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Understanding Sampling Distributions and Estimation Methods

Explore the concept of sampling distributions and point estimation methods for data collection and analysis. Learn how to select samples and estimate population characteristics with practical examples and Excel demonstrations.

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Understanding Sampling Distributions and Estimation Methods

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  1. Slides by JOHN LOUCKS St. Edward’s University

  2. Sampling Distribution of • Sampling Distribution of Chapter 7Sampling and Sampling Distributions • Selecting a Sample • Point Estimation • Introduction to Sampling Distributions • Other Sampling Methods

  3. Introduction An element is the entity on which data are collected. A population is the set of all the elements of interest. A sample is a subset of the population. A frame is a list of the elements that the sample will be selected from. The reason we select a sample is to collect data to answer a research question about a population.

  4. Introduction The sample results provide only estimates of the values of the population characteristics. The reason is simply that the sample contains only a portion of the population. With proper sampling methods, the sample results can provide “good” estimates of the population characteristics.

  5. Selecting a Sample • Sampling from a Finite Population • Sampling from a Process

  6. Sampling from a 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.

  7. Sampling from a Finite Population • 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.

  8. Sampling from a Finite Population • Example: St. Andrew’s College St. Andrew’s College received 900 applications for admission in the upcoming year from prospective students. The applicants were numbered, from 1 to 900, as their applications arrived. The Director of Admissions would like to select a simple random sample of 30 applicants.

  9. Sampling from a Finite Population Using Excel • Example: St. Andrew’s College Step 1: Assign a random number to each of the 900 applicants. Excel’s RAND function generates random numbers between 0 and 1 Step 2: Select the 30 applicants corresponding to the 30 smallest random numbers.

  10. A B Random Applicant Number 1 Number 2 1 =RAND() 3 2 =RAND() 4 3 =RAND() 5 4 =RAND() 6 5 =RAND() 7 6 =RAND() 8 7 =RAND() 9 8 =RAND() Sampling from a Finite Population Using Excel • Excel Formula Worksheet Note: Rows 10-901 are not shown.

  11. A B Random Applicant Number 1 Number 2 1 0.61021 3 2 0.83762 4 3 0.58935 5 4 0.19934 6 5 0.86658 7 6 0.60579 8 7 0.80960 9 8 0.33224 Sampling from a Finite Population Using Excel • Excel Value Worksheet Note: Rows 10-901 are not shown.

  12. Sampling from a Finite Population Using Excel • Put Random Numbers in Ascending Order Step 1Select any cell in the range B2:B901 Step 2Click the Home tab on the Ribbon Step 3In the Editing group, click Sort & Filter • Step 4Choose Sort Smallest to Largest

  13. A B Random Applicant Number 1 Number 2 12 0.00027 3 773 0.00192 4 408 0.00303 5 58 0.00481 6 116 0.00538 7 185 0.00583 8 510 0.00649 9 394 0.00667 Sampling from a Finite Population Using Excel • Excel Value Worksheet (Sorted) Note: Rows 10-901 are not shown.

  14. Sampling from a Process • 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. • Some examples of on-going processes, with infinite populations, are: • parts being manufactured on a production line • transactions occurring at a bank • telephone calls arriving at a technical help desk • customers entering a store

  15. Sampling from a Process • In the case of infinite populations, it is impossible to • obtain a list of all elements in the population. • The sampled population is such that a frame cannot be constructed. • The random number selection procedure cannot be • used for infinite populations.

  16. Sampling from a Process • A random sample from an infinite population is a sample selected such that the following conditions are satisfied. • Each of the sampled elements is independent. • Each of the sampled elements follows the same • probability distribution as the elements in the • population.

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

  18. Point Estimation • Example: St. Andrew’s College Recall that St. Andrew’s College received 900 applications 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. At a meeting in a few hours, the Director of Admissions would like to announce the average SAT score and the proportion of applicants that want to live on campus, for the population of 900 applicants.

  19. Point Estimation • Example: St. Andrew’s College However, the necessary data on the applicants have not yet been entered in the college’s computerized database. So, the Director decides to estimate the values of the population parameters of interest based on sample statistics. The sample of 30 applicants selected earlier with Excel’s RAND() function will be used.

