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Slides Prepared by JOHN S. LOUCKS St. Edward’s Edward’s University © 2006 Thomson/South-Western Slide 1 Chapter 7, Part A Sampling and Sampling Distributions Simple Random Sampling Point Estimation Introduction to Sampling Distributions Sampling Distribution of x © 2006 Thomson/South-Western Slide 2 Statistical Inference The The purpose purpose of of statistical statistical inference inference is is to to obtain obtain information information about about aa population population from from information information contained contained in in aa sample. sample. A A population population is is the the set set of of all all the the elements elements of of interest. interest. A A sample sample is is aa subset subset of of the the population. population. © 2006 Thomson/South-Western Slide 3 Statistical Inference The The sample sample results results provide provide only only estimates estimates of of the the values values of of the the population population characteristics. characteristics. With With proper proper sampling sampling methods methods,, the the sample sample results results can good” estimates can provide provide ““good” estimates of of the the population population characteristics. characteristics. A A parameter parameter is is aa numerical numerical characteristic characteristic of of aa population. population. © 2006 Thomson/South-Western Slide 4 Simple Random Sampling: Finite Population Q Finite populations are often defined by lists such as: • Organization membership roster • Credit card account numbers • Inventory product numbers Q 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. © 2006 Thomson/South-Western Slide 5 Simple Random Sampling: 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. © 2006 Thomson/South-Western Slide 6 Simple Random Sampling: Infinite Population Q Q 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. © 2006 Thomson/South-Western Slide 7 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. © 2006 Thomson/South-Western Slide 8 Point Estimation In In point point estimation estimation we we use use the the data data from from the the sample sample to to compute compute aa value value of of aa sample sample statistic statistic that that serves serves as as an an estimate estimate of of aa population population parameter. parameter. We We refer refer to to x as as the the point point estimator estimator of of the the population population mean mean μμ.. ss is is the the point point estimator estimator of of the the population population standard standard deviation deviation σσ.. p is is the the point point estimator estimator of of the the population population proportion proportion pp.. © 2006 Thomson/South-Western Slide 9 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 absolute value of the difference between an unbiased point estimate and the corresponding population parameter is called the sampling error. Sampling error is the result of using a subset of the population (the sample), and not the entire population. Statistical methods can be used to make probability statements about the size of the sampling error. © 2006 Thomson/South-Western Slide 10 Sampling Error Q The sampling errors are: |x − μ | for sample mean |s − σ | for sample standard deviation | p − p| for sample proportion © 2006 Thomson/South-Western Slide 11 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. © 2006 Thomson/South-Western Slide 12 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. © 2006 Thomson/South-Western Slide 13 Example: St. Andrew’s We will now look at three alternatives for obtaining the desired information. Q Conducting a census of the entire 900 applicants Q Selecting a sample of 30 applicants, using a random number table Q Selecting a sample of 30 applicants, using Excel © 2006 Thomson/South-Western Slide 14 Conducting a Census Q 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. Q We will assume for the moment that conducting a census is practical in this example. © 2006 Thomson/South-Western Slide 15 Conducting a Census Q Q Population Mean SAT Score xi ∑ μ= = 990 900 Population Standard Deviation for SAT Score σ= Q 2 ( x − μ ) ∑ i 900 = 80 Population Proportion Wanting On-Campus Housing 648 p= = .72 900 © 2006 Thomson/South-Western Slide 16 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. © 2006 Thomson/South-Western Slide 17 Simple Random Sampling: Using a Random Number Table Q Taking a Sample of 30 Applicants • Because 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. © 2006 Thomson/South-Western Slide 18 Simple Random Sampling: Using a Random Number Table Q 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.) © 2006 Thomson/South-Western Slide 19 Simple Random Sampling: Using a Random Number Table Q Use of Random Numbers for Sampling 3-Digit Applicant Random Number Included in Sample No. 744 744 436 No. 436 865 No. 865 790 No. 790 835 No. 835 902 Number exceeds 900 190 No. 190 836 No. 836 . . . and so on © 2006 Thomson/South-Western Slide 20 Simple Random Sampling: Using a Random Number Table Q Sample Data Random No. Number 1 744 2 436 3 865 4 790 5 835 . . . . 