Business statistics, 6e, 2005, groebner CH06

Business Statistics: A Decision-Making Approach 6th Edition Chapter Introduction to Sampling Distributions Business Statistics: A Decision-Making Approach, 6e © 2005 PrenticeHall, Inc Chap 6-1 Chapter Goals After completing this chapter, you should be able to:  Define the concept of sampling error  Determine the mean and standard deviation _ for the sampling distribution of the sample mean, x  Determine the mean and standard deviation for_the sampling distribution of the sample proportion, p Describe the Central Limit Theorem and its importance Business Statistics: A Decision_ _  Apply sampling distributions for both x and p Making Approach, 6e © 2005 Prentice-Hall, Inc Chap 6-2  Sampling Error Sample Statistics are used to estimate Population Parameters  ex: X is an estimate of the population mean, μ  Problems:  Different samples provide different estimates of the population parameter Sample results have potential variability, thus sampling error exits Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 6-3  Calculating Sampling Error  Sampling Error: The difference between a value (a statistic) computed from a sample and the corresponding value (a parameter) computed from a population Example: (for the mean) Sampling Error  x - μ where: x sample mean Business Statistics: A Decisionμ population mean Making Approach, 6e © 2005 Prentice-Hall, Inc Chap 6-4 Review  Population mean: x  μ i N Sample Mean: x  x i n where: μ = Population mean x = sample mean xi = Values in the population or sample Business Statistics: ANDecision= Population size Making Approach, 6en© 2005 size = sample Prentice-Hall, Inc Chap 6-5 Example If the population mean is μ = 98.6 degrees and a sample of n = temperatures yields a sample mean of x = 99.2 degrees, then the sampling error is x  μ 98.6  99.2  0.6 degrees Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 6-6 Sampling Errors  Different samples will yield different sampling errors  The sampling error may be positive or negative ( x may be greater than or less than μ)  The expected sampling error decreases as the sample size increases Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 6-7 Sampling Distribution  A sampling distribution is a distribution of the possible values of a statistic for a given size sample selected from a population Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 6-8 Developing a Sampling Distribution  Assume there is a population …  Population size N=4  Random variable, x, is age of individuals  Values of x: 18, 20, 22, 24 (years) Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc A B C D Chap 6-9 Developing a Sampling Distribution (continued ) Summary Measures for the Population Distribution: x  μ P(x) i N 18  20  22  24  21 σ  (x  μ) i N 2 2.236 Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc 18 20 22 24 A B C D Uniform Distribution Chap 6-10 x Sampling Distribution Properties (continued )  For sampling with replacement: As n increases, Larger sample size σ x decreases Smaller sample size Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc μ Chap 6-19 x If the Population is not Normal  We can apply the Central Limit Theorem:    Even if the population is not normal, …sample means from the population will be approximately normal as long as the sample size is large enough …and the sampling distribution will have μ x μ and Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc σ σx  n Chap 6-20 Central Limit Theorem As the sample size gets large enough… n↑ Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc the sampling distribution becomes almost normal regardless of shape of population Chap 6-21 x If the Population is not Normal (continued ) Population Distribution Sampling distribution properties: Central Tendency μ x μ Variation σ σx  n x μ Sampling Distribution (becomes normal as n increases) Smaller sample size Business(Sampling Statistics:with A Decisionreplacement) Making Approach, 6e © 2005 Prentice-Hall, Inc Larger sample size μx Chap 6-22 x How Large is Large Enough?  For most distributions, n > 30 will give a sampling distribution that is nearly normal  For fairly symmetric distributions, n > 15  For normal population distributions, the sampling distribution of the mean is always normally distributed Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 6-23 Example  Suppose a population has mean μ = and standard deviation σ = Suppose a random sample of size n = 36 is selected  What is the probability that the sample mean is between 7.8 and 8.2? Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 6-24 Example (continued ) Solution:  Even if the population is not normally distributed, the central limit theorem can be used (n > 30)  … so the sampling distribution of approximately normal   … with mean x is = μx …and standard deviation Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc σ σx   0.5 n 36 Chap 6-25 Example (continued ) Solution (continued):   μ μ  7.8 - 8.2 -  x P(7.8  μ x  8.2)  P    σ   36 n 36    P(-0.4  z  0.4)  0.3108 Population Distribution ??? ? ?? ? ? ? ? ? Sampling Distribution Standard Normal Distribution Sample ? A DecisionBusiness Statistics: Making Approach, 6e x © 2005 7.8 μ 8 Prentice-Hall, Inc .1554 +.1554 Standardize μx 8 8.2 x -0.4 μz 0 0.4 Chap 6-26 z Population Proportions, p p = the proportion of population having some characteristic  Sample proportion ( p ) provides an estimate of p: x number of successes in the sample p  n sample size  If two outcomes, p has a binomial distribution Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 6-27 Sampling Distribution of p  Approximated by a normal distribution if:  P( p ) np 5 n(1  p) 5 Sampling Distribution p where μ p p and p(1  p) σp  n Business Statistics: A DecisionMaking Approach, 6e ©(where 2005 p = population proportion) Prentice-Hall, Inc Chap 6-28 z-Value for Proportions Standardize p to a z value with the formula: p p p p z  σp p(1  p) n If sampling is without replacement and n is greater than 5% of the population size, then σ must use p the finite population correction Business Statistics: A Decisionfactor: Making Approach, 6e © 2005  Prentice-Hall, Inc p(1  p) N  n σp  n N Chap 6-29 Example  If the true proportion of voters who support Proposition A is p = 4, what is the probability that a sample of size 200 yields a sample proportion between 40 and 45?  i.e.: if p = and n = 200, what is P(.40 ≤ p ≤ 45) ? Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 6-30 Example  Find σ p : if p = and n = 200, what is P(.40 ≤ p ≤ 45) ? (continued ) p(1  p) 4(1  4) σp   .03464 n 200 Convert to 45  40   40  40 P(.40 p .45)  P z   standard 03464   03464 normal: Business Statistics: A Decision P(0 z 1.44) Making Approach, 6e © 2005 Prentice-Hall, Inc Chap 6-31 Example  (continued ) if p = and n = 200, what is P(.40 ≤ p ≤ 45) ? Use standard normal table: P(0 ≤ z ≤ 1.44) = 4251 Standardized Normal Distribution Sampling Distribution 4251 Standardize Business Statistics: 40 A Decision.45 p Making Approach, 6e © 2005 Prentice-Hall, Inc 1.44 z Chap 6-32 Chapter Summary    Discussed sampling error Introduced sampling distributions Described the sampling distribution of the mean   For normal populations Using the Central Limit Theorem Described the sampling distribution of a proportion  Calculated probabilities using sampling distributions Business Statistics: A Decision Discussed sampling from finite populations Making Approach, 6e © 2005  Prentice-Hall, Inc Chap 6-33 ... have μ x μ and Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc σ σx  n Chap 6-20 Central Limit Theorem As the sample size gets large enough… n↑ Business Statistics:... of the population parameter Sample results have potential variability, thus sampling error exits Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 6-3  Calculating... computed from a population Example: (for the mean) Sampling Error  x - μ where: x sample mean Business Statistics: A Decisionμ population mean Making Approach, 6e © 2005 Prentice-Hall, Inc