Estimating the Population Mean - when a sample average is calculated we can create an interval centered on this average

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1 6. Cofidece Iterval for the Populatio Mea p58 Estimatig the Populatio Mea - whe a sample average is calculated we ca create a iterval cetered o this average x-bar - at a predetermied level of cofidece we ca claim that the true mea ( ie populatio mea ) will lie withi this iterval A) Give Populatio Stadard Deviatio eg) A recet sample of 36 customers provided a average store shoppig time of 7.1 miutes ( s =.1 miutes ). A) Calculate the 99% cofidece iterval give =. miutes -the 7.1 miutes is a sample average ad oly a estimate of the true average -how close is 7.1 miutes to the true average x-bar = 7.1 miutes pop. mea - at a predetermied level of cofidece, we are certai that the µ will lie i the iterval - traditioal levels of cofidece: 90%, 95%, 99% 4516CImeazS /07/04

2 -cofidece, like probability, is related to the area uder the Normal curve -this is the cosequece of the Cetral Limit Theorem -the curve is created by plottig a large umber of sample meas 0.5% 99% cofidece pop. mea 0.5% - level of cofidece is level of sigificace = = measure of the risk of beig wrog = 0.01 = The z is determied from the ormal probability distributio table A( z ) = or ; z = if both ad s is available, always use x z [ [ x + z x! z 7.1! ! miutes [ [ 8.1 miutes - we ca be 99% cofidet that the true average shoppig time is betwee 6. ad 8.1 miutes 4516CImeazS04 01/07/04

3 b) Recalculate the cofidece iterval at the 5% sigificace level.5% 95% cofidece pop. mea.5% - level of cofidece is level of sigificace a = = measure of the risk of beig wrog = 0.05 = 0.05 the z is determied from the ormal probability distributio table A( z ) = 0.05 or ; z = 1.96 x z [ [ x + z 7.1! ! mi [ [ 7.8 mi - we ca be 95% cofidet that the true average repait period is betwee 6.4 ad 7.8 years 4516CImeazS /07/04

4 c) Recalculate the cofidece iterval at the 10% sigifigace level. 90% cofidece 5% 5% pop. mea - level of cofidece is level of sigificace a = = measure of the risk of beig wrog a/ = 0.10/ = 0.05 the z is determied from the ormal probability distributio table A( z ) = 0.05 or ; z = x z [ [ x + z 7.1! ! mi [ [ 7.7 mi - we ca be 95% cofidet that the true average shoppig time is betwee 6.5 ad 7.7 miutes 4516CImeazS /07/04

5 ote: the 99% C.I. 6. mi < < 8.1 mi the 95% C.I. 6.4 mi < < 7.8 mi the 90% C.I. 6.5 mi < < 7.7 mi - as the level of cofidece is icreased, what happes to the rage of the cofidece iterval? - as the degree of cofidece is icreased, the iterval grows larger - as the iterval grows larger, the iterval becomes less iformative What is the size of the 100% cofidece iterval? eg) A report claims that teeagers o average drik 49.1 L of soft driks ( = 5.6 L ) from Jue to September. A researcher wishes to test this report with a 95% cofidece iterval. She fids from a sample of 55 teeagers that they o average drik 47.8 L. What ca she coclude? Sice her value of 47.8 L is differet from 49.1 L the the 49.1 L value must ow be wrog. - No, the 47.8 L is oly a sample value beig compared to a established value of 49.1 L - aother sample of 55 may provide a value differet from 47.8 x z [ [ x + z [ [ [ [ L [ [ 49.3 L - sice 49.1 L falls withi the iterval the her sample does ot provide sufficiet evidece to show that 49.1 L is wrog. 4516CImeazS /07/04

6 MAXIMUM ERROR ( MARGIN OF ERROR ) - the populatio mea will be off by at most this amout o either side of the sample mea B = z B m as m B m as o eg) A shoppig mall survey idicated that the average distace travelled by a sample of 4 customers is.6 km ( = 0.8 km ). What is the margi of error at the 95% cofidece level? B = z = = 0.4 km we are 95% cofidet that the average travel distace of.6 km is off by at most 0.4 km 4516CImeazS /07/04

7 SAMPLE SIZE - to determie sample size, rearrage the maximum error formula B = z B = z = z B = z B eg) Fid the sample size required to estimate the average aual RRSP deposit. We wish to be 95% cofidet that the result will be off by o more tha $100. A bak maager estimates that deposits rage from $500 to $5000. estimated = = = = $115 4 z B 1.96 % 115 = 486. u 487 deposits use a sample size of at least 487 delivery times What affect does chagig M, z ad have o sample size? as B icreases, falls as z icreases, icreases s is itrisic to the populatio, the aalyst has o cotrol over s for people data ote that the accuracy of a calculatio varies as the square root of sample size, thus small to moderate icreases i sample size will greatly improve accuracy but as the sample size becomes large, the rate of improvemet i accuracy dimiishes HOMEWORK read p58-60 Page , 5.53, CImeazS /07/04

8 Sectio 5.4 Cofidece Iterval for a Populatio Mea 5.49 The illustratio shows 99% ad 99% cofidece itervals for based o usig the same data. Which oe A or B is the 95% cofidece iterval? Explai your aswer. Iterval A is the 95% iterval because a 95% cofidece iterval is geerally shorter tha a 99% cofidece iterval. Sice the 95% iterval allows for less cofidece this allows for a shorter lie A cosumer research group sampled 100 hadheld video games, all of the same make ad model. The sample mea life (hours of operatio before failure) was 560 hours. Assumig the stadard deviatio is 35 hours, costruct a 90% cofidece iterval estimate based o x of the true mea life spa of the video games. A( z 0.10 ) = 0.05 or ; = z x! z 560! ! < < h < < h x - we ca be 90% cofidet that the true average life of the video game is betwee 554. ad hours. Is a 95% cofidece iterval wider or arrower tha the iterval you obtaied? Explai your aswer. A 95% cofidece iterval would be wider because the margi of error would be multiplied by 1.96 istead of To study the birth weightsof ifats whose mothers smoke, a physicia records the weights of 100 ewbors whose mothers smoke. Fid a 98% cofidece iterval based o x for the mea birth weight of childre of smokig mothers if x was foud to be 6.1 pouds. Use =.1pouds. A( z 0.0 ) = 0.01 or ; =.33 z x! z 6.1! ! < < lb < < 6.59 lb 4516CImeazS /07/04

9 What sample size is required so that the margi of error i determiig the mea birth weight is oly 0.3 pouds?.3 =.33*.1/ so = ±.58*35/ 100 The cofidece iterval is to The iterval does cotai 500. The advertiser s claim is a valid claim ± 1.645*(000)/ 40 The cofidece iterval is to The Cetral Limit Theorem says that the z-iterval ca be applied to o-ormal populatios if the sample size is sufficietly large. I the case of severe skewess or symmetric log tails, a sample size of 40 may ot be adequate. Therefore, it is a good idea to always check for skewess ad log tails ±.776*3.7/ 5 The cofidece iterval is 65.1 to CImeazS /07/04