2. QM 2113 - Spring 2002 Business Statistics Sampling Concepts

3. Student Objectives  Discuss the issues involved in data gathering  Summarize various methods of sampling  Define what a “random” sample is  Explain the concept of sampling error  Calculate the expected sampling error for any given random sampling process  Determine probabilities associated with the sampling distribution of the mean  Explain the Central Limit Theorem

4. But Before We Get Started...  Turn in your work: – Midterm exam – Collect homework Chapters 1 & 6 Looking only for “reasonable attempt”  Midterm exam comments – Clearly not enough prior practice! First or second time for some techniques Should –Work homework when assigned –Rework until proficient – Preparation: every conceivable analysis Salary, age, salary vs. age,... Gender, computers, gender vs. computers,... And so on...

5. Basics of Data Gathering  Remember the purpose: decision making concerning some population or process  Therefore, the sample must be capable of supporting inferences upon which those decisions are to be based  Sampling approaches – Nonstatistical: based upon judgement – Convenience: can result in bias – Statistical: provide foundation for inference  Randomness: each element in the process or population has an equal chance of being selected  Note: random ≠ representative

6. Recall: Parameters versus Statistics  Descriptive numerical measures calculated from the entire population are called parameters. – Quantitative data:  and  – Qualitative data:  (proportion)  Corresponding measures for a sample are called statistics. – Quantitative data: x-bar and s – Qualitative data: p

7. Random Sampling  We need to get a little theoretical for a while here, but not too long  Consider a set of numbers; call it our population of interest – The average summarizes where on a number line the values tend to be – The standard deviation provides a measure of how different the values are  Suppose we select a subset (i.e., a sample) – It can also be described by the average and standard deviation – But those values aren’t the same

8. Sampling Process Population or Process Sample Parameter Statistic

9. Some Things to Note  Sampling error – Difference between parameter and statistic – Exact amount always unknown, but...  If we had taken a different sample, the statistic would have been a different value  Thus, the outcomes (i.e., statistics) of sampling are random variables themselves – Expected value for x-bar – Expected variation for x-bar

10. It Turns Out That...  The expected value for x-bar is the population or process average (i.e.,  )  The expected variation in x-bar from one sample average to another is – The average (expected) sampling error – Known as the standard error of the mean – Summarized by the sampling distribution Possible values for x-bar Probabilities associated with those values  If the sample size is large enough, the sampling distribution is normal (CLT)

11. Let’s Work Some Problems  Exercises assigned – Exercise 1 – Exercise 2 – Exercise 8 – Exercise 12  Consider the implications  Why is this important?  INFERENCE!!!

12. Homework  Rework (as necessary) exercises assigned from Chapter 6  Finish reading Chapter 6  Additional problems from Chapter 6  Rethink the midterm exam and bring one question to class