10/27/09 Lecture 15 1STOR 155 Introductory StatisticsLecture 15: Sampling Distributions for Sample MeansThe UNIVERSITY of NORTH CAROLINAat CHAPEL HILL10/27/09 Lecture 15 2Review• Sampling distribution for a sample count:– Binomial experiments– Binomial distribution– Normal approximation– Continuity correction• Sampling distribution for a sample proportion:– Exact calculation via binomial distribution– Normal approximation No continuity correction• How about a sample mean?10/27/09 Lecture 15 3Diversification• A basic principle of investment is that diversification reduces risk. • That is, buying several securities, (for example stocks), rather than just one, reduces the variability of the return on an investment. • The following figures show two distributions of returns in 1987.10/27/09 Lecture 15 4• Distribution of returns for all 1815 stocks on the NYSE for the entire year 1987.• The mean return was –3.5% and the distribution shows a very wide spread.10/27/09 Lecture 15 5• Distribution of returns for all possible portfolios that invested equal amounts in each of 5 stocks in 1987.• The mean is still –3.5%, but the variability is much less.10/27/09 Lecture 15 6Averages vs. IndividualsThe investment example shows that• Averages are less variable than individual observations;• Averages are closer to ``normal’’ than individual observations.Why?10/27/09 Lecture 15 7• The sampling distribution of for samples of size 10 compared with the distribution of a single observation.X10/27/09 Lecture 15 810/27/09 Lecture 15 9Sampling Distribution ofX10/27/09 Lecture 15 10The distributions of for (a) 1 obs. (b) 2 obs. (c) 10 obs. (d) 25 obs. --- correction for ``skewness’’X10/27/09 Lecture 15 11• The amount of soda pop in each bottle is normally distributed with a mean of 32.2 ounces and a standard deviation of 0.3 ounces.• Find the probability that a bottle bought by a customer will contain more than 32 ounces.• Answer: Let X denote the amount of soda in a bottle.7486.0)67.()3.02.32323.02.32()32(zPXPXPm = 32.2x = 32Soda Pop10/27/09 Lecture 15 12• Find the probability that a box of four bottles will have a mean of more than 32 ounces of soda per bottle.• Answer: The random variable here is the mean amount of soda per bottle, which is normally distributed with a mean of 32.2 and standard deviation of 0.3 / 2 = 0.15. Hence9082.0)33.1()15.02.323215.02.32()32(ZPXPXPSoda Pop10/27/09 Lecture 15 13• The average weekly income of graduates one year after graduation is $600. Suppose the distribution of weekly income has a standard deviation (SD) of $100.• What is the probability that 25 randomly selected graduates have an average weekly income of less than $550?• Answer: According to CLT, approximately has a normal distribution with mean 600 and SD 100 / 5 =20..0062.0)5.2()2060055020600()550(ZPXPXPWeekly IncomeX10/27/09 Lecture 15 14• Suppose we actually found a random sample of 25 graduates, with an average weekly income of $550.• What can you say about the validity of the claim that the average weekly income is $600?• Answer:– If the population mean is $600, then the probability to have observed a sample mean of $550 is very low (0.0062). The evidence provided by the sample suggests that the assumed average weekly income $600 is unjustified.– It’s more reasonable to believe that the population mean is actually smaller than $600. Then a sample mean of $550 becomes more probable.Weekly Income (continued)Sum of independent normal random variables• Fact: If X , Y are independent normal random variables and a, b are constants, then aX+bY also follows a normal distribution with meanE(aX+bY) = a E(X) + b E(Y)and varianceUse this fact to solve Problem 5.60 (c) (d) on page 349.10/27/09 Lecture 15 1522222YXbYaXba10/27/09 Lecture 15 16Take Home Message• Sampling distribution for a sample mean– Mean for a sample mean– Variance for a sample mean– Central Limit Theorem– Normal distribution
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