Sampling and Sampling Distributions Simple Random Sampling Point
Sampling and Sampling Distributions • Simple Random Sampling • Point Estimation • Sampling Distribution
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.
Statistical Inference The sample results provide only estimates of the values of the population characteristics. With proper sampling methods, the sample results can provide “good” estimates of the population characteristics. A parameter is a numerical characteristic of a population.
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 Good Sample is a Random Sample 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.
Simple Random Sampling: Finite Population n Replacing each sampled element before selecting subsequent elements is called sampling with replacement. n Sampling without replacement is the procedure used most often. n In large sampling projects, computer-generated random numbers are often used to automate the sample selection process.
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. n 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.
Simple Random Sampling: Infinite Population In the case of infinite populations, it is impossible to obtain a list of all elements in the population. The random number selection procedure cannot be used for infinite populations.
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. We refer to mean . as the point estimator of the population s is the point estimator of the population standard deviation . is the point estimator of the population proportion p.
Sampling Error n When the expected value of a point estimator is equal to the population parameter, the point estimator is said to be unbiased. n The absolute value of the 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. n Statistical methods can be used to make probability statements about the size of the sampling error.
Sampling Error n The sampling errors are: for sample mean for sample standard deviation for sample proportion
Example: 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.
Example: St. Andrew’s 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 Excel
Conducting a Census n If the relevant information for the entire 900 applicants was in the college’s database, the population parameters of interest could be calculated using the formulas presented in Chapter 3. n We will assume for the moment that conducting a census is practical in this example.
Conducting a Census • Population Mean SAT Score • Population Standard Deviation for SAT Score • Population Proportion Wanting On-Campus Housing
Simple Random Sampling n Now suppose that the necessary information on the current year’s applicants was not yet entered in the college’s database. n Furthermore, the Director of Admissions must obtain estimates of the population parameters of interest for a meeting taking place in a few hours. n She decides a sample of 30 applicants will be used. n The applicants were numbered, from 1 to 900, as their applications arrived.
Simple Random Sampling: Using a Random Number Table • Taking a Sample of 30 Applicants • 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 the textbook’s random number table, and continue into the fourth column as needed. (See Table 7. 1, p. 275)
Simple Random Sampling: Using a Random Number Table • Taking a Sample of 30 Applicants • • 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. (We will go through all of column 3 and part of column 4 of the random number table, encountering in the process five numbers greater than 900 and one duplicate, 835. )
Use of Random Numbers for Sampling See 3 rd column of Table 7. 1, p. 275 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. 190 836 No. 836. . . and so on
Simple Random Sampling: Using a Random Number Table • Sample Data Random No. Number 1 744 2 436 3 865 4 790 5 835. . SAT Live On. Score Campus Applicant Conrad Harris 1025 Yes Enrique Romero 950 Yes Fabian Avante 1090 No Lucila Cruz 1120 Yes Chan Chiang 930 No. . . 30 Emily Morse 1010 498 No
Using Excel to Select a Simple Random Sample • Taking a Sample of 30 Applicants • 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 has the same probability of being included.
Using Excel to Select a Simple Random Sample • Formula Worksheet Note: Rows 10 -901 are not shown.
Using Excel to Select a Simple Random Sample n Value Worksheet Note: Rows 10 -901 are not shown.
Using Excel to Select a Simple Random Sample n Put Random Numbers in Ascending Order Step 1 Select cells A 2: A 901 Step 2 Select the Data menu Step 3 Choose the Sort option Step 4 When the Sort dialog box appears: Choose Random Numbers in the Sort by text box Choose Ascending Click OK
Using Excel to Select a Simple Random Sample • Value Worksheet (Sorted) Note: Rows 10 -901 are not shown.
Point Estimates • as Point Estimator of • s as Point Estimator of • as Point Estimator of p Note: Different random numbers would have identified a different sample which would have resulted in different point estimates.
Summary of Point Estimates Obtained from a Simple Random Sample Population Parameter = Population mean Parameter Value Point Estimator Point Estimate 990 = Sample mean SAT score 997 SAT score = Population std. 80 deviation for SAT score p = Population proportion wanting campus housing . 72 s = Sample std. deviation for SAT score = Sample proportion wanting campus housing 75. 2 . 68
Sampling Distribution of • Process of Statistical Inference Population with mean =? The value of is used to make inferences about the value of . A simple random sample of n elements is selected from the population. The sample data provide a value for the sample mean.
Sampling Distribution of The sampling distribution of is the probability distribution of all possible values of the sample mean where: = the population mean
Sampling Distribution of Standard Deviation of Finite Population Infinite Population • A finite population is treated as being infinite if n/N <. 05. • • is the finite correction factor. is referred to as the standard error of the mean.
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 probability 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 probability distribution.
Central Limit Theorem In selecting simple random samples of size n, the sampling distribution of the mean can be approximated by a normal probability distribution as the sample size becomes large.
Sampling Distribution of for SAT Scores
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?
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 to the left of the upper endpoint. P(z <. 68) =. 7517
Sampling Distribution of for SAT Scores Cumulative Probabilities for the Standard Normal Distribution
Sampling Distribution of for SAT Scores Sampling Distribution of Area =. 7517 990 1000
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 to the left of the lower endpoint. P(z < -. 68) = P(z >. 68) = 1 - P(z <. 68) = 1 -. 7517 =. 2483
Sampling Distribution of Area =. 2483 980 990 for SAT Scores
Sampling Distribution of for SAT Scores 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: P(980 < < 1000) =. 5034
Sampling Distribution of for SAT Scores Sampling Distribution of Area =. 5034 980 990 1000
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