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Welcome to BUAD 310

Welcome to BUAD 310. Instructor: Kam Hamidieh Lecture 12, Monday March 3, 2014. Agenda & Announcement. Today: Cover parts of Chapters 13 & 14 Pass back the exams and go over the exam (at the end) Next time: Start Chapter 15. Statistical Inference.

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Welcome to BUAD 310

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  1. Welcome to BUAD 310 Instructor: Kam Hamidieh Lecture 12, Monday March 3, 2014

  2. Agenda & Announcement • Today: • Cover parts of Chapters 13 & 14 • Pass back the exams and go over the exam (at the end) • Next time: • Start Chapter 15 BUAD 310 - Kam Hamidieh

  3. Statistical Inference Use a small group of units to make some conclusions or inference about a larger group Population (Characteristics Unknown) Sample BUAD 310 - Kam Hamidieh

  4. Populations and Samples • Population – a group of individuals (or things) that we would like to know something about • Sample – a representative subset of a population. • Why do you need a sample? • Studying a population is too expensive and time-consuming, impractical, and often impossible to reach! • If a sample is representative of the population, then by observing the sample we can learn something about the population. BUAD 310 - Kam Hamidieh

  5. Parameters • Parameter- a characteristic of the population in which we have a particular interest • The value of the parameter is not known in general because we can not reach the entire population. • Often denoted with Greek letters (µ, ) • Examples: • The proportion of the population that would vote for candidate XYZ. • The population mean salary of all CEOs of privately owned companies in the US. BUAD 310 - Kam Hamidieh

  6. Statistics • Statistic – a number that can be computed from the sample data without making use of unknown parameters. • We use a statistic to estimate an unknown parameter. • Example: • The sample proportion that would vote for candidate XYZ. • The sample mean salary of all CEOs of privately owned companies in the US. BUAD 310 - Kam Hamidieh

  7. Example The international Air Transport Association surveys business travelers in order to develop quality ratings for transatlantic gateway airports. Suppose a random sample of a 50 business travelers is selected and each traveler is asked to provide a rating for LAX. The rating scale is 1,2,3,…,9,10. The mean of the 50 ratings is found to be 6.8. • Population? • Sample? • Parameter? • Statistic? Population of all business travelers world wide 50 business travelers, sampled in the survey Population mean rating for LAX based on all business travelers - this is unknown. Sample mean rating for LAX based on 50 business travelers - this is known. BUAD 310 - Kam Hamidieh

  8. In Class Exercise 1 We are interested in knowing about the mean cost per night of hotel room in NY. Based on a random sample of size 45, the mean cost per night of hotel room in NY is $275 (SmartMoney, March, 2009). Identify the following: • Population? • Sample? • Parameter? • Statistic? BUAD 310 - Kam Hamidieh

  9. Inference for the Population Mean Population mean unknown:  Sample Compute the sample mean… Now is our point estimates of . BUAD 310 - Kam Hamidieh

  10. Inference for the Population Mean • gives us an estimate of the population mean μ. • is a random variable. Why? • How do we quantify the uncertainty around? Why care? BUAD 310 - Kam Hamidieh

  11. μ = ? 200K 250K 100K 150K 300K 200K …… 250K BUAD 310 - Kam Hamidieh

  12. μ = ? 200K 250K 100K 150K 300K 200K …… 250K BUAD 310 - Kam Hamidieh

  13. μ = ? 200K 250K 100K 150K 300K 200K …… 250K BUAD 310 - Kam Hamidieh

  14. μ = ? 200K 250K 100K 150K 300K 200K …… 250K The collection of all the possible values of X bar is called the sampling distribution of the sample mean. BUAD 310 - Kam Hamidieh

  15. The Sampling Distribution • The sampling distribution of the sample mean is: • Keep taking random samples of the same size from a population • Find the sample mean for each sample • Collect all and display their distribution. • In one sentence: The distribution of possible values of the sample mean for repeated samples of the same size from your population of interest is called the sampling distributionof the sample mean. BUAD 310 - Kam Hamidieh

  16. How many distributions? There are two: • The population has a distribution. This is the distribution of the values for all members of a population. For example, in the CEO case, we can think of all the CEOs of private companies. • If we were to take smaller subsets of this population (a random sample) and collect all the possible sample mean values, then this would be the sampling distribution of the sample mean. BUAD 310 - Kam Hamidieh

