Presentation on theme: "Chapter 7 Introduction to Sampling Distributions"— Presentation transcript:
1 Chapter 7 Introduction to Sampling Distributions Business Statistics:A Decision-Making Approach8th EditionChapter 7 Introduction to Sampling Distributions
2 Chapter Goals After completing this chapter, you should be able to: Define the concept of sampling errorDetermine the mean and standard deviation for the sampling distribution of the sample mean, xDetermine the mean and standard deviation for the sampling distribution of the sample proportion, pDescribe the Central Limit Theorem and its importanceApply sampling distributions for both x and p____
3 Sampling Error Population Parameters Sample Statistics are used to estimatePopulation Parametersex: X is an estimate of the population mean, μProblems:Different samples provide different estimates of the population parameterSample results have potential variability, thus sampling error exits
4 Calculating Sampling Error The difference between a value (a statistic) computed from a sample and the corresponding value (a parameter) computed from a populationExample: (for the mean)where:Always present just because you sample!
5 ExampleIf the population mean is μ = 98.6 degrees and a sample of n = 5 temperatures yields a sample mean of = degrees, then the sampling error is
6 Sampling Errors The sampling error may be positive or negative ( may be greater than or less than μ)The size of the error depends on the sample selectedi.e., a larger sample does not necessarily produce a smaller error if it is not a representative sampleDownload “Sampling Distributions using Excel”First sheet: “Sampling Error”
7 Sampling Distribution A sampling distribution is a distribution of all possible values of a statistic for a given sample size that has been randomly selected from a population.Download “Sampling Distributions using Excel”Second sheet: “Sampling Dist”
8 Sampling Distribution Properties For any population,The average value of all possible sample means computed from all possible random samples of a given size from the population is equal to the population mean (call “Unbiased Estimator”):See “Sampling Distributions using Excel”Considered an “unbiased” estimatorTheorem 1
9 Sampling Distribution Properties (continued)The standard deviation of the possible sample means computed from all random samples of size n is equal to the population standard deviation divided by the square root of the sample size (call Standard Error):Try using “Sampling Distributions using Excel” Not Equal!Also called the standard errorTheorem 2
10 Finite Population Correction Virtually all survey research, sampling is conducted without replacement from populations that are of a finite size N.In these cases, particularly when the sample size n is not small in comparison with the population size N (i.e., more than 5% of the population is sampled)so that n/N > 0.05, a finite population correction factor (fpc) is used to define both the standard error of the mean and the standard error of the proportion.
12 Finite Population Correction Apply the Finite Population Correction (fpc) if:The sample size is greater than 5% of population size.Only with sampling without replacementThenSee fpc by “Sampling Distributions using Excel”
13 Sampling Distribution Properties (continued)As the sample size is increased, the StdDev of the sampling distribution is reduced…..That is, the potential for extreme sampling error is reduced when larger sample size are used.Graphical illustration: next slide
14 Sampling Distribution Properties (continued)The value of becomes closer to μ as n increases):PopulationxSmall sample sizeAs n increases,decreasesLarger sample size
15 If the Population is Normal If a population is normal with mean μ and standard deviation σ, the sampling distributionof is also normally distributed withandTheorem 3As n increases the data behaves more like a normal distribution
16 z-value for Sampling Distribution of x z-value for the sampling distribution of :where: = sample mean= population mean= population standard deviationn = sample size
17 ExampleSuppose that a population is known to be normally distributed with μ = 2,000 and σ = 230 and random sample of size n = 8 is selected.Because the population is normally distributed, the sampling distribution for the mean will also be normally distributed.What is the probability that the sample mean will exceed 2,100?Convert to Z valueUsing Excel: 1-NORMSDIST(1.23)Answer on the website
18 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 haveandTheorem 4
19 If necessary, watch the Video Central Limit TheoremIf necessary, watch the Videothe sampling distribution becomes almost normal regardless of shape of populationAs the sample size gets large enough…n↑
20 How Large is Large Enough? For most distributions, n > 30 will give a sampling distribution that is nearly normalFor fairly symmetric distributions, n > 15 is sufficientFor normal population distributions, the sampling distribution of the mean is always normally distributed
21 ExampleSuppose a population (not normally distributed) has mean μ = 8 and standard deviation σ = 3 and random sample of size n = 36 (greater than 30) is selected.What is the probability that the sample mean is between 7.8 and 8.2?
22 Example Solution (continued) -- find z-scores: x (continued) z Population DistributionSampling DistributionStandard Normal Distribution??????????SampleStandardize??xz
23 Sampling Distribution of a Proportion Try this by yourself!In many instances, the objective of sample is to estimate a population proportion.An accountant may be interested in determining the proportion of accounts payable balances that are correct.A production supervisor may wish to determine the percentage of product that is defect free.A marketing research department might want to know the proportion of potential customers who will purchase a prticular product.
24 Population Proportions Example If the true proportion of voters who support Proposition A is π (population proportion) = 0.4, what is the probability that a sample of size 200 yields a sample proportion between 0.40 and 0.45?i.e.: if π = 0.4 and n = 200, what isP(0.40 ≤ p ≤ 0.45) ?
25 Example if π = .4 and n = 200, what is P(0.40 ≤ p ≤ 0.45) ? Find : (continued)if π = .4 and n = 200, what isP(0.40 ≤ p ≤ 0.45) ?Find :Convert to standard normal (z-values):
26 Standardized Normal Distribution Example(continued)if π = 0.4 and n = 200, what isP(0.40 ≤ p ≤ 0.45) ?Use standard normal table: P(0 ≤ z ≤ 1.44) =Standardized Normal DistributionSampling Distribution0.4251Standardize0.400.451.44pz
27 Population Proportions, π π = the proportion of the population havingsome characteristicSample proportion ( p ) provides an estimateof π :If two outcomes, p is a binomial distribution
28 Sampling Distribution of p Approximated by a normal distribution if:whereandSampling DistributionP( p ).3.2.1pTheorem 5(where π = population proportion)
29 z-Value for Proportions Standardize p to a z value with the formula:If sampling is without replacement and n is greater than 5% of the population size, then must use the finite population correction factor:
30 Using the Sample Distribution for Proportions Determine the population proportion, pCalculate the sample proportion, pDerive the mean and standard deviation of the sampling distributionDefine the event of interestIf np and n(1-p) are both > 5, then covert p to z-valueUse standard normal table (Appendix D) to determine the probability
31 Chapter Summary Discussed sampling error Introduced sampling distributionsDescribed the sampling distribution of the meanFor normal populationsUsing the Central Limit Theorem (normality unknown)Described the sampling distribution of a proportionCalculated probabilities using sampling distributionsDiscussed sampling from finite populations
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