1 1 Slide IS 310 – Business Statistics IS 310 Business Statistics CSU Long Beach

2 2 Slide IS 310 – Business Statistics Sampling and Sampling Distributions In many instances, one cannot study an entire population. Main reasons are cost, time and effort involved in studying the entire population. Often, it is not even necessary to study each and every element of the population. Consider a manufacturing assembly line that produces thousands or millions of items of a product. To determine the quality of this product, is it necessary to inspect each item of the product? The answer is obviously no. In such a case, one selects a subset of the population, called a sample, and inspects each item in the sample. Based on the findings from the sample, one makes conclusion about the entire population. For example, if one finds 3 percent of the items in the sample as defective, the conclusion is made that 3 percent of the items in the population is defective.

3 3 Slide IS 310 – Business Statistics Sampling and Sampling Distribution Consider another example. Goodyear tire manufacturer wants to know the mean (or average) life of its new brand of tires. One way is testing and wearing out each tire manufactured. Obviously, this does not make sense. Goodyear takes a sample of tires, tests and wears out each of these tires and then calculates the mean (or average) life of the sampled tires. Suppose, the mean life is calculated as 42,000 miles. Based on this sample, it is concluded that the mean life all new brand of tires (that is population) is 42,000 miles.

4 4 Slide IS 310 – Business Statistics Sampling and Sampling Distribution In the previous two examples, we dealt with Mean (or average) and Mean (or average) and Proportion Proportion

5 5 Slide IS 310 – Business Statistics How to Select a Sample There are several methods to select a sample from a population. One of the most common sampling methods is Simple Random Sampling. This sampling is accomplished in many ways: using a random number table or putting all names in a hat and pulling a name from the hat until sample size is reached. Refer to Table 7.1 (10-Page 261; 11-Page 269). This is a Random Number Table. Refer to Table 7.1 (10-Page 261; 11-Page 269). This is a Random Number Table.

6 6 Slide IS 310 – Business Statistics 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 that population of size N is a sample selected such that each possible sample of size n has the same each possible sample of size n has the same probability of being selected. probability of being selected.

7 7 Slide IS 310 – Business Statistics Simple Random Sampling: Finite Population In large sampling projects, computer-generated In large sampling projects, computer-generated random numbers are often used to automate the random numbers are often used to automate the sample selection process. sample selection process. Sampling without replacement is the procedure Sampling without replacement is the procedure used most often. used most often. 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.

8 8 Slide IS 310 – Business Statistics Sample and Point Estimation n Now that we know how to select a sample, let’s use the sample to estimate population characteristics (mean, and proportion). Using sample data to estimate a population mean or proportion is known as n Point Estimation

9 9 Slide IS 310 – Business Statistics 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.

10 Slide IS 310 – Business Statistics Point Estimation n Example Problem n Refer to Table 7.2 (10-Page 265; 11-Page 274). Using the sample data of this table, we can calculate the point estimates for population mean, population standard deviation and population proportion. n _ _ n x = 51,814 s = 3,348 p = 0.63

11 Slide IS 310 – Business Statistics Sampling Distributions n If we take several samples and calculate the point estimates, these estimates will be different. Each sample will provide a different value for: n _ _ n x s and p n Refer to Table 7.4 (10-Page 268; 11-Page 277). n Since these values are different, they are random variables. They have means or expected values, standard deviations and probability distributions.

12 Slide IS 310 – Business Statistics Sampling Distributions n _ n If we consider the case of mean ( x ), the probability n _ _ n distribution of x is called Sampling Distribution of x.

13 Slide IS 310 – Business Statistics _ Sampling Distribution of x _ Sampling Distribution of x n _ n Now that we know that x have different values, what n _ n are the Expected Value of x and its standard deviation? n _ n E( x ) = µ Formula 7.1 (10-Page 270; 11-Page 279) n σ = σ / √ n Formula 7.2 (10-Page 271; 11-Page 280) n _ This is called standard error of the n x mean

14 Slide IS 310 – Business Statistics Central Limit Theorem n Central Limit Theorem is a very important concept in statistics. n If we select random samples of size n from a population, the sampling distribution of the sample n _ n mean (x) can be approximated by a normal distribution as the sample size becomes large

15 Slide IS 310 – Business Statistics Use of Central Limit Theorem n Problem #26 (10-Page 279; 11-Page 288) n Given: µ = $939 σ = 245 n = 30 (1 st case) n  =  / √n = 245/√30 = n - n x _ n a. P( 914 < x < 964) n Convert 914 and 964 to z-values n z = (914 – 939)/44.71 = z = (964 – 939)/44.71 = 0.56 n _ n P( 914 < x < 964) = P( < z < 0.56) = n b. The probability value increases with a larger sample size.

