1. 1. 1Slide Sampling and Sampling Distributions n Simple Random Sampling n Point Estimation n Introduction to Sampling Distributions n Sampling Distribution of n Properties of Point Estimators x n = 100 n = 30
2. 2. 2Slide Statistical Inference n The purpose of statistical inference is to obtain information about a population from information contained in a sample. n A population is the set of all the elements of interest. n A sample is a subset of the population. n The sample results provide only estimates of the values of the population characteristics. n A parameter is a numerical characteristic of a population. n With proper sampling methods, the sample results will provide good estimates of the population characteristics.
3. 3. 3Slide Simple Random Sampling n Finite Population A simple random sample from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected. Replacing each sampled element before selecting subsequent elements is called sampling with replacement. Sampling without replacement is the procedure used most often. In large sampling projects, computer-generated random numbers are often used to automate the sample selection process.
4. 4. 4Slide Point Estimation n In point estimation we use the data from the sample to compute a value of a sample statistic that serves as an estimate of a population parameter. n We refer to as the point estimator of the population mean . n s is the point estimator of the population standard deviation . x
5. 5. 5Slide Sampling Error n The absolute difference between an unbiased point estimate and the corresponding population parameter is called the sampling error. n Sampling error is the result of using a subset of the population (the sample), and not the entire population to develop estimates. n The sampling errors are: for sample mean | s - | for sample standard deviation || x
6. 6. 6Slide Example: St. Andrews St. Andrews University receives 900 applications annually from prospective students. The application forms contain a variety of information including the individuals scholastic aptitude test (SAT) score and whether or not the individual desires on-campus housing.
7. 7. 7Slide Example: St. Andrews The director of admissions would like to know the following information: the average SAT score for the applicants, and the proportion of applicants that want to live on campus. We will now look at three alternatives for obtaining the desired information. Conducting a census of the entire 900 applicants Selecting a sample of 30 applicants, using a random number table Selecting a sample of 30 applicants, using computer-generated random numbers
8. 8. 8Slide n Taking a Census of the 900 Applicants SAT Scores Population Mean Population Standard Deviation Applicants Wanting On-Campus Housing ix 990 900 ix 2 ( ) 80 900 Example: St. Andrews
9. 9. 9Slide Example: St. Andrews n Take a Sample of 30Applicants Using a Random Number Table Since the finite population has 900 elements, we will need 3-digit random numbers to randomly select applicants numbered from 1 to 900. We will use the last three digits of the 5-digit random numbers in the third column of a random number table. The numbers we draw will be the numbers of the applicants we will sample unless the random number is greater than 900 or the random number has already been used. We will continue to draw random numbers until we have selected 30 applicants for our sample.
10. 10. 10Slide Example: St. Andrews n Use of Random Numbers for Sampling 3-Digit Applicant Random Number Included in Sample 744 No. 744 436 No. 436 865 No. 865 790 No. 790 835 No. 835 902 Number exceeds 900 190 No. 190 436 Number already used etc. etc.
11. 11. 11Slide n Sample Data Random No. Number Applicant SAT Score On- Campus 1 744 Connie Reyman 1025 Yes 2 436 William Fox 950 Yes 3 865 Fabian Avante 1090 No 4 790 Eric Paxton 1120 Yes 5 835 Winona Wheeler 1015 No . . . . . 30 685 Kevin Cossack 965 No Example: St. Andrews
12. 12. 12Slide Example: St. Andrews n Take a Sample of 30 Applicants Using Computer- Generated Random Numbers Excel provides a function for generating random numbers in its worksheet. 900 random numbers are generated, one for each applicant in the population. Then we choose the 30 applicants corresponding to the 30 smallest random numbers as our sample. Each of the 900 applicants have the same probability of being included.
13. 13. 13Slide Using Excel to Select a Simple Random Sample n Formula Worksheet A B C D 1 Applicant Number SAT Score On-Campus Housing Random Number 2 1 1008 Yes =RAND() 3 2 1025 No =RAND() 4 3 952 Yes =RAND() 5 4 1090 Yes =RAND() 6 5 1127 Yes =RAND() 7 6 1015 No =RAND() 8 7 965 Yes =RAND() 9 8 1161 No =RAND() Note: Rows 10-901 are not shown.
14. 14. 14Slide Using Excel to Select a Simple Random Sample n Value Worksheet A B C D 1 Applicant Number SAT Score On-Campus Housing Random Number 2 1 1008 Yes 0.41327 3 2 1025 No 0.79514 4 3 952 Yes 0.66237 5 4 1090 Yes 0.00234 6 5 1127 Yes 0.71205 7 6 1015 No 0.18037 8 7 965 Yes 0.71607 9 8 1161 No 0.90512 Note: Rows 10-901 are not shown.
15. 15. 15Slide Using Excel to Select a Simple Random Sample n Value Worksheet (Sorted) A B C D 1 Applicant Number SAT Score On-Campus Housing Random Number 2 12 1107 No 0.00027 3 773 1043 Yes 0.00192 4 408 991 Yes 0.00303 5 58 1008 No 0.00481 6 116 1127 Yes 0.00538 7 185 982 Yes 0.00583 8 510 1163 Yes 0.00649 9 394 1008 No 0.00667 Note: Rows 10-901 are not shown.
16. 16. 16Slide n Point Estimates as Point Estimator of s as Point Estimator of as Point Estimator of p n Note: Different random numbers would have identified a different sample which would have resulted in different point estimates. x p ix x 29,910 997 30 30 ix x s 2 ( ) 163,996 75.2 29 29 p 20 30 .68 Example: St. Andrews
17. 17. 17Slide Sampling Distribution of n Process of Statistical Inference Population with mean = ? A simple random sample of n elements is selected from the population. x The sample data provide a value for the sample mean .x The value of is used to make inferences about the value of . x
18. 18. 18Slide n The sampling distribution of is the probability distribution of all possible values of the sample mean . n Expected Value of E( ) = where: = the population mean Sampling Distribution of x x x x x
19. 19. 19Slide n If we use a large (n > 30) simple random sample, the central limit theorem enables us to conclude that the sampling distribution of can be approximated by a normal probability distribution. n When the simple random sample is small (n < 30), the sampling distribution of can be considered normal only if we assume the population has a normal probability distribution. x x Sampling Distribution ofx
20. 20. 20Slide n Sampling Distribution of for the SAT Scoresx Example: St. Andrews x n 80 14.6 30 E x( ) 990 x
21. 21. 21Slide n Sampling Distribution of for the SAT Scores What is the probability that a simple random sample of 30 applicants will provide an estimate of the population mean SAT score that is within plus or minus 10 of the actual population mean ? In other words, what is the probability that will be between 980 and 1000? x Example: St. Andrews x
22. 22. 22Slide n Sampling Distribution of for the SAT Scores Using the standard normal probability table with z = 10/14.6= .68, we have area = (.2518)(2) = .5036 x Sampling distribution of x 1000980 990 Area = .2518Area = .2518 Example: St. Andrews x
23. 23. 23Slide End