Introduction to Probability and Statistics
Introduction to Probability and Statistics STAT 20
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This 3 page Class Notes was uploaded by Floy Kub on Thursday October 22, 2015. The Class Notes belongs to STAT 20 at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/226733/stat-20-university-of-california-berkeley in Statistics at University of California - Berkeley.
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Date Created: 10/22/15
Introduction to Inference Con dence Intervals o A Typical Inference Problem 0 The 95 Con dence Interval 7 De nition 7 Calculating a 95 Con dence Interval 0 Interpretation of Con dence Intervals o The General Form of a Con dence Interval 0 Finding 2 0 Factors Affecting CI Length 1 Recall from the previous lecture that U W The distribution is exact if the population xNm distribution is normal and approximately correct for large n in other cases by the CLT Thus 039 7 039 P 71967 X 1967 095 0 w lt 1 Rearranging terms we have 7 039 7 039 P X 7196 lt u lt X 196 095 In other words there is 95 probability that the random interval 7 a 7 a X71967 X 1967 lt W 1 x will cover In In our example i 1100 a 100 and n 100 Therefore the 95 con dence interval for u is 1100 7196 x 101100 196 x 10 1080411196 A Typical Inference Problem Suppose we want to nd out about the mean lifetime In of a certain brand of light bulbs Suppose that the true mean u is unknown but we know perhaps from previous studies that the SD 0 of the light bulb lifetime is 100 hours In order to estimate the population mean u we 0 Take a SRS of 100 light bulbs 0 Calculate the mean lifetime in the sample to be 1100 hours What can we say about the population mean EX 1 1302 100N100 10 o X 7 In Law of Large Numbers X amp Nu 10 CLT Calculating a 95 Con dence Interval For the time being we ll continue to assume that a is known To calculate a 95 con dence interval for the population mean u 1 Take a random sample of size n and calculate the sample mean i to If n is large enough i amp N In by the CLT OJ The con dence interval is given by 039 039 771967 T 1967 35 W M w Interpretation of Con dence Intervals Suppose we repeat the following procedure multiple times 1 Draw a random sample of size n 2 Calculate a 95 con dence interval for the sample 95 of the interuals thus constructed will couer the true unknown population mean Density curve of 5 General Form of a Con dence Interval In general a CI for a parameter has the form estimate i margin of error where the margin of error is determined by the con dence level 1 7 oz the population SD 0 and the sample size n A l 7 oz con dence interval for a parameter 9 is an interval computed from a SRS by a method with probability 1 7 oz of containing the true 9 For a random sample of size n drawn from a population of unknown mean u and known SD 0 a l 7 oz CI for u is a Here 2 is the critical value selected so that a ii standard Normal density has area 1 7 oz between 72 and 2 The quantity 2an then is the margin error If the population distribution is normal the interval is exact Otherwise it is approximately correct for large n Example Consider estimating the speed of light using 64 measurements with sample mean i 298 054 kms Assume we know from previous experience that the SD of measurements made using the same procedure is 60 cms What is a 95 CI for the true speed of light Incorrect 0 There is a 95 probability that the true speed of light lies in the interval 2980393 298068 0 In 95 of all possible samples the true speed of light lies in the interval 2980393 298068 Correct 0 There is 95 con dence that the true speed of light lies in the interval 2980393 298068 0 There is 95 probability that the true speed of light lies in the random interval a 7 1962 0 196 o If we repeatedly draw samples and calculate con dence intervals using this procedure 95 of these intervals will cover the true speed of light 6 Finding 2 For a given con dence level 1 7 oz how do we nd 2 Let Z N N0 l 1125 um Pp g 29 17a 42gt Pzlt7z Thus for a given con dence level 1 7 oz we can look up the corresponding 2 value on the Normal table Common 2 values Con dence Level 90 95 99 2 1645 196 2576 Examples 1 The quality control section of a company which makes bags of sugar knows that the weight of the sugar bags is NW 0 with a 3g A random sample of 30 bags yields a sample mean of 998g What is a 90 con dence interval for u ls the sugar buying public being defrauded on 1kg bags of sugar to Suppose a SRS of 500 is drawn from the population of California high school juniors Their mean SAT math test score is 461 Assume a is known to be 100 What is a 95 CI for the mean SAT math test score of the population of Californian high school juniors What about a 70 Cl Factors Affecting CI Length It is not hard to see that the length of a Cl is given by Length of the CI 22 What happens to the length of the Cl if c we decrease increase the con dence level 0 we decreaseincrease the sample size n o a is largersmaller If we know before conducting a study that want a speci c con dence level and a speci c margin of error m7 we need to adjust the sample size 2 20 n m Note that this calculation requires the same assumptions we have been using all along namely that the sample is an SRS7 i has normal distribution or n is suf ciently large that it is approximately normal7 and a is known
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