Chapter 4 Simple Random Sampling Definition of Simple Random Sample (SRS) and how to select a SRS Estimation of population mean and total; sample size for estimating population mean and total Estimation of population proportion; sample size for estimating population proportion Comparing estimates

Simple Random Samples Desire 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?

Example Select a sample of high school students as follows: Flip a fair coin If heads, select all female students in the school as the sample If tails, select all male students in the school as the sample Each student has an equal chance to be in the sample Every sample a single gender, not representative Each individual in the population has an equal chance to be selected. Is this good enough? NO!!

Simple Random Sample A 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? 1/2 NO! 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 can generate random digits (e.g., Excel “=random()”, ran# button on calculator).

Example: simple random sample Academic dept wishes to randomly choose a 3-member committee from the 28 members of the dept 00 Abbott 07 Goodwin 14 Pillotte 21 Theobald 01 Cicirelli 08 Haglund 15 Raman 22 Vader 02 Crane 09 Johnson 16 Reimann 23 Wang 03 Dunsmore 10 Keegan 17 Rodriguez 24 Wieczoreck 04 Engle 11 Lechtenb’g 18 Rowe 25 Williams 05 Fitzpat’k 12 Martinez 19 Sommers 26 Wilson 06 Garcia 13 Nguyen 20 Stone 27 Zink

Solution Use a random number table; read 2-digit pairs until you have chosen 3 committee members For example, start in row 121: 71487 09984 29077 14863 61683 47052 62224 51025 Garcia (07) Theobald (22) Johnson (10) Your calculator generates random numbers; you can also generate random numbers using Excel

Sampling Variability Suppose we had started in line 145? 19687 12633 57857 95806 09931 02150 43163 58636 Our sample would have been 19 Rowe, 26 Williams, 06 Fitzpatrick

Variability is OK; bias is bad!! Sampling Variability Samples 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 sample Using Excel tools Using 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 mean Recall the sample variance

4.3 Estimating the variance of the sample mean

4.3 Estimating the variance of the sample mean

4.3 Example Population {1, 2, 3, 4}; n = 2, equal weights Sample Pr. of sample s2 {1, 2} 1/6 1.5 0.5 0.125 {1, 3} 2.0 0.500 {1, 4} 2.5 4.5 1.125 {2, 3} {2, 4} 3.0 {3, 4} 3.5

4.3 Example Population {1, 2, 3, 4}; =2.5, 2 = 5/4; n = 2, equal weights Sample Pr. of sample s2 {1, 2} 1/6 1.5 0.5 0.125 {1, 3} 2.0 0.500 {1, 4} 2.5 4.5 1.125 {2, 3} {2, 4} 3.0 {3, 4} 3.5

4.3 Example Population {1, 2, 3, 4}; =2.5, 2 = 5/4; n = 2, equal weights Sample Pr. of sample s2 {1, 2} 1/6 1.5 0.5 0.125 {1, 3} 2.0 0.500 {1, 4} 2.5 4.5 1.125 {2, 3} {2, 4} 3.0 {3, 4} 3.5

4.3 Example Summary Population {1, 2, 3, 4}; =2.5, 2 = 5/4; n = 2, equal weights Sample Pr. of sample s2 {1, 2} 1/6 1.5 0.5 0.125 {1, 3} 2.0 0.500 {1, 4} 2.5 4.5 1.125 {2, 3} {2, 4} 3.0 {3, 4} 3.5

4.3 Margin of error when estimating the population mean 

t distributions Very similar to z~N(0, 1) Sometimes called Student’s t distribution; Gossett, brewery employee Properties: i) symmetric around 0 (like z) ii) degrees of freedom 

Student’s t Distribution P(t < -2.2281) = .025 P(t > 2.2281) = .025 .95 .025 .025 t10 -2.2281 2.2281

Standard normal P(z < -1.96) = .025 P(z > 1.96) = .025 .95 .025 1.96

Student’s t Distribution -3 -2 -1 1 2 3 Z t Figure 11.3, Page 372

Student’s t Distribution Degrees of Freedom -3 -2 -1 1 2 3 Z t1 Figure 11.3, Page 372

Student’s t Distribution Degrees of Freedom -3 -2 -1 1 2 3 Z t1 t7 Figure 11.3, Page 372

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 Values Conf. level n = 30 z = 1.645 90% t = 1.6991 z = 1.96 95% t = 2.0452 z = 2.33 98% t = 2.4620 z = 2.58 99% 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 equation Bad news: Need to know s We don’t know n so we don’t know the degrees of freedom to find t*n-1

A Way Around this Problem: Use the Standard Normal

Sampling distribution of y Confidence level .95

Estimating s Previously collected data or prior knowledge of the population If the population is normal or near-normal, then s can be conservatively estimated by s  range 6 99.7% of obs. Within 3  of the mean

We want to be 95% confident that we are within .5 inch of , so Example: sample size to estimate mean height µ of NCSU undergrad. male students We want to be 95% confident that we are within .5 inch of , so MOE = .5; z*=1.96 Suppose previous data indicates that s is about 2 inches. n= [(1.96)(2)/(.5)]2 = 61.47 We should sample 62 male students

Example: Sample Size to Estimate a Population Mean -Textbooks Suppose 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 footballs The manufacturer of NFL footballs uses a machine to inflate new footballs The 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 psi So range =13.2 – 12.8 = .4;   range/6 = .4/6 = .067 . . . 1 2 3 48

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 6 Use 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)=80.96 Calculate ybar = mean number of lakes per county, multiply by N = 87.

4.5 Estimation of population proportion p Interested 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

Population 1 Population 2 Parameters: µ1 and 12 Parameters: µ2 and 22 (values are unknown) (values are unknown) Sample size: n1 Sample size: n2 Statistics: x1 and s12 Statistics: x2 and s22 Estimate µ1 µ2 with x1 x2

An estimate of the degrees of freedom is Sampling distribution model for ? Shape? df An estimate of the degrees of freedom is min(n1 − 1, n2 − 1).

4.6 Comparing Estimates: Comparing Means

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

4.6 Comparing Estimates: Comparing Proportions, Two Cases Difference between two polls Differences within a single poll question Comparing proportions for a single poll question, horse-race polls (dependent proportions) Difference of proportions between 2 independent polls

4.6 Comparing Estimates: Comparing Proportions in Two Independent Polls

4.6 Comparing Estimates: Comparing Proportions in Two Independent Polls

Multinomial Sampling Situation 4.6 Comparing Estimates: Comparing Dependent Proportions in a Single Poll Multinomial Sampling Situation Typically 3 or more choices in a poll

Worksheet http://packpoll.com/

End of Chapter 4