Chapter 7 Sampling and Sampling Distributions

Chapter 7Sampling and Sampling DistributionsCopyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/IrwinSampling and Sampling Distributions7.1 Random Sampling7.2 The Sampling Distribution of the Sample Mean7.3 The Sampling Distribution of the Sample Proportion7.4 Stratified Random, Cluster, and Systematic Sampling (Optional)7.5 More about Surveys and Errors in Survey Sampling (Optional)7.6 Deviation of the Mean and Variance of the Sample Mean (Optional)7-*7.1 Random SamplingIf we select n elements from a population in such a way that every set of n elements in the population has the same chance of being selected, the then elements we select are said to be a random sampleIn order to select a random sample of n elements from a population, we make n random selections from the populationOn each random selection, we give every element remaining in the population for that selection the same chance of being chosenLO7-1: Explain the concept of random sampling and select a random sample.7-*With or Without ReplacementWe can sample with or without replacementWith replacement, we place the element chosen on any particular selection back into the populationWe give this element a chance to be chosen on any succeeding selectionWithout replacement, we do not place the element chosen on a particular selection back into the populationWe do not give this element a chance to be chosen on any succeeding selectionIt is best to sample without replacementLO7-17-*7.2 Sampling Distribution of the Sample MeanThe sampling distribution of the sample mean x is the probability distribution of the population of the sample means obtainable from all possible samples of size n from a population of size NLO7-2: Describe and use the sampling distribution of the sample mean.7-*General ConclusionsIf the population of individual items is normal, then the population of all sample means is also normalEven if the population of individual items is not normal, there are circumstances when the population of all sample means is normal (Central Limit Theorem)LO7-27-*Central Limit TheoremNow consider a non-normal populationStill have: mx=m and sx=s/nExactly correct if infinite population Approximately correct if population size N finite but much larger than sample size nBut if population is non-normal, what is the shape of the sampling distribution of the sample mean?The sampling distribution is approximately normal if the sample is large enough, even if the population is non-normal (Central Limit Theorem)LO7-3: Explain and use the Central Limit Theorem.7-*The Central Limit Theorem ContinuedNo matter the probability distribution that describes the population, if the sample size n is large enough, the population of all possible sample means is approximately normal with mean mx=m and standard deviation sx=s/nFurther, the larger the sample size n, the closer the sampling distribution of the sample mean is to being normalIn other words, the larger n, the better the approximationLO7-37-*7.3 Sampling Distribution of the Sample ProportionThe probability distribution of all possible sample proportions is the sampling distribution of the sample proportionIf a random sample of size n is taken from a population p-hat, then the sampling distribution of the sample proportion isApproximately normal, if n is largeHas a mean that equals ρHas standard deviation Where ρ is the population proportion and p̂ is the sampled proportionLO7-4: Describe and use the sampling distribution of the sample proportion.7-*7.4 Stratified Random, Cluster, and Systematic Sampling (Optional)Divide the population into non-overlapping groups (strata) of similar unitsSelect a random sample from each stratumCombine the random samples to make full sampleAppropriate when the population consists of two or more different groups so that:The groups differ from each other with respect to the variable of interestUnits within a group are similar to each otherDivide population into strata by age, gender, incomeLO7-5: Describe the basic ideas of stratified random, cluster, and systematic sampling (optional).7-*7.5 More about Surveys and Errors in Survey Sampling (Optional)Dichotomous questions ask for a yes/no responseMultiple choice questions give the respondent a list of of choices to select fromOpen-ended questions allow the respondent to answer in their own wordsLO7-6: Describe basic types of survey questions, survey procedures, and sources of error.7-*7.6 Derivation of the Mean and the Variance of the Sample Mean (Optional)Given the following:The mean of xi, denoted μxi, is μ is the mean of the population from which xi will be randomly selectedThat is μx1 = μx2 = = μxn = μThe variance of xi, denoted σ2xi, is σ2 the variance of population from which xi will be randomly selectedThat is σ2x1 = σ2x2 = = σ2xn = σ2Sample mean  is the average of the xi’sWe can prove that μ = μ7-*