1. 1 1 Slide © 2005 Thomson/South-Western Chapter 7, Part A Sampling and Sampling Distributions Sampling Distribution of Sampling Distribution of Introduction to Sampling Distributions Introduction to Sampling Distributions Point Estimation Point Estimation Simple Random Sampling Simple Random Sampling

2. 2 2 Slide © 2005 Thomson/South-Western The purpose of statistical inference is to obtain The purpose of statistical inference is to obtain information about a population from information information about a population from information contained in a sample. contained in a sample. The purpose of statistical inference is to obtain The purpose of statistical inference is to obtain information about a population from information information about a population from information contained in a sample. contained in a sample. Statistical Inference A population is the set of all the elements of interest. A population is the set of all the elements of interest. A sample is a subset of the population. A sample is a subset of the population.

3. 3 3 Slide © 2005 Thomson/South-Western The sample results provide only estimates of the The sample results provide only estimates of the values of the population characteristics. values of the population characteristics. The sample results provide only estimates of the The sample results provide only estimates of the values of the population characteristics. values of the population characteristics. 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, the sample results With proper sampling methods, the sample results can provide “good” estimates of the population can provide “good” estimates of the population characteristics. characteristics. With proper sampling methods, the sample results With proper sampling methods, the sample results can provide “good” estimates of the population can provide “good” estimates of the population characteristics. characteristics. Statistical Inference

4. 4 4 Slide © 2005 Thomson/South-Western 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 of size n from a finite population of size N is a sample selected such population of size N is a sample selected such that each possible sample of size n has the same that each possible sample of size n has the same probability of being selected. probability of being selected.

5. 5 5 Slide © 2005 Thomson/South-Western 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.

6. 6 6 Slide © 2005 Thomson/South-Western 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. 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.

7. 7 7 Slide © 2005 Thomson/South-Western Sampling Error n The sampling errors are: for sample proportion for sample standard deviation for sample mean

8. 8 8 Slide © 2005 Thomson/South-Western 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.

9. 9 9 Slide © 2005 Thomson/South-Western 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.

10. 10 Slide © 2005 Thomson/South-Western 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

11. 11 Slide © 2005 Thomson/South-Western 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.

12. 12 Slide © 2005 Thomson/South-Western Conducting a Census n Population Mean SAT Score n Population Standard Deviation for SAT Score n Population Proportion Wanting On-Campus Housing

13. 13 Slide © 2005 Thomson/South-Western 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 

14. 14 Slide © 2005 Thomson/South-Western PopulationParameterPointEstimatorPointEstimateParameterValue  = Population mean SAT score SAT score 990997  = 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.72.68 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

15. 15 Slide © 2005 Thomson/South-Western 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  = ? Sampling Distribution of

16. 16 Slide © 2005 Thomson/South-Western The sampling distribution of is the probability distribution of all possible values of the sample mean. Sampling Distribution of where:  = the population mean  = the population mean E ( ) =  Expected Value of

17. 17 Slide © 2005 Thomson/South-Western Sampling Distribution of 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

18. 18 Slide © 2005 Thomson/South-Western 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.

19. 19 Slide © 2005 Thomson/South-Western Sampling Distribution of for SAT Scores SamplingDistributionof

20. 20 Slide © 2005 Thomson/South-Western 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

21. 21 Slide © 2005 Thomson/South-Western 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

22. 22 Slide © 2005 Thomson/South-Western Sampling Distribution of for SAT Scores Cumulative Probabilities for the Standard Normal Distribution the Standard Normal Distribution

23. 23 Slide © 2005 Thomson/South-Western Sampling Distribution of for SAT Scores 990SamplingDistributionof1000 Area =.7517

24. 24 Slide © 2005 Thomson/South-Western 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 . 7517 = 1  P ( z <.68) Sampling Distribution of for SAT Scores

25. 25 Slide © 2005 Thomson/South-Western Sampling Distribution of for SAT Scores 980990 Area =.2483 SamplingDistributionof

26. 26 Slide © 2005 Thomson/South-Western 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

27. 27 Slide © 2005 Thomson/South-Western 1000980990 Sampling Distribution of for SAT Scores Area =.5034 SamplingDistributionof

28. 28 Slide © 2005 Thomson/South-Western 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:

29. 29 Slide © 2005 Thomson/South-Western Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of With n = 30, With n = 100,

30. 30 Slide © 2005 Thomson/South-Western Recall that when n = 30, P (980 < < 1000) =.5034. Recall that when n = 30, P (980 < < 1000) =.5034. 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) =.7888. Now, with n = 100, P (980 < < 1000) =.7888. 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.

31. 31 Slide © 2005 Thomson/South-Western Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of1000980990 Area =.7888 SamplingDistributionof