Statistics - Sampling | Statistics

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Study GuideStatistics–Sampling1.Quiz: Sampling Distributions1. QuestionThe best description of thesampling distribution of a sample statisticis:Answer Choices• the distribution of the values of the statistic for all individuals in the sample.• the distribution of the values of the statistic for some samples, with the same size, selected from thepopulation.• the distribution of the values of the statistic for all possible samples, with the same size, selectedfrom the population.Correct Answerthe distribution of the values of the statistic for all possible samples, with the same size,selected from the population.Why This Is CorrectAsampling distributionis formed by takingall possible random samplesof the same size from apopulation and calculating the statistic (like the mean) for each sample.This shows how the statistic behaves across repeated sampling.2. QuestionSampling variability means that:Answer Choices• the value of the sample statistic will vary for the samples.• the sample statistic will vary from the population parameter.• the value of the sample statistic will vary for different sample sizes.

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Study GuideCorrect Answerthe value of the sample statistic will vary for the samples.Why This Is CorrectEven when all samples come from thesame populationand have thesame size, the statistic (like amean or proportion) will not be exactly the same each time.This natural change from sample to sample is calledsampling variability.3. QuestionWhich isnot truefor the mean of the sampling distribution?Answer Choices• It is the mean of the statistic for all of the samples in the distribution.• It is the same as the population parameter.• It depends on the sample size.Correct AnswerIt depends on the sample size.Why This Is CorrectThe mean of the sampling distribution of the sample mean is:equal to the population mean (μ)This valuedoes not changewhen the sample size changes.Whatdoeschange with sample size is thespread (standard deviation)of the sampling distribution.2.Random and Systematic Error (Bias)When we use asampleto estimate something about awhole population, our answer may not beperfect.

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Study GuideThis happens becausetwo main types of errorcan occur:1.Random error (sampling variability)2.Systematic error (bias)Let’s understand both in a simple way.1) Random Error (Sampling Variability)Random error happens naturally because samples can differ from each other.Even if we take samples correctly, the results may change a little each time.What it looks like:•Some sample results will bebelowthe true population value.•Some sample results will beabovethe true population value.•This happens just because ofchancein sampling.Example idea (from the chapter):In the previous section,ten sample meanswere taken.They were not all the same as thetrue population mean:•Some sample means were smaller•Some were largerEven theaverage of those ten sample meanswas a little lower than the true population mean.But if we tookten new samples again, the average of those new sample means might behigherthan the true population mean.Key point:Random error causes values toscatter around the true value, but not always equally each time.2) Systematic Error (Bias)Systematic error (also called bias) happens when the method of sampling is unfair.Instead of being “sometimes high and sometimes low,” it pushes results inone direction repeatedly.

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Study GuideWhat bias means:•The sample resultsconsistently overestimatethe true value,or•The sample resultsconsistently underestimatethe true valueExample of bias:Suppose your list of magazine subscribers comes from a database ofair travelers.That would create a problem because:•People withlower incomeareless likely to travel by air•Many lower-income subscribers mightnot even be includedin the list•So they won’t have a chance to be selected in the sampleAs a result, the sample would likely give an income average that istoo highcompared to the trueaverage income ofall subscribers.Key point:Bias shifts resultsaway from the true value in a consistent direction.Understanding Figure 1 (Dot Plots)Figure 1 shows how random error and bias affect sample results.Each dot plot shows where the sample results fall compared to thetrue population value.Important observation:•Thetwo dot plots on the rightshowsystematic error (bias)Their dots are not centered near the true value.•Thetwo dot plots on the leftarecentered close to the true valueThese donotshow bias.Simple rule from the figure:•Center close to true value = low bias•Center far from true value = high bias•Dots spread out a lot = high variability•Dots tightly grouped = low variability

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Study GuideFigure 1.Random (sampling) error and systematic error (bias) distort the estimation ofpopulation parameters from sample statistics.3.Central Limit TheoremWhen we take samples from a population, we often study thesampling distribution of the samplemean(written as (x̄).This helps us understand how sample means behave when we takemany samplesof the same size.3.1What if the population is normal?If the population of all magazine subscribers isnormally distributed, then thesampling distributionof the sample meanswill also benormal.This makes working with probabilities easier, because we can use thenormal curveright away.3.2What if the population is NOT normal?This is where theCentral Limit Theorem (CLT)becomes very important.

