Slides by JOHN LOUCKS St. Edward’s University
Sampling Distribution of Chapter 7, Part ASampling and Sampling Distributions • Simple Random Sampling • Point Estimation • Introduction to Sampling Distributions
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.
Statistical Inference A sample mean provides an estimate of a population mean, and a sample proportion provides an estimate of a population proportion. Finally, a sample variance provides an estimate of a population variance. As with all estimates, we can expect some estimation error. In this chapter, we will provide the basis for determining how large that error might be by using confidence intervals and our knowledge of the sampling distribution for sample means, proportions, and variances.
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.
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.
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. • In practice, a population being studied is usually • considered infinite if it involves an ongoing process • that makes listing or counting every element in the • population impossible.
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 to avoid • selection bias. • Each of the elements remaining in the population has • the same probability of being selected.
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. • The number of different simple random samples of • size n that you can select from a finite population of • size N is: • N! • n!(N-n)!
We refer to as the point estimator of the population mean . is the point estimator of the population proportion p. Point Estimation To estimate the value of a population parameter, we compute a corresponding characteristic of the sample, referred to as a sample statistic. 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. s is the point estimator of the population standard deviation .
Sampling Error • When the expected 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.
for sample mean for sample standard deviation for sample proportion Sampling Error • The sampling errors are:
Example: 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.
Example: 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.
Example: St. Andrew’s We will now look at two alternatives for obtaining the desired information. • Conducting a census of the entire 900 applicants • Selecting a sample of 30 applicants, using Excel
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 example.
Conducting a Census • Population Mean SAT Score • Population Standard Deviation for SAT Score • Population Proportion Wanting On-Campus Housing
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.
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 • textbook’s random number table, and continue • into the fourth column as needed.
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.)
Simple Random Sampling:Using a Random Number Table • Use of Random Numbers for Sampling 3-Digit Random Number Applicant Included in Sample 744 No. 744 436 No. 436 865 No. 865 790 No. 790 835 No. 835 902 Number exceeds 900 190 No. 190 836 No. 836 . . . and so on
Simple Random Sampling:Using a Random Number Table • Sample Data Random Number SAT Score Live On- Campus No. Applicant 1 744 Conrad Harris 1025 Yes 2 436 Enrique Romero 950 Yes 3 865 Fabian Avante 1090 No 4 790 Lucila Cruz 1120 Yes 5 835 Chan Chiang 930 No . . . . . . . . . . 30 498 Emily Morse 1010 No
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 example, Excel’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.
as Point Estimator of • as Point Estimator of p Point Estimation • s as Point Estimator of Note:Different random numbers would have identified a different sample which would have resulted in different point estimates.
= Sample mean SAT score = Sample pro- portion wanting campus housing Summary of Point Estimates Obtained from a Simple Random Sample Population Parameter Parameter Value Point Estimator Point Estimate m = Population mean SAT score 990 997 80 s = Sample std. deviation for SAT score 75.2 s = Population std. deviation for SAT score .72 .68 p = Population pro- portion wanting campus housing
Sampling Distribution of The value of is used to make inferences about the value of m. The sample data provide a value for the sample mean . • Process of Statistical Inference A simple random sample of n elements is selected from the population. Population with mean m = ?
Sampling Distribution of Expected Value of E( ) = The sampling distribution of is the probability distribution of all possible values of the sample mean . where: = the population mean
Sampling Distribution of Standard Deviation of • is the finite correction factor. • is referred to as the standard error of the • mean. Finite Population Infinite Population • A finite population is treated as being • infinite if n/N< .05. • In general, the term standard error refers to • the standard deviation of a point estimator.
Central Limit Theorem 1. In selecting simple random samples of size n from a population, the sampling distribution of the sample mean can be approximated by a normal distribution as the sample size becomes large. 2. The practical reason we are interested in the sampling distribution of the sample mean is that it can be used to provide probability information about the difference between the sample mean and the population mean. 3. The mean of all possible values of the sample mean is equal to the population mean regardless of the sample size. 4. As the sample size is increased, the standard error of the mean decreases. As a result, the larger sample size provides a higher probability that the sample mean is within a specified distance of the population mean.
Form of the Sampling Distribution of When the population has a normal distribution, the sampling distribution of is normally distributed for any sample size. In most applications, the sampling distribution of can be approximated by a normal distribution whenever the sample is size 30 or more. In cases where the population is highly skewed or outliers are present, samples of size 50 may be needed.
Sampling Distribution of for SAT Scores Sampling Distribution of
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 ? 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. z = (1000 - 990)/14.6= .68 Step 2: Find the area under the curve to the left of the upper endpoint. P(z< .68) = .7517
Sampling Distribution of for SAT Scores Cumulative Probabilities for the Standard Normal Distribution
Sampling Distribution of for SAT Scores Sampling Distribution of Area = .7517 990 1000
Sampling Distribution of for SAT Scores Step 3: Calculate the z-value at the lower endpoint of the interval. z = (980 - 990)/14.6= - .68 Step 4: Find the area under the curve to the left of the lower endpoint. P(z< -.68) = .2483
Sampling Distribution of for SAT Scores Sampling Distribution of Area = .2483 980 990
Sampling Distribution of for SAT Scores P(980 << 1000) = .5034 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) = .7517 - .2483 = .5034 The probability that the sample mean SAT score will be between 980 and 1000 is:
Sampling Distribution of for SAT Scores Sampling Distribution of Area = .5034 980 990 1000
Relationship Between the Sample Size and the Sampling Distribution of • E( ) = m regardless of the sample size. In our example, E( ) remains at 990. • 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 is decreased to: • Suppose we select a simple random sample of 100 applicants instead of the 30 originally considered.
Relationship Between the Sample Size and the Sampling Distribution of With n = 100, With n = 30,
Relationship Between the Sample Size and the Sampling Distribution of • Recall that when n = 30, P(980 << 1000) = .5034. • 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) = .7888. • 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.
Relationship Between the Sample Size and the Sampling Distribution of Sampling Distribution of Area = .7888 980 990 1000