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Lecture4_print - Chapters 5 17 Sampling Distribution of...

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Chapters 5 & 17: Sampling Distribution of Sample Mean and Quality Control Readings: Section 5.2 and Chapter 17 September 10, 2009 1 Sampling Distribution of Sample Mean Sampling Distribution Parameter: describes population Statistic: describes the sample; sampling variability Sampling distribution A probability distribution that characterize the sampling variability The distribution of values taken by a sample statistic, e.g., sample mean, across all possible samples of the same size from the population The sample mean ¯ x is most often used as an estimate of the population mean μ . For a particular sample, sample mean ¯ x is a fixed number Imagine taking repeated samples of size n from the population and calculating sample mean for each of them: ¯ x 1 , ¯ x 2 , . . . How are these sample means distributed? How closely to μ is ¯ x ? Sampling Distribution of ¯ X = X 1 + X 2 + ... + X n n If X follows a normal distribution with mean μ and standard deviation σ , then ¯ X is also normally distributed with mean μ ¯ X = μ and standard deviation σ ¯ X = σ n What if X is not normally distributed? When randomly sampling from any population with mean μ and standard deviation σ , when n is large enough ( > 30), the sampling distribution of ¯ X is approximately normal: ¯ X N ( μ, σ n ) Formula to use for Z : Z = ¯ X - μ ¯ X σ ¯ X = ¯ X - μ σ/ n 1
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Central Limit Theorem (CLT) Example 1 : In a certain population of fish, the lengths of individual fish follow a normal distribution with mean 54mm and standard deviation 4.5mm. a. What is the probability that a randomly chosen fish is between 51mm and 60mm long? b. Suppose we sample 4 fish. What is the mean and standard deviation of the mean length of these 4 fish? c. What is the probability that the mean length of the 4 fish is between 51mm and 60mm?
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