Chapter 7 Sampling and Sampling Distributions

1. Sampling Distribution of • Sampling Distribution of p Chapter 7 Sampling and Sampling Distributions • Simple Random Sampling • Point Estimation • Introduction to Sampling Distributions

2. Statistical Inference The purpose of statistical inference is to obtain information about a population from information contained in a sample. A population is the set of all the elements of interest. A sample is a subset of the population.

3. Statistical Inference The sample results provide only estimates of the values of the population characteristics. A parameter is a numerical characteristic of a population.

4. Simple Random Sampling:Finite Population • Finite populations are often defined by lists such as: • Organization membership roster • Credit card account numbers • Inventory product numbers • A simple random sample of size n 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.

5. Simple Random Sampling:Finite Population • Replacing each sampled element before selecting • subsequent elements is called sampling with • replacement. • Computer-generated random numbers are often used to automate the sample selection process.

6. Simple Random Sampling:Infinite Population • Infinite populations are often defined by an ongoing process whereby the elements of the population consist of items generated as though the process would operate indefinitely. • A simple random sample from an infinite population is a sample selected such that the following conditions are satisfied. • Each element selected comes from the same • population. • Each element is selected independently.

7. We refer to as the point estimator of the population mean . is the point estimator of the population proportion p. Point Estimation s is the point estimator of the population standard deviation .

8. Sampling Error • When the expected value of a point estimator is equal • to the population parameter, the point estimator is said • to be unbiased. • The difference between an unbiased point estimate and the corresponding population parameter is called the sampling error. • Statistical methods can be used to make probability • statements about the size of the sampling error.

9. Example: St. Andrew’s St. Andrew’s College receives 900 applications annually from prospective students. The application form contains a variety of information including the individual’s scholastic aptitude test (SAT) score and whether or not the individual desires on-campus housing.

10. Example: St. Andrew’s The director of admissions would like to know the following information: • the average SAT score for the 900 applicants, and • the proportion of applicants that want to live on campus.

11. Conducting a Census • If the relevant data for the entire 900 applicants were in the college’s database, the population parameters of interest could be calculated using the formulas presented in Chapter 3. • We will assume for the moment that conducting a census is practical in this example.

12. Conducting a Census • Population Mean SAT Score • Population Standard Deviation for SAT Score • Population Proportion Wanting On-Campus Housing

13. Simple Random Sampling • Now suppose that the necessary data on the current year’s applicants were not yet entered in the college’s database. • Furthermore, the Director of Admissions must obtain • estimates of the population parameters of interest for • a meeting taking place in a few hours. • She decides a sample of 30 applicants will be used. • The applicants were numbered, from 1 to 900, as • their applications arrived.

14. Simple Random Sampling:Using a Random Number Table • Use of Random Numbers for Sampling 3-Digit Random Number Applicant Included in Sample 744 No. 744 436 No. 436 865 No. 865 790 No. 790 835 No. 835 902 Number exceeds 900 190 No. 190 836 No. 836 . . . and so on

15. Simple Random Sampling:Using a Random Number Table • Sample Data Random Number SAT Score Live On- Campus No. Applicant 1 744 Conrad Harris 1025 Yes 2 436 Enrique Romero 950 Yes 3 865 Fabian Avante 1090 No 4 790 Lucila Cruz 1120 Yes 5 835 Chan Chiang 930 No . . . . . . . . . . 30 498 Emily Morse 1010 No

16. as Point Estimator of  • as Point Estimator of p Point Estimation • s as Point Estimator of  Note:Different random numbers would have identified a different sample which would have resulted in different point estimates.

17. = Sample mean SAT score = Sample pro- portion wanting campus housing Summary of Point Estimates Obtained from a Simple Random Sample Population Parameter Parameter Value Point Estimator Point Estimate m = Population mean SAT score 990 997 80 s = Sample std. deviation for SAT score 75.2 s = Population std. deviation for SAT score .72 .68 p = Population pro- portion wanting campus housing

18. Sampling Distribution of The value of is used to make inferences about the value of m. The sample data provide a value for the sample mean . • Process of Statistical Inference A simple random sample of n elements is selected from the population. Population with mean m = ?

19. Sampling Distribution of Expected Value of E( ) =  The sampling distribution of is the probability distribution of all possible values of the sample mean . where:  = the population mean

20. Sampling Distribution of Standard Deviation of • is the finite correction factor. • is referred to as the standard error of the • mean. Finite Population Infinite Population • A finite population is treated as being • infinite if n/N< .05.

