Introduction To Statistics Sampling & Sampling Distributions: Basics
Sampling & Sampling Distributions
Basics
Peter Wludyka / samp1
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Goal of Sampling • to gain knowledge about some process, phenomenon or population • types of knowledge – knowledge about some (population) parameter such as • the mean • the percent of a population having a characteristic
– knowledge of the distribution of itms in a population so that one can make statements such as • diamters of XYZ Ball Bearings are normally distributed • the lifetimes of batteries are exponentially distributed Peter Wludyka / samp1
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Random Sampling • Each item in the population is equally likely to be chosen • this means that – each item being sampled is from the same population – the measurements are independent
Peter Wludyka / samp1
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Peter Wludyka / samp1.ppt
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Introduction To Statistics Sampling & Sampling Distributions: Basics
Theory for Sampling Distribution of the Mean • For random samples of size n – the average value of the sample mean is the population mean; that is E ( X ) = µ = E( X ) = µ X
– the standard deviation of the sample mean is the population standard deviation divided by the square root of the sample size. S D( X ) = σ X =
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σ = the standard error o f the mean n
These statements are true for any poluation; no matter what shape it may have. Peter Wludyka / samp1
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Samples From Normal Populations • When sampling from a Normal population the sample mean is itself Normally distributed NORMAL POPULATION
NORMAL SAMPLE MEAN
E ( X ) = µX = E ( X ) = µ • with mean • and standard deviation SD ( X ) = σ = X
σ = the standard error o f the mean n Peter Wludyka / samp1
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Central Limit Theorem • For large samples the sample mean is approximately normally distributed. – “large” means around 30 but often 5 or 6 is enough. – the more the population “resembles” the normal the smaller sample size that is needed to be able to rely on the normality of the sample mean.
Peter Wludyka / samp1
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Peter Wludyka / samp1.ppt
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Introduction To Statistics Sampling & Sampling Distributions: Basics
What do we know, then? • We know that in a lot of cases we can think of the sample mean as being normally distributed. • We know the mean and standard deviation of the distribution of sample means.
Peter Wludyka / samp1
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Practical Consequences • • • • •
I want knowledge about a certain power of daily wear contact lenses. In particular, I want to know about the center thickness. How do I gain knowledge? A sample of 100 lenses has mean 14.2mm and standard deviation 0.52 mm. What does this tell me about the center thickness process? The standard error (based on sample) is
s . = SEM = SD ( X ) = σ = σ ≈ = 052 0.052 X
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n n normally Since the sample mean is approximately 100 distributed we know that 95% of the time the sample mean will be within two standard errors of the population mean. Therefore the population mean is likely to be in the interval s X ± 2SEM or X ± 2 n
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14.2 ± 2( 0.052) or 14.2 ± 0.104 or (14.096, 14.304)
Roll of the Sample Size in Estimation of the Mean Sample Size & the Standard Error of the Mean +/- 2 SEM n SEM LCL UCL mean 2 0.368 13.465 14.935 14.2 4 0.260 13.680 14.720 Sdev 9 0.173 13.853 14.547 0.52 16 0.130 13.940 14.460 20 0.116 13.967 14.433 25 0.104 13.992 14.408 50 0.074 14.053 14.347 100 0.052 14.096 14.304 200 0.037 14.126 14.274 500 0.023 14.153 14.247 1000 0.016 14.167 14.233 Peter Wludyka / samp1
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What happens to the SEM as the sample size increases? What happens to the interval estimate for the process mean? How large a sample is required to make a good guess for the process mean? Using your sample size, is the process on target (14.0 mm average)? 9
Peter Wludyka / samp1.ppt
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Introduction To Statistics Sampling & Sampling Distributions: Basics
Roll of the Sample Size in Estimation of the Mean Sample Size & the Standard Error of the Mean +/- 2 SEM n SEM LCL UCL mean 2 0.141 13.917 14.483 14.2 4 0.100 14.000 14.400 Sdev 9 0.067 14.067 14.333 0.2 16 0.050 14.100 14.300 20 0.045 14.111 14.289 25 0.040 14.120 14.280 50 0.028 14.143 14.257 100 0.020 14.160 14.240 200 0.014 14.172 14.228 500 0.009 14.182 14.218 1000 0.006 14.187 14.213
point????
Peter Wludyka / samp1
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What happens to the SEM as the sample size increases? What happens to the interval estimate for the process mean? How large a sample is required to make a good guess for the process mean? Using your sample size, is the process on target (14.0 mm average)? 10
Simulating Random Samples using EXCEL • GoTo EXCEL(sBREAD1.xls … sheet descrp)
– weight of one pound loaves of bread. • GoTo EXCEL(sBREAD1.xls … sheet CLT)
– the Central Limit Theorem • GoTo EXCEL(SimSam1.xls …) – sheet NORMAL1 for samples from normal ; observe fluctuations is summary stats.
Peter Wludyka / samp1
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Sampling Distribution of a Proportion (%) • What percentage of the contact lenses in a Lot (#AF34) are nonconforming? – Select a random sample of 400 lenses; 40 are nonconforming. What conclusions can you draw? – If another sample of 400 is selected will there be exactly 40 nonconforming? By how much will the result differ?
• What percentage of packages are being mislabeled? – Select a random sample of 100 lenses; 20 are mislabled. What conclusions can you draw?
• In each case we want to draw conclusions about a population (process) based on a sample. In each case, what do you want to determine. Peter Wludyka / samp1
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Peter Wludyka / samp1.ppt
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Introduction To Statistics Sampling & Sampling Distributions: Basics
Sampling Distribution of a Proportion (%) •
What percentage of the contact lenses in a Lot (#AF34) are nonconforming?
– Model: Let X = number nonconforning in sample; n = sample size.
X 40 = = 040 . n 400
Sample Proportion = What % of lot is nonconforming??
Peter Wludyka / samp1
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Sampling Distribution of a Proportion (%) What percentage of packages are being mislabeled?
– Model: Let X = number mislabeled in sample; n = sample size.
Sample Proportion =
X 20 = = 0.20 n 100
What % of all packages are mislabeled? Peter Wludyka / samp1
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Theory for the Sampling Distribution of a Proportion (%) • Theory The sample proportion p$ =
X number of successes has = n sample size
mean = p p( 1-p) . n For large samples (b y the Central Limit Theorem) the statsitic p$ has an approximately normal distribution (with the above mean an d SD). standard d eviation =
• Practice – z-interval used to estimate p – z-test performedPetertoWludyka test/ statinf4 p
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Peter Wludyka / samp1.ppt
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Introduction To Statistics Sampling & Sampling Distributions: Basics
Using EXCEL to Simulate Samples from the Binomial • The Binomial Model decsribes the situation in which we are sampling from a population and calculating the proportion with a charateristic p. • GoTo EXCEL(SimSam2.xls … – sheet p-hat obersve variation in the sample proportion from repeated sampling – sheet for PhatSam1 to observe the sampling distribution of the sample proportion.
