University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Inferential Statistics

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Chapter 7 _________________________ Sampling and Sampling Distributions

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Chapter 7, Part A Sampling and Sampling Distributions Sampling Distribution of the sample mean Sampling Distribution of the sample mean Sampling Distributions: An Introduction Sampling Distributions: An Introduction Point Estimation Point Estimation Simple Random Sampling Simple Random Sampling

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Chapter 7, Part B Sampling and Sampling Distributions Other Sampling Methods Other Sampling Methods Sampling Distribution of the sample Proportion Sampling Distribution of the sample Proportion

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) The purpose of statistical inference is to obtain information about The purpose of statistical inference is to obtain information about a population from information contained in a sample. a population from information contained in a sample. E.g., Current Media Predictions about Presidential Election E.g., Current Media Predictions about Presidential Election The purpose of statistical inference is to obtain information about The purpose of statistical inference is to obtain information about a population from information contained in a sample. a population from information contained in a sample. E.g., Current Media Predictions about Presidential Election E.g., Current Media Predictions about Presidential Election Statistical Inference A population is the set of all the elements of interest. A population is the set of all the elements of interest. E.g., The people eligible for voting on Nov. 4 E.g., The people eligible for voting on Nov. 4 A population is the set of all the elements of interest. A population is the set of all the elements of interest. E.g., The people eligible for voting on Nov. 4 E.g., The people eligible for voting on Nov. 4 A sample is a subset of the population. A sample is a subset of the population. E.g., The number different media use to make E.g., The number different media use to make A sample is a subset of the population. A sample is a subset of the population. E.g., The number different media use to make E.g., The number different media use to make

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) A parameter is a numerical characteristic of a A parameter is a numerical characteristic of a population. population. A parameter is a numerical characteristic of a A parameter is a numerical characteristic of a population. population. With proper sampling methods, results from samples With proper sampling methods, results from samples can provide “good” estimates of the population can provide “good” estimates of the population characteristics ( Parameter ). characteristics ( Parameter ). With proper sampling methods, results from samples With proper sampling methods, results from samples can provide “good” estimates of the population can provide “good” estimates of the population characteristics ( Parameter ). characteristics ( Parameter ). Statistical Inference

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Simple Random Sampling: Finite Population n Finite populations are often defined by lists such as: Organization membership roster Organization membership roster Credit card account numbers Credit card account numbers Inventory product numbers Inventory product numbers n A simple random sample is a sample selected such that each possible element of the population has the same probability of being selected.

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Simple Random Sampling: Finite Population In large sampling projects, computer-generated In large sampling projects, computer-generated random numbers are often used to automate the random numbers are often used to automate the sample selection process. sample selection process. Sampling without replacement is the procedure Sampling without replacement is the procedure used most often. used most often. Replacing each sampled element before selecting Replacing each sampled element before selecting subsequent elements is called sampling with subsequent elements is called sampling with replacement. replacement.

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n 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. Simple Random Sampling: Infinite Population

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Simple Random Sampling: Infinite Population n A simple random sample from an infinite population is a sample selected such that the following conditions is a sample selected such that the following conditions are satisfied. are satisfied. Each element selected comes from the same Each element selected comes from the same population. population. Each element is selected independently. Each element is selected independently. In infinite populations, it is impossible to obtain a list of all elements in the population. In infinite populations, it is impossible to obtain a list of all elements in the population.

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Simple Random Sampling: Infinite Population Thus the random number selection procedure cannot be Thus the random number selection procedure cannot be used for infinite populations. used for infinite populations. In the case of infinite populations, it is impossible to In the case of infinite populations, it is impossible to obtain a list of all elements in the population. obtain a list of all elements in the population.

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) s is the point estimator of the population standard s is the point estimator of the population standard deviation . deviation . s is the point estimator of the population standard s is the point estimator of the population standard deviation . deviation . In point estimation we use the data from the sample In point estimation we use the data from the sample to compute a value of a sample statistic that serves to compute a value of a sample statistic that serves as an estimate of a population parameter. as an estimate of a population parameter. In point estimation we use the data from the sample In point estimation we use the data from the sample to compute a value of a sample statistic that serves to compute a value of a sample statistic that serves as an estimate of a population parameter. as an estimate of a population parameter. Point Estimation We refer to as the point estimator of the population We refer to as the point estimator of the population mean . mean . We refer to as the point estimator of the population We refer to as the point estimator of the population mean . mean . is the point estimator of the population proportion p. is the point estimator of the population proportion p.

