Chapter 7, Part A Sampling and Sampling Distributions

Transcription

1 Slides Prepared by JOHN S. LOUCKS St. Edward s University Slide 1 Chapter 7, Part A Sampling and Sampling Distributions Simple Random Sampling Point Estimation Introduction to Sampling Distributions Sampling Distribution of Slide 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. Slide 3 1

2 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. Slide 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. 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. Slide 6 2

3 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. 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. Slide 8 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 as the point estimator of the population mean µ. s is the point estimator of the population standard deviation σ. p is the point estimator of the population proportion p. Slide 9 3

4 Sampling Error When the epected 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. Slide 10 Sampling Error The sampling errors are: µ for sample mean s σ for sample standard deviation p p for sample proportion Slide 11 Eample: 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. Slide 12 4

5 Eample: 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. Slide 13 Eample: 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 Ecel Slide 14 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 eample. Slide 15 5

6 Conducting a Census Population Mean SAT Score i µ = = Population Standard Deviation for SAT Score 2 ( i µ ) σ = = Population Proportion Wanting On-Campus Housing 648 p = = 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. Slide 17 Simple Random Sampling: Using a Random Number Table 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 tetbook s random number table, and continue into the fourth column as needed. Slide 18 6

7 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.) Slide 19 Simple Random Sampling: Using a Random Number Table Use of Random Numbers for Sampling 3-Digit Applicant Random Number Included in Sample 744 No No No No No Number eceeds No No and so on Slide 20 Sample Data Simple Random Sampling: Using a Random Number Table No. Random Number Applicant SAT Score Live On- Campus Conrad Harris 1025 Yes Enrique Romero 950 Yes Fabian Avante 1090 No Lucila Cruz 1120 Yes Chan Chiang 930 No Emily Morse 1010 No Slide 21 7

8 Simple Random Sampling: Using a Computer Taking a Sample of 30 Applicants Computers can be used to generate random numbers for selecting random samples. For eample, Ecel 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. Slide 22 as Point Estimator of µ i 29,910 = = = s as Point Estimator of σ 2 ( i ) 163,996 s = = = p as Point Estimator of p Point Estimation p = =.68 Note: Different random numbers would have identified a different sample which would have resulted in different point estimates. Slide 23 Summary of Point Estimates Obtained from a Simple Random Sample Population Parameter µ = Population mean SAT score Parameter Value Point Estimator = Sample mean SAT score Point Estimate σ = Population std. deviation for SAT score p = Population proportion wanting campus housing 80 s = Sample std. deviation for SAT score p = Sample proportion.68 wanting campus housing Slide 24 8

9 Sampling Distribution of Process of Statistical Inference Population with mean µ =? A simple random sample of n elements is selected from the population. The value of is used to make inferences about the value of µ. The sample data provide a value for the sample mean. Slide 25 Sampling Distribution of The sampling distribution of is the probability distribution of all possible values of the sample mean. Epected Value of E( ) = µ where: µ = the population mean Slide 26 Sampling Distribution of Standard Deviation of Finite Population σ N n σ = ( ) n N 1 Infinite Population σ σ = n A finite population is treated as being infinite if n/n <.05. ( N n )/( N 1 ) is the finite correction factor. σ σ is referred to as the standard error of the mean. Slide 27 9

10 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 approimated 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. Slide 28 Sampling Distribution of for SAT Scores Sampling Distribution of σ 80 σ = = = 14.6 n 30 E ( ) = 990 Slide 29 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 µ? In other words, what is the probability that will be between 980 and 1000? Slide 30 10

11 Sampling Distribution of for SAT Scores Step 1: Calculate the z-value at the upper endpoint of the interval. z = ( )/14.6=.68 Step 2: Find the area under the curve to the left of the upper endpoint. P(z <.68) =.7517 Slide 31 Sampling Distribution of for SAT Scores Cumulative Probabilities for the Standard Normal Distribution z Slide 32 Sampling Distribution of for SAT Scores Sampling Distribution of σ = =14.6 Area = Slide 33 11

12 Sampling Distribution of for SAT Scores Step 3: Calculate the z-value at the lower endpoint of the interval. z = ( )/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) = =.2483 Slide 34 Sampling Distribution of for SAT Scores Sampling Distribution of σ = =14.6 Area = Slide 35 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) = =.5034 The probability that the sample mean SAT score will be between 980 and 1000 is: P(980 < < 1000) =.5034 Slide 36 12

13 Sampling Distribution of for SAT Scores Sampling Distribution of σ = =14.6 Area = Slide 37 Relationship Between the Sample Size and the Sampling Distribution of Suppose we select a simple random sample of 100 applicants instead of the 30 originally considered. E( ) = µ regardless of the sample size. In our eample, E( ) remains at 990. Whenever the sample size is increased, the standard error of the mean σ is decreased. With the increase in the sample size to n = 100, the standard error of the mean is decreased to: σ 80 σ = = = 8.0 n 100 Slide 38 Relationship Between the Sample Size and the Sampling Distribution of With n = 100, σ = 8 With n = 30, σ = =14.6 E ( ) = 990 Slide 39 13

14 Relationship Between the Sample Size and the Sampling Distribution of Recall that when n = 30, P(980 < < 1000) = 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) = 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. Slide 40 Relationship Between the Sample Size and the Sampling Distribution of Sampling Distribution of σ = =88 Area = Slide 41 End of Chapter 7, Part A Slide 42 14