21. Form of the Sampling Distribution of 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 distribution. 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 distribution.

22. Sampling Distribution of for SAT Scores Sampling Distribution of

23. Sampling Distribution of for 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 +/-10 of the actual population mean  ? In other words, what is the probability that will be between 980 and 1000?

24. Sampling Distribution of for SAT Scores Step 1: Calculate the z-value at the upper endpoint of the interval. z = (1000 - 990)/14.6= .68 Step 2: Find the area under the curve between µ and Z P(µ ≤ z< .68) = .2517

25. Sampling Distribution of for SAT Scores Sampling Distribution of Area = .2517 990 1000

26. Sampling Distribution of for SAT Scores Step 3: Calculate the z-value at the lower endpoint of the interval. z = (980 - 990)/14.6= - .68 Step 4: Find the area under the curve between µ and lower endpoint. P(µ ≥ z ≥ - .68) = .2517

27. Sampling Distribution of for SAT Scores Sampling Distribution of Area = .2517 .2517 980 990 1000

28. Sampling Distribution of for SAT Scores P(980 << 1000) = .5034 Step 5: Calculate the area under the curve between the lower and upper endpoints of the interval. P(-.68 <z< .68) = .2517 + .2517 = .5034 The probability that the sample mean SAT score will be between 980 and 1000 is:

29. Sampling Distribution of for SAT Scores Sampling Distribution of Area = .5034 980 990 1000

30. Relationship Between the Sample Size and the Sampling Distribution of • E( ) = m regardless of the sample size. In our example, E( ) remains at 990. • Whenever the sample size is increased, the standard error of the mean is decreased. With the increase in the sample size to n = 100, the standard error of the mean is decreased to: • Suppose we select a simple random sample of 100 applicants instead of the 30 originally considered.

31. Relationship Between the Sample Size and the Sampling Distribution of With n = 100, With n = 30,

32. Relationship Between the Sample Size and the Sampling Distribution of • Recall that when n = 30, P(980 << 1000) = .5034. • We follow the same steps to solve for P(980 << 1000) when n = 100 as we showed earlier when n = 30. • Now, with n = 100, P(980 << 1000) = .7888. • Because the sampling distribution with n = 100 has a smaller standard error, the values of have less variability and tend to be closer to the population mean than the values of with n = 30.

33. Relationship Between the Sample Size and the Sampling Distribution of Sampling Distribution of Area = .7888 980 990 1000

34. Sampling Distribution of The sample data provide a value for the sample proportion . The value of is used to make inferences about the value of p. • Making Inferences about a Population Proportion A simple random sample of n elements is selected from the population. Population with proportion p = ?

35. Sampling Distribution of Expected Value of The sampling distribution of is the probability distribution of all possible values of the sample proportion . where: p = the population proportion

36. Sampling Distribution of Standard Deviation of is referred to as the standard error of the proportion. Finite Population Infinite Population

37. Form of the Sampling Distribution of The sampling distribution of can be approximated by a normal distribution whenever the sample size is large. The sample size is considered large whenever these conditions are satisfied: np> 5 n(1 – p) > 5 and

38. Sampling Distribution of • Example: St. Andrew’s College Recall that 72% of the prospective students applying to St. Andrew’s College desire on-campus housing. What is the probability that a simple random sample of 30 applicants will provide an estimate of the population proportion of applicant desiring on-campus housing that is within plus or minus .05 of the actual population proportion?

39. Sampling Distribution of Sampling Distribution of

40. Sampling Distribution of Step 1: Calculate the z-value at the upper endpoint of the interval. z = (.77 - .72)/.082 = .61 Step 2: Find the area under the curve between µ and the upper endpoint. P(µ ≤ z< .61) = .2291

41. Sampling Distribution of Sampling Distribution of Area = .2291 .72 .77

42. Sampling Distribution of Step 3: Calculate the z-value at the lower endpoint of the interval. z = (.67 - .72)/.082 = - .61 Step 4: Find the area under the curve between µ and the lower endpoint. P(z< -.61) = P(µ ≥ z> -.61) = .2291

43. Sampling Distribution of Sampling Distribution of .2291 Area = .2291 .72 .77 .67

44. Sampling Distribution of P(.67 << .77) = .4582 Step 5: Calculate the area under the curve between the lower and upper endpoints of the interval. P(-.61 <z< .61) = .2291 + .2291 = .4582 The probability that the sample proportion of applicants wanting on-campus housing will be within +/-.05 of the actual population proportion :

45. Sampling Distribution of Sampling Distribution of Area = .4582 .67 .72 .77