Chapter 8 Sampling Methods and the Central Limit Theorem

Our Objectives Explain why a sample is often the only feasible way to learn something about a population. Describe methods to select a sample. Define and construct a sampling distribution of the sample mean. Explain the central limit theorem. Use the central limit theorem to find probabilities of selecting possible sample means from a specified population.

Sample A portion or part of the population of interest. A tool to infer something about a population. In many cases, sampling is more feasible than studying the entire population.

Reasons to Sample Contacting the whole population may be time consuming. Cost of studying all the items in a population may be prohibitive. Physical impossibility of checking all items in the population. Destructive nature of some tests. Sample results are adequate.

Types of Sampling Simple random sampling Systematic random sampling Stratified random sampling Cluster sampling

Simple Random Sample A sample selected so that each item or person in the population has the same chance of being included. It is the most widely used type of sampling. A table of random numbers is an efficient way to select members of the sample.

Systematic Random Sample A random starting point is selected, and then every k th member of the population if selected. K is calculated as the population size divided by the sample size. (If K not whole #, round down.) If physical order is related to the population characteristic, then systematic random sampling should not be used. Ex. Invoices are filed in increasing $ order.

Stratified Random Sample A population is divided into subgroups, called strata, and a sample is randomly selected from each stratum. Ex. A study comparing high ROI firms to low ROI firms on advertising expenditures. StratumROI# of FirmsRelative Frequency Number Sampled 130% (.02*50) % (.10*50) % (.54*50) 40-10% (.33*50) 5Deficit5.011 (.01*50) Total

Cluster Sample A population is divided into clusters using naturally occurring geographic or other boundaries. Then, clusters are randomly selected and a sample is collected by randomly selecting from each cluster. It is used when we want to reduce the cost of sampling a population scattered over a large geographic area. Ex. Divide state into counties (primary units), randomly select 3 counties, then randomly select people from the primary units.

Sampling Error The difference between a sample statistic and its corresponding population parameter. Sampling errors are random and occur by chance. To make accurate predictions based on sample results, we need to first develop sampling distributions of the sample means.

Sampling Distribution of the Sample Mean A probability distribution of all possible sample means of a given sample size. Sample means vary from sample to sample. Ex. We organize the means of all possible samples of 2 employees (out of 100) into a probability distribution.

Example Hourly earnings of employees at company X EmployeeHourly Earnings Joe$7 Sam7 Sue8 Bob8 Jan7 Art8 Ted9

Example (cont’d) The population mean is: µ=($ ) / 7 = $7.71 To arrive at the sampling distribution for samples of size 2, all possible samples of 2 are selected and their means computed. There are 21 possible combinations of 2 employees out of 7 total # of employees: Or 7! / 2!(7-2)! = 21

Possible Combinations

Example (cont’d) Sampling distribution of the sample mean for n=2 Sample Mean # of MeansProbability $ (3/21) (9/21) (6/21) (3/21) Total

Example (cont’d) The mean of the sampling distribution of the sample mean= sum of various sample means / sum of # of samples. or

Shapes of Population & Sampling Distribution of S.M.

Example (cont’d) The following observation are made: Mean of population = mean of sampling distribution = 7.71 Spread in distribution of sample mean is less than spread of population values (distribution of sample means values range from 7.00 to 8.50; population values range from 7.00 to 9.00). The shape of the sampling distribution of sample mean and the shape of the population are different. The first tends to be bell shaped and closer to a normal distribution.

Population vs. Sampling Distribution of Sample Mean 1. The mean of the sample means is exactly equal to population mean. 2. The dispersion of the sampling distribution of sample means is narrower than the population dispersion. 3. The sampling distribution of sample means tends to become bell-shaped and to approximate the normal probability distribution.

The Central Limit Theorem If all samples of a particular size are selected from any population, the sampling distribution of the sample mean is approximately a normal distribution. This approximation improves with larger samples. We can reason about the distribution of the sample mean with no information about the shape of the population distribution from which the sample is taken. The central limit theorem is true for all distributions.

Central Limit Theorem (cont’d) The standard deviation of the sampling distribution of the sample mean is called the standard error of the mean. Standard error of the mean: σ= SD of the population; n= number of observations is each sample.

Conclusions About C.L.T. 1. The mean of the distribution of sample means will be exactly equal to the population mean if we are able to select all possible samples of the same size from a given population. Even if we do not select all samples, we can expect the mean of the sampling distribution of the sample means to be close to the population mean. 2. There will be less dispersion in the sampling distribution of the sample mean than in the population. When we increase the sample size, the standard error of the mean decreases.

Using Sampling Distribution of the Sample Mean To convert any normal distribution to a standard normal distribution we used the formula: z= (x-µ)/σ Most business decisions are made on the basis of sampling rather than one observation. We can change the above formula to reflect the results of a sampling distribution of the sample mean.

Using Sampling Distribution of the Sample Mean (cont’d) Finding the z value of when the population SD is known: Finding the z value of when the population SD is unknown:

Example Suppose distribution of amount of cola in Cola Inc. bottles follows a normal distribution. Company records show that the mean amount per bottle is 31.2 oz and population SD is 0.4 oz. The quality assurance supervisor selected 16 bottles today from the filling line. The mean amount of cola in these bottles was oz. Is this an unlikely result? Is it likely the process is putting too much soda in the bottles? Is the sampling error of 0.18 oz unusual?

Example (cont’d) Using the formula for finding z values when SD is known (0.4 oz): From the table, we get the probability corresponding to a z value of This probability is equal to The probability of a z value greater than 1.80 is equal to ( ) We can conclude that it is unlikely, less than 4% chance, that we select a sample of 16 bottles from that population and find the sample mean equal to or greater than 31.38

Example- Area Under Standard Normal Distribution Curve

Homework 12 th edition: 27, 29 (pg. 277), 33 (pg. 278), 37 (pg.279). 13 th edition: 27, 29 (pg.287), 33, 37 (pg. 288).