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Chapter 6 Sample mean In statistics, we are often interested in finding the population mean (µ): Average Household Income in San Diego Area Average Student’s SAT score in CSUSM Average hours per week a person spend on the beach Average temperature in San Diego Most of the time, we only study a sample sample mean (x). Collecting information for the population may be too costly or just impossible. We cannot get the exact information, but an estimation. The estimation varies depending the actual sample we get. BUS304 – Chapter 6 Sample mean 1 A simple example to observe: Assume the population is composed of the following 4 people: A: John, age =26 A B C D B: Mary, age = 24 C: Cindy, age = 22 D: Mike, age = 28 μ=? σ=? BUS304 – Chapter 6 Sample mean 2 Sample: Use random sampling technique (with replacement) to select 2 of them. If Cindy and John were selected What is the mean age for the sample? John: 26 and Cindy: 22 Sample mean? BUS304 – Chapter 6 Sample mean 3 Sampling Error sampling error = x For example, the sampling error in the above example is 24 – 25 = -1 Note the order! Sample mean first, population mean second. The sampling error has a sign. “+” means the sample mean is too high “-” means the sample mean is too low. What is the probability to get a positive sampling error? probability to get a negative one? BUS304 – Chapter 6 Sample mean 4 Example Problem 6.4 (page 234, in homework) BUS304 – Chapter 6 Sample mean 5 Understanding Sampling error Why does sampling error exist? Because you only examined a part of the population to get the sample mean What if sample size gets larger? Sample mean should get closer to population mean (The more you sampled, the more reliable your sample mean is as an estimation of population.) BUS304 – Chapter 6 Sample mean 6 Exercise Problem 6.1 and 6.8 BUS304 – Chapter 6 Sample mean 7 Sample mean Sample mean may not has the same value as population mean. Sampling errors always exist. Sample mean varies when you get different samples. Before you sample, the sample mean has uncertain value It can be treated as a “random variable” BUS304 – Chapter 6 Sample mean 8 The Distribution of sample mean Sample mean is a random variable (before Sampling). Hence, it has a distribution. What should be the mean of this distribution? How will the variance of this distribution change if the sample size gets larger? What does this distribution look like? Symmetric? Skewed? BUS304 – Chapter 6 Sample mean 9 Distribution of sample mean Example: In the population of the four people (assume the selection is with replacement. That is, the two people being selected could be the same) The second person selected The first person selected John (26) Mary (24) Cindy (22) Mike (28) John (26) Mary (24) Cindy (22) Mike (28) BUS304 – Chapter 6 Sample mean 10 Probability Distribution What is a probability distribution? How many ways to present a probability distribution? How to assign probabilities? BUS304 – Chapter 6 Sample mean 11 Central Limit Theorem Theorem: IF (1) the population has mean and variance 2, and (2)the sample size n is large enough, THEN: the sample mean x follows a normal distribution, mean: μ x =μ 2 σ variance: σ 2x = n σ2 σ standard deviation: σ x = n n BUS304 – Chapter 6 Sample mean 12 Example Assuming a sample of 50 is selected from a population. The population mean is 500, and variance is 25, determine the mean, variance, and the standard deviation of sample mean. sample size: n=50 population mean: μ=500 population variance: 2 25 μ x =μ=500 2 σ 25 2 σx = =0.5 n 50 σ x = σ 2x = 0.5 0.707 BUS304 – Chapter 6 Sample mean 13 Comments on the CLT Two scenarios: 1. The population is normal: sample size doesn’t have to be large to guarantee the sample mean distribution. 2. The population is not normal sample size has to be large. The larger the better! The theorem is just an approximation. The expected value (mean) of sample mean is always the population mean What does it mean? The variance of the distribution is small when the population variance itself is small; or the sample size is large. BUS304 – Chapter 6 Sample mean 14 Concepts clarification In chapter 3 (numerical measures), we compute the following given a set of sample data Sample mean: x Sample variance: s sample statistics 2 Sample standard deviation: s In this chapter, we are examining the characteristics of sample mean x It is a random variable The mean of the sample mean μ x The variance of the sample mean: σ 2x The standard deviation of sample mean: σ x They are population parameters! BUS304 – Chapter 6 Sample mean 15 Determine the probability What is the probability a sample mean falls into a certain range? e.g. P(3<x<5)? 1. Sample mean is normally distributed! 2. In order to figure out the probability, we should use normal table 3. In order to use normal table, we need to know the z scores… 4. In order to know the z-scores, we need to know the mean and standard deviation of this normal distribution 5. Do we? Yes, we can!! Population parameters (, 2) Sample size n Central Limit Theorem (CLT) BUS304 – Chapter 6 Sample mean 16 Exercise Problem 6.23 (a) P249 BUS304 – Chapter 6 Sample mean 17 Summary In this chapter, we studied Sample mean Sample mean varies based on different sample. It can be regarded as a random variable When the sample size is large enough, it has a mean of and standard deviation /n. We can compute the probability that the sample mean falls in a range using normal table. BUS304 – Chapter 6 Sample mean 18