16 Slide IS 310 – Business Statistics Differences Between Chapter 6 and Chapter 7 n Chapter 6: n P(100 < x < 200) n Use the following formula: n z = (x - µ)/σ n Chapter 7: n _ n P( 100 < x < 200) n Use the following formula: n _ n z = [(x - µ)/(σ/√n)]

17 Slide IS 310 – Business Statistics Differences Between Chapter 6 and Chapter 7 n Sample Problem: n The regular savings accounts of a large bank have a mean balance of $750 (µ = 750) and a standard deviation of $120 (σ = 120). A sample of 36 accounts is selected. n Find the following: n a. Probability of any single account balance being n between $720 and $780. n b. Probability of the mean of a sample of 36 accounts n being between $720 and $780.

18 Slide IS 310 – Business Statistics Differences Between Chapter 6 and Chapter 7 n In first part of the problem, we deal with Chapter 6 n P( 720 < x < 780) = ? n z = ( )/120 z = ( )/120 n = = 0.25 n P(-0.25 < z < 0.25) = n In the second part of the problem, we deal with Chapter 7 n _ n P(720 < x < 780) = ? n z = ( )/(120/√36) z = ( )/(120/√36) n = -1.5 = 1.5 n P(-1.5 < z < 1.5) =

19 Slide IS 310 – Business Statistics Relationship Between Sample Size and Sampling Distribution n If we look at Formula 7.2 (Page 280), we know that the standard error of the mean will be lower if we increase the size of the sample. Lower the standard error of the mean, the better is the estimate of the population mean. n Using the example of EAI managers, let’s use a sample size of 100 rather than 30. The standard error of the mean is reduced to 400 from

20 Slide IS 310 – Business Statistics Sample Problem Problem # 15 (10-Page ; 11-Page 276) a. Point estimate of the mean cost per treatment with Herceptin _ x = ( ) / 10 = 4550 x = ( ) / 10 = 4550 b. Point estimate of the standard deviation of the cost per treatment with Herceptin Cost Per Sample Mean Deviation Squared Deviation Cost Per Sample Mean Deviation Squared Deviation Treatment from Mean from Mean Treatment from Mean from Mean , , , , ,056, ,056, ,594, ,594, ,359,889 S = 9,068,620 / (10-1) ,359,889 S = 9,068,620 / (10-1) ,761 = 1,007, ,761 = 1,007, ,900 S = ,900 S = , , , , , ,696

21 Slide IS 310 – Business Statistics More Sample Problem Problem # 16 (10-Page 267; 11-Page 276) Given: n = 50 a. Estimate of the proportion of Fortune 500 companies based in NY = 5/50 = 0.1 or 10 percent c. Estimate of the proportion of Fortune 500 companies not based in NY, CA, MN or WI = 36/50 = 0.72 or 72 percent

22 Slide IS 310 – Business Statistics Other Sampling Methods n Stratified Random Sampling n Population is divided into groups, called strata. n Samples are selected from each strata. Useful in n applications where populations are diverse. n Examples are household incomes. n Cluster Sampling n If population is spread over a large geographical n area, cluster sampling is ideal. Think about n universities in the US. If we want to select a n sample from all universities, Cluster Sampling can be n employed.

23 Slide IS 310 – Business Statistics Other Sampling Methods n Systematic Sampling n If we select every nth element from a population, we are using Systematic Sampling. Useful in assembly line where every 10 th or 15 th element can be chosen to make a sample. n Convenience Sampling n When we select a sample mainly for convenience reasons, we are using Convenience Sampling. Think about a professor who chooses students in a study to form a sample. n Judgment Sampling n When an expert selects a sample using his judgment, this is known as Judgment Sampling.

24 Slide IS 310 – Business Statistics End of Chapter 7, Part A