  20. A B C D Random Applicant SAT On-Campus Number Number Score Housing 1 2 12 0.00027 1107 No 3 773 0.00192 1043 Yes 4 408 0.00303 991 Yes 5 58 0.00481 1008 No 6 116 0.00538 1127 Yes 7 185 0.00583 982 Yes 8 510 0.00649 1163 Yes 9 1008 No 394 0.00667 Point Estimation Using Excel • Excel Value Worksheet (Sorted) Note: Rows 10-31 are not shown.

  21. 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.

  22. Point Estimation Once all the data for the 900 applicants were entered in the college’s database, the values of the population parameters of interest were calculated. • Population Mean SAT Score • Population Standard Deviation for SAT Score • Population Proportion Wanting On-Campus Housing

  23. = 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

  24. Practical Advice The target population is the population we want to make inferences about. The sampled population is the population from which the sample is actually taken. Whenever a sample is used to make inferences about a population, we should make sure that the targeted population and the sampled population are in close agreement.

  25. 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 = ?

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

  27. Sampling Distribution of • Standard Deviation of We will use the following notation to define the standard deviation of the sampling distribution of . s = the standard deviation of s = the standard deviation of the population n = the sample size N = the population size

  28. Sampling Distribution of • Standard Deviation of • is the finite population • 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.

  29. Sampling Distribution of When the population has a normal distribution, the sampling distribution of is normally distributed for any sample size. In most applications, the sampling distribution of can be approximated by a normal distribution whenever the sample is size 30 or more. In cases where the population is highly skewed or outliers are present, samples of size 50 may be needed.

  30. Sampling Distribution of Sampling Distribution of for SAT Scores • Example: St. Andrew’s College

  31. Sampling Distribution of In other words, what is the probability that will be between 980 and 1000? • Example: St. Andrew’s College 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  ?

  32. Sampling Distribution of • Example: St. Andrew’s College 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 to the left of the upper endpoint. P(z< .68) = .7517

  33. Sampling Distribution of Cumulative Probabilities for the Standard Normal Distribution • Example: St. Andrew’s College

  34. Sampling Distribution of Sampling Distribution of for SAT Scores • Example: St. Andrew’s College Area = .7517 990 1000

  35. Sampling Distribution of • Example: St. Andrew’s College 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 to the left of the lower endpoint. P(z< -.68) = .2483

  36. Sampling Distribution of for SAT Scores Sampling Distribution of for SAT Scores • Example: St. Andrew’s College Area = .2483 980 990

  37. Sampling Distribution of for SAT Scores P(980 << 1000) = .5034 • Example: St. Andrew’s College Step 5: Calculate the area under the curve between the lower and upper endpoints of the interval. P(-.68 <z< .68) = P(z< .68) -P(z< -.68) = .7517 - .2483 = .5034 The probability that the sample mean SAT score will be between 980 and 1000 is:

  38. Sampling Distribution of for SAT Scores Sampling Distribution of for SAT Scores • Example: St. Andrew’s College Area = .5034 980 990 1000

  39. 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 from 14.6 to: • Example: St. Andrew’s College • Suppose we select a simple random sample of 100 applicants instead of the 30 originally considered.

  40. Relationship Between the Sample Size and the Sampling Distribution of With n = 100, With n = 30, • Example: St. Andrew’s College

  41. 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. • Example: St. Andrew’s College

  42. Relationship Between the Sample Size and the Sampling Distribution of Sampling Distribution of for SAT Scores • Example: St. Andrew’s College Area = .7888 980 990 1000

  43. 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 = ?

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

  45. Sampling Distribution of • Standard Deviation of • is referred to as the standard error of • the proportion. • is the finite population • correction factor. Finite Population Infinite Population

  46. 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

  47. Form of the Sampling Distribution of For values of p near .50, sample sizes as small as 10 permit a normal approximation. With very small (approaching 0) or very large (approaching 1) values of p, much larger samples are needed.

  48. 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?

  49. Sampling Distribution of • Example: St. Andrew’s College For our example, with n = 30 and p = .72, the normal distribution is an acceptable approximation because: np = 30(.72) = 21.6 > 5 and n(1 - p) = 30(.28) = 8.4 > 5

  50. Sampling Distribution of Sampling Distribution of • Example: St. Andrew’s College

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