Sampling Distributions

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Chapter Sampling Distributions Introduction Generally, we are interested in population parameters When the census is impossible, we draw a sample from the population, then construct sample statistics, that have close relationship to the population parameters Introduction Samples are random, so the sample statistic is a random variable As such it has a sampling distribution 8.1 Sampling Distribution of the Mean Example 1: A die is thrown infinitely many times Let X represent the number of spots showing on any throw The probability distribution of X is x p(x) 1/6 1/6 1/6 1/6 1/6 1/6 E(X) = 1(1/6) +2(1/6) + 3(1/6)+………………….= 3.5 V(X) = (1-3.5)2(1/6) + (2-3.5)2(1/6) +….…… …= 2.92 Suppose we want to estimate  from the mean of a sample of size n = What is the distribution of x ? Throwing a die twice – sample mean these are the means of each These are And all the possible pairs of values for pair the throws The distribution of x when n = Calculating the relative frequency of each value of x we have the following results Frequency1 1/36 Relative freq (1+1)/2 = 1.5 2.0 2/36 3/36 2.5 4/36 (1+2)/2 = 1.5 (2+1)/2 = 1.5 3.0 5/36 3.5 6/36 (1+3)/2 = (2+2)/2 = (3+1)/2 = 4.0 5/36 4.5 4/36 5.0 5.5 6.0 3/36 2/36 1/36 Notice there are 36 possible pairs of values: 1,1 1,2 … 1,6 2,1 2,2 … 2,6 ……………… 6,1 6,2 … 6,6 n 5 n 10 n 25  x 3.5  x 3.5  x 3.5  2x  .5833 (  )  2x  x .2917 (  ) 10  2x  .1167 (  ) 25 x x As the sample size changes, the mean of the sample mean does not change! n 5 n 10 n 25  x 3.5  x 3.5  x 3.5  2x  .5833 (  )  2x  x .2917 (  ) 10  2x  .1167 (  ) 25 x x As the sample size increases, the variance of the sample mean decreases! Demonstration: Why is the variance of the sample mean is smaller than the population variance? Mean = 1.5 Mean = Mean = 2.5 Population 1.5 2.5 Compare thetake range of the population Let us samples to the range of the sample mean of two observations 10 The Central Limit Theorem If a random sample is drawn from any population, the sampling distribution of the sample mean is: – Normal if the parent population is normal, – Approximately normal if the parent population is not normal, provided the sample size is sufficiently large The larger the sample size, the more closely the sampling distribution of x will resemble a normal distribution 11 The mean of X is equal to the mean of the parent population μ x μx The variance of X is equal to the parent population variance divided by ‘n’ x σ σ  n x 12 n Sampling Distribution Population distribution 30 30 50 70 90 120 Census n Sampling Distribution Normal Pop distribution Example 2: The amount of soda pop in each bottle is normally distributed with a mean of 32.2 ounces and a standard deviation of ounces Find the probability that a bottle bought by a customer will contain more than 32 ounces 0.7486 P(x  32) x = 32  = 32.2 x  μ 32  32.2 P(x  32) P(  ) P(z   67) 0.7486 σx 16 Find the probability that a carton of four bottles will have a mean of more than 32 ounces of soda per bottle x   32  32.2 P( x  32) P(  ) x P( z   1.33 ) 0.9082 P(x  32) x 32  x 32.2 17 Example 3: The average weekly income of B.B.A graduates one year after graduation is $600 Suppose the distribution of weekly income has a standard deviation of $100 What is the probability that 35 randomly selected graduates have an average weekly income of less than $550? x  μ 550  600 P(x  550) P(  ) σx 100 35 P(z   2.97) 0.0015 18 19 ... Samples are random, so the sample statistic is a random variable As such it has a sampling distribution 8.1 Sampling Distribution of the Mean Example 1: A die is thrown infinitely many times... parent population variance divided by ‘n’ x σ σ  n x 12 n Sampling Distribution Population distribution 30 30 50 70 90 120 Census n Sampling Distribution Normal Pop distribution Example 2: The... provided the sample size is sufficiently large The larger the sample size, the more closely the sampling distribution of x will resemble a normal distribution 11 The mean of X is equal to the
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