  17. Importance of Sampling Distribution • Sampling distributions provide the key for telling us how close the sample mean (or any statistic) falls to the unknown population mean (or any population parameter) we’d like to know about. • Some questions: • What will this distribution look like? • How does it relate to the original population where the random sample was drawn? BUAD 310 - Kam Hamidieh

  18. THE CLT The Central Limit Theorem: Suppose we draw a random sample of size n from a population with mean μ and a standard deviation of σ. When n is large, the sampling distribution of the sample mean is approximately normal: Behaves like BUAD 310 - Kam Hamidieh

  19. Sampling Distribution Demo http://onlinestatbook.com/stat_sim/sampling_dist/index.html (Use Chrome; it works better.) BUAD 310 - Kam Hamidieh

  20. Lessons Learned • Every statistics has a sampling distribution. • The sampling distribution of the sample mean is: • Exactly normal when the population is normal • Approximately normal if the population distribution is not normal and n is large • In either case: • The mean of the sampling distribution matches the population mean: • The standard deviation of the sampling distribution of the sample mean is: BUAD 310 - Kam Hamidieh

  21. Standard Error • The standard error is defined as • We can interpret the standard error of the mean as the average distance of the possible sample mean values (for repeated samples of the same size n) from the population mean population. • What does a small or large SE indicate? BUAD 310 - Kam Hamidieh

  22. Example • Let X be normally distributed with a mean 5 and a sd of 10. Then the distribution of the sample mean with n = 30 will be exactly normal with • Let X be uniform (0,1). The mean is ½ and sdis square root of 1/12. The distribution of the sample mean with n = 30 will be approximately normal with: BUAD 310 - Kam Hamidieh

  23. Example An insurance company looks at the records for millions of homeowners and sees that the mean loss is μ=$250, and the standard deviation of losses is σ = $1000. The company plans to sell fire insurance for $250 plus enough to cover expenses and some profit. Questions: • If the company sells 10,000 policies what is the approximate probability that the average loss in a year will be greater than $280? Assume that 10,000 policy holders behave randomly. • (Tricky!) What is the probability that the total of the 10,000 policies is greater than $2.8 million? • Explain why selling many thousands of such policies is a safe business. BUAD 310 - Kam Hamidieh

  24. Related to The Previous Example Discuss the idea of selling hurricane insurance. See: http://usnews.nbcnews.com/_news/2013/09/03/20249654-20000-a-year-for-flood-insurance-sandy-survivors-face-tough-rebuilding-choices?lite BUAD 310 - Kam Hamidieh

  25. Yet Another Example (More Involved) You are playing Red and Black in roulette. (A roulette wheel has 38 pockets: 18 red, 18 black, 2 green.) The house takes all the money on green. You pick either red or black; if the ball lands in the color you pick, you win a dollar. Otherwise you lose a dollar. Suppose you play 100 times. What is the chance that you win$25 or more? BUAD 310 - Kam Hamidieh

  26. In Class Exercise 2 What is wrong with the following statements? • If the standard deviation of a population is 10, then the standard deviation of the mean for a random sample of 30 observations from this population will be 10 as well. • When taking a random sample from a population, larger sample sizes will result in larger standard deviation of the sample mean. • The mean of a sampling distribution of changes when the sample size changes. • For large n, the distribution of observed values will be approximately normal. • The CLT states that for large n, μ (the population mean) is approximately normal. BUAD 310 - Kam Hamidieh

  27. IID Random Variables (Time Permitting) • A collection of random variables which are independent from each other and have identical distribution are called IID. • Example: Suppose we measure the heights of n=10 women:X1 = height of first woman,X2 = height of second woman, …X10 = height of 10th woman. • Saying that X1, …, X10 are IID means that that all the X’s have the same distribution, for example N(64, 2.5), and knowing that X1 = 66, does not change the probabilities for X2. BUAD 310 - Kam Hamidieh

  28. What is exactly a random sample? (Time Permitting) • Simple Random Sample (SRS) versus a Random Sample: • SRS: A SRS of size n consists of n things from the population chosen in a way so that every set of n things has an equal chance of being selected. • Random Sample: You assume that the population has a certain distribution and you draw from this distribution. • Random Sample of size n = Collection of n IID random variables from the population of interest. BUAD 310 - Kam Hamidieh

  29. Useful Relationships for IID (Time Permitting) You need not memorize this! BUAD 310 - Kam Hamidieh

  30. Next Time • Chapter 15: Statistical Inference and Confidence Intervals. BUAD 310 - Kam Hamidieh

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