Econ 3790: Business and Economics Statistics. Instructor: Yogesh Uppal
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1 Econ 3790: Business and Economics Statistics Instructor: Yogesh Uppal
2 Chapter 7, Part A Sampling and Sampling Distributions Simple Random Sampling Point Estimation Introduction to Sampling Distributions Sampling Distribution of
3 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.
4 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.
5 Simple Random Sampling: Finite Population 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.
6 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.
7 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. s is the point estimator of the population standard deviation. p is the point estimator of the population proportion p.
8 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.
9 The sampling errors are: Sampling Error for sample mean s for sample standard deviation p p for sample proportion
10 Air Quality Eample Let us suppose that the population of air quality data consists of 191 observations. How would you determine the following population parameters: mean, standard deviation, proportion of cities with good air quality.
11 Air Quality Eample How about picking a random sample from this population representing the air quality? We shall use SPSS to do this random sampling for us. How would you use this sample to provide point estimates of the population parameters?
12 Summary of Point Estimates Obtained from a Simple Random Sample Population Parameter = Population mean SAT score Parameter Point Point Value Estimator Estimate 40.9 = Sample mean. SAT score = Population std. deviation for SAT score 20.5 s = Sample std. deviation for SAT score.. p = Population pro- portion.62 p = Sample pro-. portion wanting campus housing
13 Process of Statistical Inference Population with mean =? A simple random sample of n elements is selected from the population. The value of is used to make inferences about the value of. The sample data provide a value for the sample mean.
14 Sampling Distribution of The sampling distribution of is the probability distribution of all possible values of the sample mean. Epected Value of where: E( ( ) = = the population mean
15 Sampling Distribution of Standard Deviation of Finite Population Infinite Population ( ) n N N n 1 n A finite population is treated as being infinite if n/n <.05. ( N n )/( N 1 ) is the finite correction factor. is also referred to as the standard error of the mean.
16 The Shape of Sampling Distribution of If the shape of the distribution of in the population is normal, the shape of the sampling distribution of is normal as well. If the shape of the distribution of in the population is approimately normal, the shape of the sampling distribution of is approimately normal as well. If the shape of the population is not approimately normal then If n is small, the shape of the sampling distribution of is unpredictable. If n is large (n 30), the shape of the sampling distribution of can be assumed to be approimately normal.
17 Sampling Distribution of for the air quality eample when the population is (almost) infinite Sampling Distribution of n E( ) 40.9
18 Sampling Distribution of for the air quality eample when the population is finite Sampling Distribution of n Nn N E( ) 40.9
19 Relationship Between the Sample Size and the Sampling Distribution of E( ( ) = regardless of the sample size. In our eample, E( ( ) remains at 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 decreases.
20 Relationship Between the Sample Size and the Sampling Distribution of With n With n E( ) 40.9
21 Sampling Distribution of If we use a large random sample (n>30), then the sampling distribution of can be approimated by the normal distribution. If the sample is small, then the sampling distribution of can be normal only if we assume that our population has a normal distribution.
22 Sampling Distribution of for the air quality Inde when n = 5. What is the probability that a simple random sample of 5 applicants will provide an estimate of the population mean air quality inde that is within +/-2 2 of the actual population mean, μ? In other words, what is the probability that will be between 38.9 and 42.9?
23 Sampling Distribution of for the air quality Inde when n = 100. What is the probability that a simple random sample of 100 applicants will provide an estimate of the population mean air quality inde that is within +/-2 2 of the actual population mean, μ?
24 Relationship Between the Sample Size and the Sampling Distribution of 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 = 5. Basically, a given interval with smaller standard error (larger n) will cover more area under the normal curve than the same interval with larger standard error (smaller n).
25 Chapter 7, Part B Sampling and Sampling Distributions Sampling Distribution of p
26 Sampling Distribution of p Making Inferences about a Population Proportion Population with proportion p =? A simple random sample of n elements is selected from the population. p The value of is used to make inferences about the value of p. The sample data provide a value for the sample proportion. p
27 Sampling Distribution of p The sampling distribution of p is the probability distribution of all possible values of the sample proportion p. Epected Value of p E ( p ) p where: p = the population proportion
28 Sampling Distribution of p Standard Deviation of Finite Population Infinite Population p ( 1 p ) p n N N n 1 p ( 1 p ) p n is referred to as the standard error of the proportion. p
29 Form of Sampling Distribution of p The sampling distribution of p can be approimated by a normal distribution whenever the sample size is large: Central Limit Theorem (CLT). The sample size is considered large whenever these conditions are satisfied: np > 5 and n(1 p) > 5
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Math 1431 Section 1485 MWF 1 1pm SR 117 Dr. Melahat Almus almus@math.uh.edu http://www.math.uh.edu/~almus COURSE WEBSITE: http://www.math.uh.edu/~almus/1431_sp16.html Visit my website regularly for announcements