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Study GuideTheCentral Limit Theoremsays:Even if the population distribution isstrongly non-normal, the sampling distribution of the mean willbecomeapproximately normalas long as the sample size islarge enough.A common rule is:If ( n > 30 ), the sampling distribution of the mean is approximately normal.Because of this, we can still usenormal probability methods(like Z-scores and the normal curve)forlarge samples, even if the original population is not normal.3.3Mean of the Sampling DistributionThe CLT also tells us something powerful about theaverage of all sample means.The mean of the sampling distribution of sample means is:This means:•The average of all sample means equals the population mean.•So, the sample mean is anunbiased estimatorof the population mean.3.4Standard Deviation of the Sampling Distribution (Standard Error)The standard deviation of the sampling distribution of sample means is:This value is also called the:Standard Error of the MeanIt tells us how much the sample mean (\bar{x} )typically variesfrom sample to sample.

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Study GuideImportant idea:•Bigger samples → smaller standard error•That means sample means becomemore consistentwhen the sample size increases.Why does the sample size matter?As the sample size ( n ) increases:•The sample mean becomesless variable•The sampling distribution becomesnarrower•We getmore reliable averagesIn simple words:The mean of many observations is less variable than the mean of only a few observations.Standard Error (A Quick Note)Every statistic has its ownstandard error, which measures the statistic’srandom variability.For the sample mean:•The standard error tells us how “spread out” the sample means are.Example 1ProblemIf the population mean number of fish caught per trip is:•Population mean: (μ= 3.2 )•Population standard deviation: (σ= 1.8 )Find themean and standard deviation of the sampling distributionfor samples of size:•( n = 40 )

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Study GuideStep 1: Find the mean of the sampling distributionSo the mean of the sampling distribution is:Step 2: Find the standard deviation (standard error)So the standard deviation of the sampling distribution is:(σx̄= 0.285 )Final Answer for Example 1•Mean of the sampling distribution:(μx̄= 3.2 )•Standard deviation (standard error):(σx̄= 0.285)4.Quiz: Central Limit Theorem1. QuestionThe Central Limit Theorem says that the sampling distribution of the sample mean is approximatelynormal if:

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Study GuideAnswer Choices• all possible samples are selected.• the sample size is large.• the standard error of the sampling distribution is small.Correct Answerthe sample size is large.Why This Is CorrectTheCentral Limit Theorem (CLT)states that when thesample size is large, the samplingdistribution of the sample mean becomesapproximately normal, even if the original population isnot normal.2. QuestionThe Central Limit Theorem says that the mean of the sampling distribution of the sample means is:Answer Choices• equal to the population mean divided by the square root of the sample size.• close to the population mean if the sample size is large.• exactly equal to the population mean.Correct Answerexactly equal to the population mean.Why This Is CorrectThe mean of the sampling distribution of (\bar{x}) is:μₓ̄ =μSo the average of all sample means isexactly the population mean.

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Study Guide3. QuestionThe Central Limit Theorem says that the standard deviation of the sampling distribution of the samplemeans is:Answer Choices• equal to the population standard deviation divided by the square root of the sample size.• close to the population standard deviation if the sample size is large.• exactly equal to the standard deviation.Correct Answerequal to the population standard deviation divided by the square root of the sample size.Why This Is CorrectThe standard deviation of the sampling distribution of (\bar{x}) is called thestandard error:σx̄=σ / √nIt becomes smaller assample size increases.4. QuestionSamples of size25are selected from a population with mean40and standard deviation7.5.The mean of the sampling distribution of sample means is:Answer Choices• 7.5• 8• 40Correct Answer40

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