GRAPHICAL METHODS FOR QUANTITATIVE DATA

Chapter 4Simple Random SamplingDefinition of Simple Random Sample (SRS) and how to select a SRSEstimation of population mean and total; sample size for estimating population mean and totalEstimation of population proportion; sample size for estimating population proportionComparing estimates

Simple Random SamplesDesire the sample to be representative of the population from which the sample is selected

Each individual in the population should have an equal chance to be selected

Is this good enough?ExampleSelect a sample of high school students as follows:Flip a fair coinIf heads, select all female students in the school as the sampleIf tails, select all male students in the school as the sampleEach student has an equal chance to be in the sampleEvery sample a single gender, not representativeEach individual in the population has an equal chance to be selected. Is this good enough?NO!!

Simple Random SampleA simple random sample (SRS) of size n consists of n units from the population chosen in such a way that every set of n units has an equal chance to be the sample actually selected. Simple Random Samples (cont.)Suppose a large History class of 500 students has 250 male and 250 female students.To select a random sample of 250 students from the class, I flip a fair coin one time.If the coin shows heads, I select the 250 males as my sample; if the coin shows tails I select the 250 females as my sample.What is the chance any individual student from the class is included in the sample?This is a random sample. Is it a simple random sample?1/2NO!Not every possible group of 250 students has an equal chance to be selected.Every sample consists of only 1 gender hardly representative.Simple Random Samples (cont.)The easiest way to choose an SRS is with random numbers.

Statistical software cangenerate random digits (e.g., Excel =random(),ran# button oncalculator).Example: simple random sampleAcademic dept wishes to randomly choose a 3-member committee from the 28 members of the dept00 Abbott07 Goodwin14 Pillotte21 Theobald01 Cicirelli08 Haglund15 Raman22 Vader02 Crane09 Johnson16 Reimann23 Wang03 Dunsmore10 Keegan17 Rodriguez24 Wieczoreck04 Engle11 Lechtenbg 18 Rowe25 Williams05 Fitzpatk12 Martinez19 Sommers26 Wilson06 Garcia13 Nguyen20 Stone27 ZinkSolutionUse a random number table; read 2-digit pairs until you have chosen 3 committee membersFor example, start in row 121:71487 09984 29077 14863 61683 47052 62224 51025Garcia (07) Theobald (22) Johnson (10)Your calculator generates random numbers; you can also generate random numbers using ExcelSampling VariabilitySuppose we had started in line 145?19687 12633 57857 95806 09931 02150 43163 58636Our sample would have been19 Rowe, 26 Williams, 06 FitzpatrickSampling VariabilitySamples drawn at random generally differ from one another.Each draw of random numbers selects different people for our sample.These differences lead to different values for the variables we measure.We call these sample-to-sample differences sampling variability.Variability is OK; bias is bad!!

Example: simple random sampleUsing Excel toolsUsing statcrunch (NFL)4.3 Estimation of population mean Usual estimator

4.3 Estimation of population mean For a simple random sample of size n chosen without replacement from a population of size N

The correction factor takes into account that an estimate based on a sample of n=10 from a population of N=20 items contains more information than a sample of n=10 from a population of N=20,000

4.3 Estimating the variance of the sample meanRecall the sample variance

4.3 Estimating the variance of the sample mean

4.3 Estimating the variance of the sample mean

4.3 ExamplePopulation {1, 2, 3, 4}; n = 2, equal weightsSamplePr. of samples2{1, 2}1/61.50.50.125{1, 3}1/62.02.00.500{1, 4}1/62.54.51.125{2, 3}1/62.50.50.125{2, 4}1/63.02.00.500{3, 4}1/63.50.50.125

4.3 ExamplePopulation {1, 2, 3, 4}; =2.5, 2 = 5/4; n = 2, equal weightsSamplePr. of samples2{1, 2}1/61.50.50.125{1, 3}1/62.02.00.500{1, 4}1/62.54.51.125{2, 3}1/62.50.50.125{2, 4}1/63.02.00.500{3, 4}1/63.50.50.125

4.3 ExamplePopulation {1, 2, 3, 4}; =2.5, 2 = 5/4; n = 2, equal weightsSamplePr. of samples2{1, 2}1/61.50.50.125{1, 3}1/62.02.00.500{1, 4}1/62.54.51.125{2, 3}1/62.50.50.125{2, 4}1/63.02.00.500{3, 4}1/63.50.50.125

4.3 Example SummaryPopulation {1, 2, 3, 4}; =2.5, 2 = 5/4; n = 2, equal weightsSamplePr. of samples2{1, 2}1/61.50.50.125{1, 3}1/62.02.00.500{1, 4}1/62.54.51.125{2, 3}1/62.50.50.125{2, 4}1/63.02.00.500{3, 4}1/63.50.50.125

4.3 Margin of error when estimating the population mean

t distributionsVery similar to z~N(0, 1)Sometimes called Students t distribution; Gossett, brewery employeeProperties:i)symmetric around 0 (like z)ii)degrees of freedom

02.2281Students t Distribution P(t > 2.2281) = .025-2.2281.025.025.95t10P(t < -2.2281) = .02501.96Standard normalP(z > 1.96) = .025-1.96.025.025.95zP(z < -1.96) = .025-3-2-10123Zt0123-1-2-3Students t Distribution Figure 11.3, Page 372

