A fancy R trick N 10 c 2 4 8 16 32 64 for n in N print sum dbinom seq 0 7 n 0 8 n by 1 n 0 75 1 0 5606259 1 0 6389116 1 0 7553899 1 0 8799318 1 0 9670862 1 0 9970046 4 Statistics 371 Fall 2004 Quantitative Observations Sampling Distributions The mean of the sampling distribution of Y Y is the same as the population mean In symbols Y The standard deviation of the sampling distribution of Y Y is smaller than the population standard deviation by a factor of n In symbols Y n If the sample size n is sufficiently large the shape of the sampling distribution of Y will be approximately normal This is the Central Limit Theorem If the population is normal a sample size of 1 suffices If the population is not normal it depends on how the population differs from normality to determine if the normal approximation is reasonably accurate Statistics 371 Fall 2004 6 Bret Larget 5 A population may be modeled as a box with numbered or colored balls We think of a sample of data as having been selected at random from this population From each sample we can calculate a sample statistic such as a sample mean The sampling distribution of the sample mean is the collection of all possible sample means that could occur by random sampling of a given sample size n The textbook refers to the thought exercise of considering all the ways a sample could have turned out as a metaexperiment Department of Statistics Sampling Distribution of Y October 4 2004 Statistics 371 Fall 2004 University of Wisconsin Madison Now consider a population where each individual is associated with a quantitative variable We can compute the sample mean from each sample The sampling distribution of Y is the collection of sample means from the meta experiment of all possible samples of size n Sampling Distributions Statistics 371 Fall 2003 Here is R code to do the previous calculation for a variety of sample sizes Statistics 371 Fall 2004 Dichotomous Observations Consider a cross of two heterozygotes Aa Aa 1 The probability distribution of the genotypes of the offspring is as follows cross Aa Aa Offspring Genotype AA Aa aa 0 25 0 50 0 25 If Y is the number of dominant offspring AA or Aa in a sample of size n 2 and if p Y n is the sample proportion then the sampling distribution of p is tabulated below Probabilities are from the binomial distribution Y 0 1 2 p 0 0 0 5 1 0 Prob 0 0625 0 3750 0 5625 Statistics 371 Fall 2004 2 Example calculation Larger Example Suppose that the weights of seeds are approximately normal with a mean of 500 mg and a standard deviation of 150 mg Find the probability that the sample mean is between 450 and 50 for a variety of sample sizes For the previous cross what is the probability that exactly 15 of 20 offspring are dominant dbinom 15 20 0 75 1 0 2023312 For n 4 we have Pr 450 Y 550 The number of dominant offspring will have a binomial distribution with n 20 and p 0 75 Y 500 550 500 450 500 Pr 150 4 150 4 150 4 Pr 0 67 Z 0 67 What is the probability that p is within 0 05 of p Translate the probability to a binomial question 0 5028 from a normal table calculation Pr 0 70 p 0 80 Pr 0 70 Y 20 0 80 Pr 14 Y 16 sum dbinom 14 16 20 0 75 1 0 5606259 Statistics 371 Fall 2004 7 Statistics 371 Fall 2004 3 Fancy R example Here is sample R code to do this calculation for several n differently than the previous example N len p for c 4 8 16 32 64 length N rep 0 len i in 1 len p i pnorm 550 500 150 sqrt N i pnorm 450 500 150 sqrt N i cbind N p N p 1 4 0 4950149 2 8 0 6542214 3 16 0 8175776 4 32 0 9406536 5 64 0 9923392 The first number in this table disagrees with the previous calculation slightly because R did not round off the z score to the nearest hundredth Statistics 371 Fall 2004 8 Exercise 5 18 Assume that height of corn plants are normally distributed with a mean 145 cm and a standard deviation of 22 cm What proportion of plants are between 135 and 155 cm pnorm 155 145 22 pnorm 135 145 22 1 0 3505637 Find Pr 135 Y 155 when n 16 pnorm 155 145 22 sqrt 16 pnorm 135 145 22 sqrt 16 1 0 9309637 Find Pr 135 Y 155 when n 36 pnorm 155 145 22 sqrt 36 pnorm 135 145 22 sqrt 36 1 0 993614 Statistics 371 Fall 2004 9
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