30 498 Applicant Conrad Harris Enrique Romero Fabian Avante Lucila Cruz Chan Chiang . . SAT Score 1025 950 1090 1120 930 . . Live OnCampus Yes Yes No Yes No . . Emily Morse 1010 No © 2006 Thomson/South-Western Slide 21 Simple Random Sampling: Using a Computer Q Taking a Sample of 30 Applicants • Computers can be used to generate random numbers for selecting random samples. • For example, Excel’s function = RANDBETWEEN(1,900) can be used to generate random numbers between 1 and 900. • Then we choose the 30 applicants corresponding to the 30 smallest random numbers as our sample. © 2006 Thomson/South-Western Slide 22 Point Estimation Q x as Point Estimator of μ x ∑ x= 29,910 = = 997 30 30 i Q s as Point Estimator of σ s= Q ∑ (x i − x )2 29 = 163, 996 = 75.2 29 p as Point Estimator of p p = 20 30 = .68 Note: Different random numbers would have identified a different sample which would have resulted in different point estimates. © 2006 Thomson/South-Western Slide 23 Summary of Point Estimates Obtained from a Simple Random Sample Population Parameter Parameter Value μ = Population mean 990 x = Sample mean 997 σ = Population std. 80 s = Sample std. deviation for SAT score 75.2 p = Population proportion wanting campus housing .72 p = Sample proportion wanting campus housing SAT score deviation for SAT score © 2006 Thomson/South-Western Point Estimator Point Estimate SAT score .68 Slide 24 Sampling Distribution of x Q Process of Statistical Inference Population with mean μ=? The value of x is used to make inferences about the value of μ. © 2006 Thomson/South-Western A simple random sample of n elements is selected from the population. The sample data provide a value for the sample mean x . Slide 25 Sampling Distribution of x The sampling distribution of x is the probability distribution of all possible values of the sample mean x . Expected Value of x E( x ) = μ where: μ = the population mean © 2006 Thomson/South-Western Slide 26 Sampling Distribution of x Standard Deviation of x Finite Population σ N −n σx = ( ) n N −1 Infinite Population σx = σ n • A finite population is treated as being infinite if n/N < .05. • ( N − n ) / ( N − 1) is the finite correction factor. • σ x is referred to as the standard error of the mean. © 2006 Thomson/South-Western Slide 27 Form of the Sampling Distribution of x If we use a large (n > 30) simple random sample, the central limit theorem enables us to conclude that the sampling distribution of x can be approximated by a normal distribution. When the simple random sample is small (n < 30), the sampling distribution of x can be considered normal only if we assume the population has a normal distribution. © 2006 Thomson/South-Western Slide 28 Sampling Distribution of x for SAT Scores Sampling Distribution of x E( x ) = 990 © 2006 Thomson/South-Western σx = σ n = 80 = 14.6 30 x Slide 29 Sampling Distribution of x 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 μ ? In other words, what is the probability that x will be between 980 and 1000? © 2006 Thomson/South-Western Slide 30 Sampling Distribution of x 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 © 2006 Thomson/South-Western Slide 31 Sampling Distribution of x for SAT Scores Cumulative Probabilities for the Standard Normal Distribution z . .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 . . . . . . . . . . .5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224 .6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549 .7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852 .8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133 .9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389 . . . . . . . . . . . © 2006 Thomson/South-Western Slide 32 Sampling Distribution of x for SAT Scores Sampling Distribution of x σ x = 14.6 Area = .7517 x 990 1000 © 2006 Thomson/South-Western Slide 33 Sampling Distribution of x 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 © 2006 Thomson/South-Western Slide 34 Sampling Distribution of x for SAT Scores Sampling Distribution of x σ x = 14.6 Area = .2483 x 980 990 © 2006 Thomson/South-Western Slide 35 Sampling Distribution of x 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 < x < 1000) = .5034 © 2006 Thomson/South-Western Slide 36 Sampling Distribution of x for SAT Scores Sampling Distribution of x σ x = 14.6 Area = .5034 980 990 1000 © 2006 Thomson/South-Western x Slide 37 Relationship Between the Sample Size and the Sampling Distribution of x Suppose we select a simple random sample of 100 applicants instead of the 30 originally considered. E(x) = μ regardless of the sample size. In our example, E( x) remains at 990. Whenever the sample size is increased, the standard error of the mean σ x is decreased. With the increase in the sample size to n = 100, the standard error of the mean is decreased to: σ 80 σx = = = 8.0 100 n © 2006 Thomson/South-Western Slide 38 Relationship Between the Sample Size and the Sampling Distribution of x With n = 100, σx = 8 With n = 30, σx = 14.6 E( x ) = 990 © 2006 Thomson/South-Western x Slide 39 Relationship Between the Sample Size and the Sampling Distribution of x Recall that when n = 30, P(980 < x < 1000) = .5034. We follow the same steps to solve for P(980 < x < 1000) when n = 100 as we showed earlier when n = 30. Now, with n = 100, P(980 < x < 1000) = .7888. Because the sampling distribution with n = 100 has a smaller standard error, the values of x have less variability and tend to be closer to the population mean than the values of x with n = 30. © 2006 Thomson/South-Western Slide 40 Relationship Between the Sample Size and the Sampling Distribution of x Sampling Distribution of x σx = 8 Area = .7888 x 980 990 1000 © 2006 Thomson/South-Western Slide 41 End of Chapter 7, Part A © 2006 Thomson/South-Western Slide 42