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Basic Statistical Concepts population–the collection of all items of interest to a researcher. sample–a subset of the population which we gather information on. A common sample is SRS (simple random sample). descriptive statistics–summarize information contained in a sample. statistical inference–generalize from the sample to the population. statistic–a numerical summary of a sample. A statistic is random. Its distribution is called sampling distribution. Descriptive Statistics frequency distribution and histogram: for summarizing quantitative data. Pn yi sample mean ȳ = i=1 n sample median: the midpoint of the ordered data. Pn (y −ȳ )2 i 2 sample variance: s = i=1n−1 . √ sample standard deviation: s = s 2 . Random Variables random variable is a mapping from every possible outcome of an experiment to real numbers. notation: X, Y, Z,... Example: Ask a student whether she/he works part time or not. S= {Yes, No}. X=1 if Yes, X=0 if No. Y = number of car accidents in a week. Flip a coin three times. Let Z=number of heads in three flips. {TTT } → Z = 0 {HTT , THT , TTH} → Z = 1 {HHT , HTH, THH} → Z = 2 {HHH} → Z = 3. W = the weight of a randomly selected athlete. Discrete Random Variables A discrete random variable takes finite or countably infinite values. The probability mass function distribution (pmf) of Z is z P(z) —————— 0 1/8 1 3/8 2 3/8 3 1/8 Note that 1.) 0P≤ P(x) ≤ 1 2.) P(x) = 1. The Mean of X The mean or expected value of a random variable X is P E (X ) = µX = xP(x). What is the mean of Z ? The variance of P X is 2 Var (X ) = σX = (x − µ)2 P(x) = E (X 2 ) − µ2 . Exercise A Computer shop builds shipments of parts it receives from various suppliers. Let X be the number of defective hard drives per shipments. It is assumed X has the following distribution. x P(x) ——————– 0 0.55 1 0.15 2 0.10 3 0.10 4 0.05 5 0.05 Find P(X ≥ 2), E (X ).