Review Sheet for MATH 160 at UA
Review Sheet for MATH 160 at UA
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Date Created: 02/06/15
Con dence intervals The basics BPS chapter 14 2006 WH Freeman and Company Objectives BPS Chapter 14 Confidence intervals the basics n Estimating with confidence I Confidence intervals for the mean u a How confidence intervals behave n Choosing the sample size Estimating with con dence Although the sample meanC is a unique number for any particular sample if you pick a different sample you will probably get a different sample mean In fact you could get many different values for the sample mean and virtually none of them would actually equal the true population mean u Sampling 39 distributlm quotw 1 quot f 272 Y of I All Populatth 5 s n g 39 Ll 3 a 1 w a f39 lm l Wham g mw 1H 7 39u 5H5 n sun tunlmnwn n i 39 39 1 quot39 t u l quot l39 t Ii z But the sample distribution is narrowerthan the population distribution by a factor of n Thus the estimates f gained from our samples are always relatively close to the population parameterJ Sample means 7 n subjects If the population is normally distributed Npo so will the sampling distribution Muchn 95 of all sample means will be within roughly 2 standard deviations 2oin of the population parameter u li I quotmm Because distances are symmetrical this implies that the population parameter u must be within roughly 2 standard deviations from the sample average 3 in 95 of all samples Red dot mean value of individual sample This reasoning is the essence of statistical inference The weight of single eggs of the brown variety is normally distributed N65g5g Think of a carton of 12 brown eggs as an SR8 of size 12 o What is the distribution of the sample means 3 Normal mean 4 standard deviation aNn N659144g 1 Find the middle 95 of the sample means distribution Roughly 1 2 standard deviations from the mean or 659 1 2889 population sample You buy a carton of 12 white eggs instead The box weighs 770g The average egg weight from that SR8 is thus 3 642g 1 Knowing that the standard deviation of egg weight is 5g what can you infer about the mean I of the white egg population There is a 95 chance that the population mean p is roughly within 1 20m of 2 or 6429 i 2889 Confidence interval A level C con dence interval for a parameter has two parts a An interval calculated 39om the data usually ofthe form estimate margin of error a A con dence level C which gives the probability that the interval will capture the true parameter value in repeated samples or the success rate for the method EULZ2722L2 57 at these i x uvz 25 v2 intervals capture was unknown u Luau Implications We don t need to take lots of gym4h u 25 random samples to rebuild the l u H l m y us my men x sampling distribution and find u at its center 22 w z 2 n Mean an lnmmln 4 m sunken Means 7 or m subjects sampe All we need is one SRS of 4 size n and relying on the W POPUIation Observations on 1 subject propertles Of the sample means distribution to infer the population mean p Reworded With 95 confidence we can say that U should be within roughly 2 standard deviations 2oNn from our sample mean 7 bar a In 95 of all possible samples of this size n u will indeed fall in our confidence interval a In only 5 of samples would 7 be farther from p sampling distribution of i Interpreting a con dence interval for a mean A confidence interval can be a Two endpoints of an interval expressed as y possibly within f m to f n f i m m m is called the margin oferror Example 114 to 126 Standard A con dence level C in normaicuwe indicates the success rate of the method that produces the interval i 39 16 ProbabilitgT Probability It represents the area under the normal curve within i m of the center of the curve Review standardizing the normal curve using 2 N645 25 Np oNn 95 997 w 68 of data 95 of data I 997 of data 57 555 62 645 67 69395 72 x 3 2 1 o 1 2 3 Z Height inches Standardized height no units Here we work with the sampling distribution and on is its standard deviation spread Rememberthat ois the standard deviation of the original population Varying con dence levels Con dence intervals contain the population mean u in C of samples Different areas under the curve give different con dence levels C standard Practical use of z 2 normaiwme u 2 is related to the chosen con dence level C C u C is the area under the standard quotmbab39mvm Probahililu u normal curve between 2 and 2 Probability M The confidence interval is thus 2 125 2mm gtllt x i Z OF Example For an 80 confidence level C 80 ofthe normal curve s area is contained in the interval How do we find specific 2 values We can use a table of zt values Table C For a particular confidence level C the appropriate 2 value is just above it z39 Ide Ila 10 1515 13351 Ellil REE 39 quotquotquot539i39rilaquotquot izitriam quot Ema quot 51375 Con dence level C Ex For a 98 confidence level z2326 We can use software In Excel NORMNVprobabilitymeanstandarddev gives 2 for a given cumulative probability Since we want the middle C probability the probability we require is 1 C2 Example For a 98 confidence level NORMINV 01 O1 232635 neg 2 Link between confidence level and margin of error The con dence level C determines the value of z in Table C The margin of error also depends on 2 mz039 Higher confidence C implies a larger margin of errorm thus less precision in our estimates A lower confidence level C produces a smaller margin of error 111 thus better precision in our estimates Different con dence intervals for the same set of measurements Density of bacteria in solution Measurement equipment has standard deviation 039 1105 bacteriam1 uid 3 measurements 24 29 and 31 105 bacteriam1 uid Mean f 28105 bacteriam1 Find the 96 and 70 CI in 96 confidence interval for the n 70 confidence interval for the true density 2 2054 and write true density 2 1036 and write k U U xiz 28i20541N3 xiz 28i10361N3 J J 28i119106 28i060106 bacteriaml bacteriaml yet 129 USN 18s SE 35 l 82 To Con dence Irma C 1 Impact of sample size The spread in the sampling distribution of the mean is a function of the number of individuals per sample The larger the sample size the smaller the standard deviation spread of the sample mean distribution But the spread only decreases at a rate P0P equal to in Standard error oxn Sample size n Sample size and experimental design You may need a certain margin of error eg drug trial manufacturing specs In many cases the population variability 039 is fixed but we can choose the number of measurements n So plan ahead what sample size to use to achieve that margin of error 2 039 2039 mz lt3 2 W m Remember though that sample size is not always stretchable at will There are typically costs and constraints associated with large samples The best approach is to use the smallest sample size that can give you useful results What sample size for a given margin of error Density of bacteria in solution Measurement equipment has standard deviation 0 1106 bacteriaml fluid How many measurements should you make to obtain a margin of error of at most 05106 bacteriaml with a confidence level of 90 For a 90 confidence interval 2 1645 gtxlt Z gtxlt Z n2 a nj 329Z108241 Using only 10 measurements will not be enough to ensure that m is no more than 05106 Therefore we need at least 11 measurements H611 LEGS 1282 1645 Lg 2 837 JEGI H811 129 9 1 5 Confidence lmral C
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