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Statistics 270 - Lecture 20 • Last Day…completed 5.1 • Today Parts of Section 5.3 and 5.4 Sampling • In chapter 1, we concerned ourselves with numerical/graphical summeries of samples (x1, x2, …, xn) from some population • Can view each of the Xi’s as random variables • We will be concerned with random samples • The xi‘s are independent • The xi‘s have the same probability distribution • Often called Definitions • A parameter is a numerical feature of a distribution or population • Statistic is a function of sample data (e.g., sample mean, sample median…) • We will be using statistics to estimate parameters (point estimates) • Suppose you draw a sample and compute the value of a statistic • Suppose you draw another sample of the same size and compute the value of the statistic • Would the 2 statistics be equal? • Use statistics to estimate parameters • Will the statistics be exactly equal to the parameter? • Observed value of the statistics depends on the sample • There will be variability in the values of the statistic over repeated sampling • The statistic has a distribution of its own • Probability distribution of a statistic is called the sampling distribution (or distribution of the statistic) • Is the distribution of values for the statistic based on all possible samples of the same size from the population? • Based on repeated random samples of the same size from the population Example • Large population is described by the probability distribution X P(X=x) 0 0.2 3 0.3 12 0.5 • If a random sample of size 2 is taken, what is the sampling distribution for the sample mean? Sampling Distribution of the Sample Mean • Have a random sample of size n • The sample mean is n xi x i1 n • What is it estimating? Properties of the Sample Mean • Expected value: • Variance: • Standard Deviation: Properties of the Sample Mean • Observations Sampling from a Normal Distribution • Suppose have a sample of size n from a N(m,s2) distribution • What is distribution of the sample mean? Example • Distribution of moisture content per pound of a dehydrated protein concentrate is normally distributed with mean 3.5 and standard deviation of 0.6. • Random sample of 36 specimens of this concentrate is taken • Distribution of sample mean? • What is probability that the sample mean is less than 3.5? Central Limit Theorem • In a random sample (iid sample) from any population with mean, m, and standard deviation, s, when n is large, the distribution of the sample mean X is approximately normal. • That is, Central Limit Theorem • For the sample total, Implications • So, for random samples, if have enough data, sample mean is approximately normally distributed...even if data not normally distributed • If have enough data, can use the normal distribution to make probability statements about x Example • A busy intersection has an average of 2.2 accidents per week with a standard deviation of 1.4 accidents • Suppose you monitor this intersection of a given year, recording the number of accidents per week. • Data takes on integers (0,1,2,...) thus distribution of number of accidents not normal. • What is the distribution of the mean number of accidents per week based on a sample of 52 weeks of data Example • What is the approximate probability that X is less than 2 • What is the approximate probability that there are less than 100 accidents in a given year?