【正文】 y nonnormal.A perfect distribution would have hadall six bars exactly equal, but even with10,000 data points, there is still somedifferences in the histogram. If a betterestimate is required, a different data setcould be constructed with exactly equalcounts of each possible oute. Try itand see if the numbers are any different. Sampling a Nonnormal Distribution – Exercise
Each person in the class is to toss a single die sixteentimes and record the data. Calculate the mean and standard deviation of eachsample of sixteen Record the means and standard deviations from eachperson in the class in a Minitab worksheet
Use Minitab’s Graphical Summary routine for analysis Stat Basic Statistics Display Descriptive Statistics…Alternately, a sample of sixteen throwsof the dice can be simulated in Minitab asfollows:Select: Calc Random Data Integer… fromthe main menuGenerate 16 rows of data in C1: Min = 1, Max= 6Analyze the Sample Data
What is the mean of the sample averages? Mean ≈
What is the standard deviation of the sample averages? Sigma ≈
Is the distribution normal? What is the pvalue?
What is the relationship between the average of thesample means and the population average?
What is the relationship between the sigma of theaverages and the sigma of the individuals?The Central Limit Theorem
Formal Definition:
If random samples of n measurements are repeatedlydrawn from a population with a finite mean μμμμand a standarddeviation σ σσ σ , then, when n is large, the relative frequencyhistogram for the sample means (calculated from therepeated samples) will be approximately normal with amean μμμμand a standard deviation equal to the populationstandard deviation, σ σσ σ , divided by the square root of n.(Note: The approximation bees more precise as nincreases.)Central Limit Theorem – Exercise
From a Minitab analysis of the uniformly distributeddata:
For an exercise, verify that the Central Limit Theorem isvalid for this uniform dataVariable N Mean StDevn=1 (Individuals) 10000 n=2 (Means) 10000 n=5 (Means) 10000 n=30 (Means) 10000 相關(guān)性及簡(jiǎn)單線性回歸:Regression amp。 x) – A Few ThoughtsPg 8 ?March 01, Breakthrough Management Group. Unpublished proprietary work available only under license. All rights reserved. March 16, 2001
Make sure the process settings cover the likely productionrange (but not too far).
Too great a range
points outside the normal range mayhave too great an effect on the model.
Too small a range
Error term may dominate the fit.
Take several replicates at each input setting (x).
Replicate runs help increase the model accuracy.
Randomize runs whenever practical.
Run order is often significant factor.
The output (y) at different inputs (x抯) is not alwaysindependent of previous settings.A good spread in the data is required for agood model. Consider two examples:All of the data is collected at the normalprocess settings. In this case, regression willtry to fit a linear model to a bination ofrandom process variation and randommeasurement variation. The results will be ofno value.The second case is when most of the datais clustered around the standard settingsexcept for a couple of points at the extremeranges. In this case, the extreme pointscontrol the fit of the model. If one of theextreme points is a flyer, then the model willbe in error due to the flyer.The ideal case is for the Black Belt tocollect a range of data throughout the processspace. 置信區(qū)間:Confidence Intervals
A population is the set of all measurements of interest to the experimenter
A sample is a subset of measurements selected from the population
An inference is a statement about a population parameter based oninformation contained in a sample Two types of inference
Estimation A poll has been devised to determine the public’s reaction to anew political scandal. The purpose is to estimate the reactionof all Americans by polling a representative sample
Hypothesis testing A vaccine for Lyme disease has been developed but the rateof negative side effects is %. A new vaccine has beendeveloped and it is desired to know if the rate of negative sideeffects is lower than %.The other branch of statistics isdescriptive. Its purpose is merely todescribe a set of measurements.Inferential statistics is used to guess whatGod knows about a population from a sample.Within inferential statistics, there are twotypes: estimation and hypothesis testing.Estimation is trying to guess the populationstatistics from a sample. Hypothesis testingconcerns evaluating a sample statistic andparing it to some hypothetical value.Estimates and the CLT
What is the best estimate of the population mean using sample data? The sample mean!
How good of an estimate is the sample mean? What factors influence the accuracy of the estimate of the meanfrom sample data? Recall that:
The variation in the distribution of sample means is a function of thevariance of the Population and the sample size!n Pop X /σ σ =What About Small Samples?
If the population standard deviation is known (it almost never is) usethe previous formula for small samples, too
If the population sigma is unknown (it usually is): The estimate for standard deviation (s) is used The tdistribution is used instead of the normal (Z) distribution
Q: What is a tdistribution? The tdistribution is a family of bellshaped (normallike)distributions that are dependent on sample size The smaller the sample size n, the wider and flatter thedistributionns t X μ ns t X n n 1 , 2 / 1 , 2 / +≤≤α αThe tdistribution is the general case forany sample where the population standarddeviation is unknown. However, with largesamples, the t and zdistributions are nearlyidentical, so e