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Sampling Error The absolute value of the difference between an The absolute value of the difference between an unbiased point estimate and the corresponding unbiased point estimate and the corresponding population parameter is called the sampling error. population parameter is called the sampling error. When the expected value of a point estimator is equal When the expected value of a point estimator is equal to the population parameter, the point estimator is said to the population parameter, the point estimator is said to be unbiased. to be unbiased.

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Sampling Error Statistical methods can be used to make probability Statistical methods can be used to make probability statements about the size of the sampling error. statements about the size of the sampling error. Sampling error is the result of using a subset of the Sampling error is the result of using a subset of the population (the sample), and not the entire population (the sample), and not the entire population. population.

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Sampling Error n The sampling errors are: for sample proportion for sample standard deviation for sample mean

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Example: St. Andrew’s St. Andrew’s College receives 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.

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Example: St. Andrew’s The director of admissions The director of admissions would like to know the following information: the average SAT score for the average SAT score for the 900 applicants, and the 900 applicants, and the proportion of the proportion of applicants that want to live on campus.

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Example: St. Andrew’s We will now look at three alternatives for obtaining the desired information. n Conducting a census of the entire 900 applicants entire 900 applicants n Selecting a sample of 30 applicants, using a random number table n Selecting a sample of 30 applicants, using Excel

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Conducting a Census n 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. n We will assume for the moment that conducting a census is practical in this example.

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Conducting a Census n Population Mean SAT Score n Population Standard Deviation for SAT Score n Population Proportion Wanting On-Campus Housing

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Simple Random Sampling The applicants were numbered, from 1 to 900, as The applicants were numbered, from 1 to 900, as their applications arrived. their applications arrived. She decides a sample of 30 applicants will be used. She decides a sample of 30 applicants will be used. Furthermore, the Director of Admissions needs Furthermore, the Director of Admissions needs estimates of the population parameters of interest for estimates of the population parameters of interest for a meeting taking place in a few hours. a meeting taking place in a few hours. Now suppose that the necessary data on the Now suppose that the necessary data on the current year’s applicants were not yet entered in the current year’s applicants were not yet entered in the college’s database. college’s database.

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Taking a Sample of 30 Applicants Simple Random Sampling: Using a Random Number Table We will use the last three digits of the 5-digit We will use the last three digits of the 5-digit random numbers in the third column of the random numbers in the third column of the textbook’s random number table, and continue textbook’s random number table, and continue into the fourth column as needed. into the fourth column as needed. Because the finite population has 900 elements, we Because the finite population has 900 elements, we will need 3-digit random numbers to randomly will need 3-digit random numbers to randomly select applicants numbered from 1 to 900. select applicants numbered from 1 to 900.

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Taking a Sample of 30 Applicants Simple Random Sampling: Using a Random Number Table The numbers we draw will be the numbers of the applicants we will sample unless 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 is greater than 900 or the random number has already been used. the random number has already been used.

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Taking a Sample of 30 Applicants Simple Random Sampling: Using a Random Number Table (We will go through all of column 3 and part of (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.) We will continue to draw random numbers until We will continue to draw random numbers until we have selected 30 applicants for our sample. we have selected 30 applicants for our sample.

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Use of Random Numbers for Sampling Simple Random Sampling: Using a Random Number Table and so on 3-Digit 3-Digit Random Number Applicant Included in Sample No. 436 No. 865 No. 790 No. 835 Number exceeds 900 No. 190 No. 836 No. 744

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Sample Data Simple Random Sampling: Using a Random Number Table 1744 Conrad Harris1025 Yes 2436 Enrique Romero 950 Yes 3865 Fabian Avante1090 No 4790 Lucila Cruz1120 Yes 5835 Chan Chiang 930 No Emily Morse 1010 No No. RandomNumber Applicant SAT Score Score Live On- Campus.....

