Chapter 7 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 Sampling Distribution of p Sampling Distribution of p

The purpose of statistical inference is to obtain information about a population from information contained in a sample. The purpose of statistical inference is to obtain information about a population from information contained in a sample. Statistical Inference A population is the set of all the elements of interest. A sample is a subset of the population.

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. Statistical Inference

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.

Simple Random Sampling: Finite Population Computer-generated random numbers are often used Computer-generated random numbers are often used to automate the sample selection process. to automate the sample selection process. 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.

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 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.

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 . 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.

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. The difference between an unbiased point estimate and The difference between an unbiased point estimate and the corresponding population parameter is called the the corresponding population parameter is called the sampling error. 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.

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.

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.

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.

Conducting a Census n Population Mean SAT Score n Population Standard Deviation for SAT Score n Population Proportion Wanting On-Campus Housing

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 must obtain Furthermore, the Director of Admissions must obtain 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.

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

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.....

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 

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

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

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

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

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.

Sampling Distribution of for SAT Scores SamplingDistributionof

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

Step 1: Calculate the z -value at the upper endpoint of the interval. the interval. z = (1000  990)/14.6=.68 P (µ ≤ z <.68) =.2517 Step 2: Find the area under the curve between µ and Z Sampling Distribution of for SAT Scores

990SamplingDistributionof1000 Area =.2517

Step 3: Calculate the z -value at the lower endpoint of the interval. the interval. Step 4: Find the area under the curve between µ and lower endpoint. lower endpoint. z = (980  990)/14.6= -.68 Sampling Distribution of for SAT Scores P (µ ≥ z ≥ -.68) =.2517

Sampling Distribution of for SAT Scores 990SamplingDistributionof1000 Area =

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) = =.5034 The probability that the sample mean SAT score will be between 980 and 1000 is: P (980 < < 1000) =.5034

Sampling Distribution of for SAT Scores Area =.5034 SamplingDistributionof

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:

Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of With n = 30, With n = 100,

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.

Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of Area =.7888 SamplingDistributionof

A simple random sample of n elements is selected from the population. Population with proportion p = ? n Making Inferences about a Population Proportion The sample data provide a value for the sample proportion. The value of is used to make inferences about the value of p. Sampling Distribution of

where: p = the population proportion The sampling distribution of is the probability distribution of all possible values of the sample proportion. Expected Value of

is referred to as the standard error of the is referred to as the standard error of the proportion. Sampling Distribution of Finite Population Infinite Population Standard Deviation of

The sampling distribution of can be approximated The sampling distribution of can be approximated by a normal distribution whenever the sample size by a normal distribution whenever the sample size is large. is large. The sample size is considered large whenever these The sample size is considered large whenever these conditions are satisfied: conditions are satisfied: np > 5 n (1 – p ) > 5 and Form of the Sampling Distribution of

Recall that 72% of the Recall that 72% of the prospective students applying to St. Andrew’s College desire on-campus housing. n Example: St. Andrew’s College Sampling Distribution of What is the probability that What is the probability that a simple random sample of 30 applicants will provide an estimate of the population proportion of applicant desiring on-campus housing that is within plus or minus.05 of the actual population proportion?

SamplingDistributionof Sampling Distribution of

Step 1: Calculate the z -value at the upper endpoint of the interval. the interval. z = (.77 .72)/.082 =.61 P (µ ≤ z <.61) =.2291 Step 2: Find the area under the curve between µ and the upper endpoint. upper endpoint. Sampling Distribution of

Area =.2291 SamplingDistributionof Sampling Distribution of

Step 3: Calculate the z -value at the lower endpoint of the interval. the interval. Step 4: Find the area under the curve between µ and the lower endpoint. lower endpoint. z = (.67 .72)/.082 = -.61 P ( z -.61) =.2291 Sampling Distribution of

Area =.2291 SamplingDistributionof Sampling Distribution of

P (.67 < <.77) =.4582 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 (-.61 < z <.61) = =.4582 The probability that the sample proportion of applicants wanting on-campus housing will be within +/-.05 of the actual population proportion : Sampling Distribution of

Area =.4582 SamplingDistributionof Sampling Distribution of