-3-2-10123Zt10123-1-2-3Students t Distribution Figure 11.3, Page 372Degrees of Freedom

-3-2-10123Zt10123-1-2-3t7Students t Distribution Figure 11.3, Page 372Degrees of Freedom

4.3 Margin of error when estimating the population mean

4.3 Margin of error when estimating the population mean Understanding confidence intervals; behavior of confidence intervals.4.3 Margin of error when estimating the population mean

Comparing t and z Critical ValuesConf.leveln = 30z = 1.64590%t = 1.6991z = 1.9695%t = 2.0452z = 2.3398%t = 2.4620z = 2.5899%t = 2.7564

4.4 Determining Sample Size to Estimate

Required Sample Size To Estimate a Population Mean If you desire a C% confidence interval for a population mean with an accuracy specified by you, how large does the sample size need to be?We will denote the accuracy by MOE, which stands for Margin of Error.Example: Sample Size to Estimate a Population Mean Suppose we want to estimate the unknown mean height of male students at NC State with a confidence interval.We want to be 95% confident that our estimate is within .5 inch of How large does our sample size need to be?Confidence Interval for

Good news: we have an equationBad news:Need to know sWe dont know n so we dont know the degrees of freedom to find t*n-1

A Way Around this Problem: Use the Standard Normal

.95Confidence levelSampling distribution of y

Estimating sPreviously collected data or prior knowledge of the populationIf the population is normal or near-normal, then s can be conservatively estimated bys range699.7% of obs. Within 3 of the meanExample: sample size to estimate mean height of NCSU undergrad. male studentsWe want to be 95% confident that we are within .5 inch of , so MOE = .5; z*=1.96Suppose previous data indicates that s is about 2 inches.n= [(1.96)(2)/(.5)]2 = 61.47We should sample 62 male students

Example: Sample Size to Estimate a Population Mean -TextbooksSuppose the financial aid office wants to estimate the mean NCSU semester textbook cost within MOE=$25 with 98% confidence. How many students should be sampled? Previous data shows is about $85.

Example: Sample Size to Estimate a Population Mean -NFL footballsThe manufacturer of NFL footballs uses a machine to inflate new footballsThe mean inflation pressure is 13.0 psi, but random factors cause the final inflation pressure of individual footballs to vary from 12.8 psi to 13.2 psi After throwing several interceptions in a game, Tom Brady complains that the balls are not properly inflated.The manufacturer wishes to estimate the mean inflation pressure to within .025 psi with a 99% confidence interval. How many footballs should be sampled?

Example: Sample Size to Estimate a Population Mean The manufacturer wishes to estimate the mean inflation pressure to within .025 pound with a 99% confidence interval. How may footballs should be sampled?99% confidence z* = 2.58; ME = .025 = ? Inflation pressures range from 12.8 to 13.2 psiSo range =13.2 12.8 = .4; range/6 = .4/6 = .067

12348. . .

Required Sample Size To Estimate a Population Mean It is frequently the case that we are sampling without replacement.

Required Sample Size To Estimate a Population Mean When Sampling Without Replacement.

Required Sample Size To Estimate a Population Mean When Sampling Without Replacement.

Required Sample Size To Estimate a Population Mean When Sampling Without Replacement.

4.3 Estimation of population total

4.3 Estimation of population total

Required Sample Size To Estimate a Population Total

4.3 Estimation of population total Estimate number of lakes in Minnesota, the Land of 10,000 Lakes.WORKSHEET 6Use Minnesota lakes data at statcrunch (number of lakes in each of 87 counties). MOE=500, 95% confidence. Estimate s =100.n0 = .154; n = N^2*n0/(1+N*n0)=80.96Calculate ybar = mean number of lakes per county, multiply by N = 87.4.5 Estimation of population proportion pInterested in the proportion p of a population that has a characteristic of interest.Estimate p with a sample proportion.http://packpoll.com/

4.5 Estimation of population proportion p

4.5 Estimation of population proportion p

4.5 Estimation of population proportion p

Required Sample Size To Estimate a Population Proportion p When Sampling Without Replacement.

4.6 Comparing Estimates

4.6 Comparing Estimates: Comparing Means

4.6 Comparing Estimates: Comparing Means

60Population 1Population 2Parameters: 1 and 12 Parameters: 2 and 22 (values are unknown) (values are unknown)

Sample size: n1 Sample size: n2

Statistics: x1 and s12 Statistics: x2 and s22

Estimate 1 2 with x1 x2

df0

Sampling distribution model for ?

An estimate of the degrees of freedom ismin(n1 1, n2 1).

Shape?

4.6 Comparing Estimates: Comparing Means

4.6 Comparing Estimates: Comparing Means (Special Case, Seldom Used)

4.6 Comparing Estimates: Comparing Proportions, Two CasesDifference between two pollsDifference of proportions between 2 independent pollsDifferences within a single poll questionComparing proportions for a single poll question, horse-race polls (dependent proportions)

4.6 Comparing Estimates: Comparing Proportions in Two Independent Polls

4.6 Comparing Estimates: Comparing Proportions in Two Independent Polls

4.6 Comparing Estimates: Comparing Dependent Proportions in a Single PollMultinomial Sampling SituationTypically 3 or more choices in a poll

Worksheethttp://packpoll.com/

End of Chapter 4