figure belw. Picture taken frm Pearsn Educatin: 2008 Pearsn Educatin, Inc.

27 t-values (cntinued) [1] T find the t-value using a calculatr (areas given t left): 2 nd > Vars > invt(0.95, 13) [2] The t-value may als be fund by using Excel: = tinv(0.10, 13) In Excel: In rder t find the t-value lying in the right-tail, yu must give the area that wuld lie in tw tails f the graph. In this case, the area in tw tails wuld be 2(0.05) = 0.10.

28 Example: t-values 1. Suppse a t-curve has 15 degrees f freedm. Determine the t-value that has an area f 0.05 t its right t its right 2. Suppse a t-curve has 12 degrees f freedm. Determine the t-value that has an area f 0.01 t its left t its left

29 Example: t-values (slutins) 1. Fr a t-curve with 15 degrees f freedm, find the t-value with 5% and 2.5% area t its right: t 0.05 = t = Fr a t-curve with 12 degrees f freedm, find the t-value with 99% and 97.5% area t its right: t 0.99 = t = 2.179

30 T-Interval Prcedure There are 3 assumptins fr using the t-interval: (1) Simple Randm Sample (2) Large sample (30 r greater) r nrmal ppulatin (3) Ppulatin standard deviatin is nt knwn If these 3 assumptins are satisfied, then the cnfidence interval fr estimating μ is s x ± t n Pint Estimate (sample mean) Margin f errr (smaller is better) Nte: df = n 1

31 Example: T-Interval The FBI cmpiles data n rbbery and prperty crimes and regularly publishes the infrmatin. A simple randm sample f pickpcket ffenses yielded the lsses, in dllars, shwn in the table belw. Use the data t find a 95% cnfidence interval fr the mean lss, μ, f all pickpcket ffenses. Table taken frm Pearsn Educatin: 2008 Pearsn Educatin, Inc.

32 Example: T-Interval (cntinued) Since the sample size, n = 25, is mderate, we first need t cnsider questins f nrmality and utliers. T d that, we may cnstruct a bxplt r stem-and-leaf plt, fr example. Yur plt shuld reveal that there are n utliers and n strng skewness (the data are nt t far frm being Nrmal). S, the t-interval prcedure may be perfrmed n the data.

33 Example: T-Interval (cntinued) Fr n = 25, we have df = 25 1 = 24. Using Excel, t = x = and s = S, a 95% cnfidence interval fr μ is calculated as ± (2. 064) Cnclusin: We are 95% cnfident that the mean lss f all pickpcket ffenses is smewhere between $ and $

34 Example: T-Interval (using Excel) [1] First enter the data in a blank clumn [2] Find the sample mean and sample standard deviatin with the fllwing cmmands and by highlighting the data: = average( ) = stdev( ) [3] With a t-value f and sample size 25, yu may nw btain the 90% cnfidence interval.

35 Example: T-Interval (using TI-83) [1] We first enter the data in a list: Stat > Edit > Enter data in L 1 [2] Nw we create a T-Interval: Stat > Tests > T-Interval > Data [3] After entering the statistics, indicate the fllwing: C-level:.90

36 Example: T-Interval Hw many hurs f sprts d Americans watch per week? T answer the questin, a randm sample f 81 individuals yielded a sample mean f 5.5 hurs with a sample standard deviatin f 1.2 hurs. Find a 95% cnfidence interval Find a 99% cnfidence interval Find the margin f errr fr bth Make yur cnclusins

37 Example: Z-Interval T-Interval Pick (cntinued) a card (slutins) D hand calculatins..assumptins.. Pick a card frm a standard 52-card deck The 95% cnfidence interval is (5.23, 5.77) The margin f errr is ±0.27 S = { A, 2,, 10, J, Q, K, A, 2,, 10, J, We are 95% cnfident that the average number f hurs Q, f sprts K, A, watched 2,, amng 10, all J, Americans Q, K, is A, between 2,, , hurs J, and Q, 5.77 K hurs. } Each card is equally likely The 99% cnfidence interval is (5.15, 5.85) The margin f errr is ±0.35 P( Ace r ) = P( Ace ) + P( ) P( Ace and ) We are 99% cnfident that the average number f hurs f sprts watched = 4/52 + amng 13/52 all Americans 1/52 = 16/52 is between hurs and 5.85 hurs.

38 Review Prblem #1 Cnsider the prblem f estimating the mean price f all cell phne bills. Suppse we select a randm sample 36 individual phne bills and btain a sample mean f $100. Assume the ppulatin standard deviatin is $6. Will we use a Z-Interval r T-Interval? Calculate a 99% cnfidence interval Calculate the margin f errr State a cnclusin

39 Slutin Cnsider the prblem f estimating the mean price f all cell phne bills. Suppse we select a randm sample 36 individual phne bills and btain a sample mean f $100. Assume the ppulatin standard deviatin is $6. Z-Interval $97.42 t $ ±2.58 We are 99% cnfident that the mean cell phne bill fr all individuals in the ppulatin is between $97.42 and $102.58

40 Review Prblem #2 Amng a certain ppulatin f business firms, suppse the distributin f the firms ages has a standard deviatin f 21 years. In planning a survey t estimate the mean age f the firms in the ppulatin, it is f interest t prduce a 94% cnfidence interval with a margin f errr less than 5. What sample size is needed t achieve this bjective? n = 7 n = 43 n = 63 n = 98

41 Slutin Amng a certain ppulatin f business firms, suppse the distributin f the firms ages has a standard deviatin f 21 years. In planning a survey t estimate the mean age f the firms in the ppulatin, it is f interest t prduce a 94% cnfidence interval with a margin f errr less than 5. What sample size is needed t achieve this bjective? n = 7 n = 43 n = 63 n = 98

42 Review Prblem #3 A statistician is analyzing a randm sample f 121 measurements frm a Nrmal ppulatin with a standard deviatin f 50. The mean f the sample is bserved at 172, and the statistician reprts a cnfidence interval f 172 ± 10 = (162, 182). What is the cnfidence-level assciated with this interval? 95.8% 96.4% 97.2% 98.3%

43 Slutin A statistician is analyzing a randm sample f 121 measurements frm a Nrmal ppulatin with a standard deviatin f 50. The mean f the sample is bserved at 172, and the statistician reprts a cnfidence interval f 172 ± 10 = (162, 182). What is the cnfidence-level assciated with this interval? 95.8% 96.4% 97.2% 98.3%

44 Summary Cnfidence Intervals A cnfidence interval estimates the value f a parameter. We dn t knw the value f μ, but C.I. s allw us t estimate its value. Basic structure f a cnfidence interval: Pint Estimate ± Margin f Errr As C.L. increases, the interval gets wider The researcher may chse any cnfidence-level Interpretatin f C.I s dne in the lng run

45 Summary Margin f Errr Margin f errr measures the spread r variability f a cnfidence interval As MOE increases, the intervals widens Margin f errr is everything after the plus r minus As sample size increases, margin f errr decreases As C.L. increases, the margin f errr increases. As C.L. decreases, the margin f errr decreases.