Presentation on theme: "Chapter 7 Sampling and Sampling Distributions"— Presentation transcript:
1 Chapter 7 Sampling and Sampling Distributions EF 507QUANTITATIVE METHODS FOR ECONOMICS AND FINANCEFALL 2008Chapter 7Sampling andSampling Distributions
2 Tools of Business Statistics Descriptive statisticsCollecting, presenting, and describing dataInferential statisticsDrawing conclusions and/or making decisions concerning a population based only on sample data
3 Populations and Samples A Population is the set of all items or individuals of interestExamples: All likely voters in the next electionAll parts produced todayAll sales receipts for NovemberA Sample is a subset of the populationExamples: 1000 voters selected at random for interviewA few parts selected for destructive testingRandom receipts selected for audit
4 Population vs. Sample Population Sample a b c d b c e f g h i j k l m no p q r s t u v wx y zb cg i no r uy
5 Why Sample? Less time consuming than a census Less costly to administer than a censusIt is possible to obtain statistical results of a sufficiently high precision based on samples.
6 Simple Random SamplesEvery object in the population has an equal chance of being selectedObjects are selected independentlySamples can be obtained from a table of random numbers or computer random number generatorsA simple random sample is the ideal against which other sample methods are compared
7 Inferential Statistics Making statements about a population by examining sample resultsSample statistics Population parameters(known) Inference (unknown, but canbe estimated fromsample evidence)
8 Inferential Statistics Drawing conclusions and/or making decisions concerning a population based on sample results.Estimatione.g., Estimate the population mean weight using the sample mean weightHypothesis Testinge.g., Use sample evidence to test the claim that the population mean weight is 75 kg.
9 Sampling Distributions A sampling distribution is a distribution of all of the possible values of a statistic for a given size sample selected from a population
10 Chapter Outline Sampling Distributions Sampling Distribution of Sample MeanSampling Distribution of Sample ProportionSampling Distribution of Sample Variance
11 Sampling Distributions of Sample Means Sampling Distribution of SampleMeanSampling Distribution of Sample ProportionSampling Distribution of Sample Variance
12 Developing a Sampling Distribution Assume there is a population …Population size N=4Random variable, X, is age of individualsValues of X:18, 20, 22, 24 (years)DACB
13 Developing a Sampling Distribution (continued)Summary Measures for the Population Distribution:P(x)0.25xA B C DUniform Distribution
14 Now consider all possible samples of size n = 2 Developing a Sampling Distribution(continued)Now consider all possible samples of size n = 216 Sample Means16 possible samples (sampling with replacement)
15 Sampling Distribution of All Sample Means Developing aSampling Distribution(continued)Sampling Distribution of All Sample MeansSample Means Distribution16 Sample Means_P(X)0.30.20.1_X(no longer uniform!!!)
16 Summary Measures of this Sampling Distribution: Developing aSampling Distribution(continued)Summary Measures of this Sampling Distribution:
17 Comparing the Population with its Sampling Distribution Sample Means Distributionn = 2_P(X)P(X)0.30.30.20.20.10.1_XA B C DX
18 Expected Value of Sample Mean Let X1, X2, Xn represent a random sample from a populationThe sample mean value of these observations is defined as
19 Standard Error of the Mean Different samples of the same size from the same population will yield different sample meansA measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean:Note that the standard error of the mean decreases as the sample size increases
20 If the Population is Normal If a population is normal with mean μ and standard deviation σ, the sampling distribution of is also normally distributed withand
21 Z-value for Sampling Distribution of the Mean Z-value for the sampling distribution of :where: = sample mean= population mean= population standard deviationn = sample size
22 Finite Population Correction Apply the Finite Population Correction if:a population member cannot be included more than once in a sample (sampling is without replacement), andthe sample is large relative to the population(n is greater than about 5% of N)Thenor
23 Finite Population Correction If the sample size n is not small compared to the population size N , then use
24 Sampling Distribution Properties Normal Population Distribution(i.e is unbiased )Normal Sampling Distribution(has the same mean)
25 Sampling Distribution Properties (continued)For sampling with replacement:As n increases,decreasesLarger sample sizeSmaller sample size
26 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.Properties of the sampling distribution:and
27 Central Limit Theoremthe sampling distribution becomes almost normal regardless of shape of populationAs the sample size gets large enough…n↑
28 If the Population is not Normal (continued)Population DistributionSampling distribution properties:Central TendencySampling Distribution(becomes normal as n increases)VariationLarger sample sizeSmaller sample size
29 How Large is Large Enough? For most distributions, n > 25 will give a sampling distribution that is nearly normalFor normal population distributions, the sampling distribution of the mean is always normally distributed
30 ExampleSuppose a population has mean μ = 8 and standard deviation σ = 3. 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?
