Chapter 8 - Chapter 8 Sampling Distribution of Sample...

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Chapter 8 Sampling Distribution of Sample Statistics A parameter is a function of population data. Parameters are numerical characteristics of a population. They are fixed numbers, i.e., their values do not change from sample to sample. Some examples are μ, σ and π. A (sample) statistic is a function of sample data. Statistics are numerical summaries of sample data (that contain no unknown parameters). Their values do change from sample to sample. Some examples are Statistics are random variables since their values change from sample to sample. Like any other random variable, we will talk about the mean of a statistic and standard deviation of a statistic (called standard error of the statistic ) as well as the distribution of a statistic. 8.1 Sampling Distribution of a Sample Statistic Sampling Distribution of a statistic is the probability distribution of that statistic. [At this point we need to use our imagination] Suppose we decide to use simple random sampling to select ALL POSSIBLE SAMPLES of some fixed size, n. (How many possible SRS’s of size n can you select from a population of size N?) After each sample of n units is selected we calculate its mean and put it in a set, . By this process we obtain a population of all sample means: (How many elements does this population have?) STA3032 Fall 2011, Chapter 8, Page 1 of 12
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Like any other population, the population of sample means has A mean, called the mean of the population of all sample means , denoted by the symbol. A variance called the variance of the population of all sample means , denoted by ; [the square root of this variance is called the standard error of the sample mean, denoted by ] and A distribution called the ( sampling) distribution of sample mean (or simply distribution of) which is the probability distribution of the means of all samples of size n, from the given population. Suppose a simple random sample of size n is selected from a population that has mean μ X and standard deviation σ X . Then, Theorem 1: for any n if , i.e., the population from which the sample is selected has a normal distribution. Now, suppose we know the population mean, μ X , and the population standard deviation σ X but do not know the distribution of the population, or know that it is not normal. Then we cannot use this theorem. Instead we will use the central limit theorem (CLT): Theorem 2: (The Central Limit Theorem): when the distribution population is not known, or known to be not normal approximately, if n is large. Using these two theorems we can write the following results: Let , be the mean of a SRS of size n selected from a population with mean μ X and standard deviation σ X , then IF σ is KNOWN and THEN IF σ is KNOWN and n ≥ 30, THEN approximately IF σ is UNKNOWN and , THEN SOME INFORMATION ABOUT THE T-DISTRIBUTION: STA3032 Fall 2011, Chapter 8, Page 2 of 12
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The t-distribution is bell-shaped, centered at zero. So it looks similar to the standard
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This note was uploaded on 11/12/2011 for the course STA 3032 taught by Professor Kyung during the Summer '08 term at University of Florida.

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Chapter 8 - Chapter 8 Sampling Distribution of Sample...

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