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Taking a Sample of 30 Applicants Then we choose the 30 applicants corresponding Then we choose the 30 applicants corresponding to the 30 smallest random numbers as our sample. to the 30 smallest random numbers as our sample. For example, Excel’s function For example, Excel’s function = RANDBETWEEN(1,900) = RANDBETWEEN(1,900) can be used to generate random numbers between can be used to generate random numbers between 1 and and 900. Computers can be used to generate random Computers can be used to generate random numbers for selecting random samples. numbers for selecting random samples. Simple Random Sampling: Using a Computer

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) as Point Estimator of  as Point Estimator of  n as Point Estimator of p Point Estimation Note: Different random numbers would have identified a different sample which would have resulted in different point estimates. s as Point Estimator of  s as Point Estimator of 

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) PopulationParameterPointEstimatorPointEstimateParameterValue  = Population mean SAT score SAT score  = Population std. deviation for deviation for SAT score SAT score 80 s = Sample std. s = Sample std. deviation for deviation for SAT score SAT score75.2 p = Population pro- portion wanting portion wanting campus housing campus housing Summary of Point Estimates Obtained from a Simple Random Sample = Sample mean = Sample mean SAT score SAT score = Sample pro- = Sample pro- portion wanting portion wanting campus housing campus housing

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Sampling Distribution of the Sample Mean: n Process of Statistical Inference The value of is used to make inferences about the value of . The sample data provide a value for the sample mean. A simple random sample of n elements is selected from the population. Population with mean  = ?

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) The sampling distribution of is the probability distribution of all possible values of the sample mean. where:  = the population mean  = the population mean E ( ) =  Expected Value of Sampling Distribution of the Sample Mean

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Finite Population Infinite Population is referred to as the standard error of the is referred to as the standard error of the mean. mean. A finite population is treated as being A finite population is treated as being infinite if n / N <.05. infinite if n / N <.05. is the finite correction factor. is the finite correction factor. Standard Deviation of Sampling Distribution of the Sample Mean:

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) 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.

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Sampling Distribution of for SAT Scores SamplingDistributionof

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) What is the probability that a simple random sample 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 In other words, what is the probability that will be between 980 and 1000? Sampling Distribution of for SAT Scores

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Step 1: Calculate the z -value at the upper endpoint of the interval. the interval. z = (1000  990)/14.6=.68 P ( z <.68) =.7517 Step 2: Find the area under the curve to the left of the upper endpoint. upper endpoint. Sampling Distribution of for SAT Scores

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Sampling Distribution of for SAT Scores Cumulative Probabilities for the Standard Normal Distribution the Standard Normal Distribution

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Sampling Distribution of for SAT Scores 990SamplingDistributionof1000 Area =.7517

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Step 3: Calculate the z -value at the lower endpoint of the interval. the interval. Step 4: Find the area under the curve to the left of the lower endpoint. lower endpoint. z = (980  990)/14.6= -.68 P ( z.68) =.2483 = 1  = 1  P ( z <.68) Sampling Distribution of for SAT Scores

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Sampling Distribution of for SAT Scores Area =.2483 SamplingDistributionof

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Sampling Distribution of for SAT Scores Step 5: Calculate the area under the curve between the lower and upper endpoints of the interval. 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

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Sampling Distribution of for SAT Scores Area =.5034 SamplingDistributionof

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of Suppose we select a simple random sample of 100 Suppose we select a simple random sample of 100 applicants instead of the 30 originally considered. applicants instead of the 30 originally considered. E ( ) =  regardless of the sample size. In our E ( ) =  regardless of the sample size. In our example, E ( ) remains at 990. example, E ( ) remains at 990. Whenever the sample size is increased, the standard Whenever the sample size is increased, the standard error of the mean is decreased. With the increase error of the mean is decreased. With the increase in the sample size to n = 100, the standard error of the in the sample size to n = 100, the standard error of the mean is decreased to: mean is decreased to:

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of With n = 30, With n = 100,

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Recall that when n = 30, P (980 < < 1000) = Recall that when n = 30, P (980 < < 1000) = Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of We follow the same steps to solve for P (980 < < 1000) We follow the same steps to solve for P (980 < < 1000) when n = 100 as we showed earlier when n = 30. when n = 100 as we showed earlier when n = 30. Now, with n = 100, P (980 < < 1000) = Now, with n = 100, P (980 < < 1000) = Because the sampling distribution with n = 100 has a Because the sampling distribution with n = 100 has a smaller standard error, the values of have less smaller standard error, the values of have less variability and tend to be closer to the population variability and tend to be closer to the population mean than the values of with n = 30. mean than the values of with n = 30.

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of Area =.7888 SamplingDistributionof

University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) End of Chapter 7, Part A