31 Example(continued)Solution:Even if the population is not normally distributed, the central limit theorem can be used (n > 25)… so the sampling distribution of is approximately normal… with mean = 8…and standard deviation
32 Example Solution (continued): (continued) Z X Population Distribution Sampling DistributionStandard Normal Distribution??????????SampleStandardize??ZX
33 Acceptance IntervalsGoal: determine a range within which sample means are likely to occur, given a population mean and varianceBy the Central Limit Theorem, we know that the distribution of X is approximately normal if n is large enough, with mean μ and standard deviationLet zα/2 be the z-value that leaves area α/2 in the upper tail of the normal distribution (i.e., the interval - zα/2 to zα/2 encloses probability 1 – α)Thenis the interval that includes X with probability 1 – α
34 Sampling Distributions of Sample Proportions Sampling Distribution of SampleMeanSampling Distribution of Sample ProportionSampling Distribution of Sample Variance
35 Population Proportions, P P = the proportion of the population havingsome characteristicSample proportion ( ) provides an estimateof P:0 ≤ ≤ 1has a binomial distribution, but can be approximated by a normal distribution when nP(1 – P) > 9
36 Sampling Distribution of P ^Sampling Distribution of PNormal approximation:Properties:andSampling Distribution.3.2.1(where P = population proportion)
37 Z-Value for Proportions Standardize to a Z value with the formula:
38 ExampleIf the true proportion of voters who support Proposition A is P = 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 P = 0.4 and n = 200, what isP(0.40 ≤ ≤ 0.45) ?
39 Example if P = 0.4 and n = 200, what is P(0.40 ≤ ≤ .45) ? Find : (continued)if P = 0.4 and n = 200, what isP(0.40 ≤ ≤ .45) ?Find :Convert to standard normal:
40 Standardized Normal Distribution Example(continued)if p = 0.4 and n = 200, what isP(0.40 ≤ ≤ 0.45) ?Use standard normal table: P(0 ≤ Z ≤ 1.44) =Standardized Normal DistributionSampling Distribution0.4251Standardize0.400.451.44Z
41 Sampling Distributions of Sample Proportions Sampling Distribution of SampleMeanSampling Distribution of Sample ProportionSampling Distribution of Sample Variance
42 Sample VarianceLet x1, x2, , xn be a random sample from a population. The sample variance isthe square root of the sample variance is called the sample standard deviationthe sample variance is different for different random samples from the same population
43 Sampling Distribution of Sample Variances The sampling distribution of s2 has mean σ2If the population distribution is normal, thenIf the population distribution is normal thenhas a 2 distribution with n – 1 degrees of freedom
44 The Chi-square Distribution The chi-square distribution is a family of distributions, depending on degrees of freedom:d.f. = n – 1Text Table 7 (p.869) contains chi-square probabilities222d.o.f. = 1d.o.f. = 5d.o.f. = 15
45 Degrees of Freedom (dof) Idea: Number of observations that are free to varyafter sample mean has been calculatedExample: Suppose the mean of 3 numbers is 8.0Let X1 = 7Let X2 = 8What is X3?If the mean of these three values is 8.0,then X3 must be 9(i.e., X3 is not free to vary)Here, n = 3, so degrees of freedom = n – 1 = 3 – 1 = 2(2 values can be any numbers, but the third is not free to vary for a given mean)
46 Chi-square ExampleA commercial freezer must hold a selected temperature with little variation. Specifications call for a standard deviation of no more than 4 degrees (a variance of 16 degrees2).A sample of 14 freezers is to be testedWhat is the upper limit (K) for the sample variance such that the probability of exceeding this limit, given that the population standard deviation is 4, is less than 0.05?
47 Finding the Chi-square Value Is chi-square distributed with (n – 1) = 13 degrees of freedomUse the the chi-square distribution with area 0.05 in the upper tail:213= (α = and 14 – 1 = 13 d.f.)probability α = .052213= 22.36
48 Chi-square Example 213 = 22.36 (α =0.05 and 14 – 1 = 13 d.f.) So: (continued)213= (α =0.05 and 14 – 1 = 13 d.f.)So:or(where n = 14)soIf s2 from the sample of size n = 14 is greater than 27.52, there is strong evidence to suggest the population variance exceeds 16.