Handout3 - Week 3 Statistics of Sampling from...

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Week 3: Statistics of Sampling from Populations (Student t distribution and chi-squared distribution) ChE, BioE, EnvE 213 Prof. Milo Koretsky 1. Definitions: Confidence Interval. The interval in which we are reasonably confident a statistical parameter of a population lies, based on a finite number of samples . Typically we speak of 95% confidence intervals. Central Limit Theorem. The sum of n independently distributed random variables will tend to be normally distributed as n becomes large. Standard Error , x σ : The spread of the sample distribution is given by, x , which is often called the standard error. This terminology reflects the fact that we usually do not know the distribution of the population so that there is uncertainty in the sample data. The variance of sample data is correlated to the degree of uncertainty. It can be shown that the population’s standard deviation, σ , is related to the standard error by n x = Therefore, we have a lower value for standard error, i.e., we can be more certain of our estimate of the mean of the population when the sample size n is larger. So in summary, we can compare the mean and standard deviation of the population to the mean and standard deviation of the sample of size n . as follows Population sample from ( i) large population or (ii) small population with replacement sample from small population, N 1 Mean μ = x = x St. Dev n x = = 1 N n N n x What is the difference between x x , , ? What is the difference between S x , , ? 2. Confidence Interval estimation from Sample Data: Large sample size from any population 1 This form applies to sampling without replacement from a small population, N , 1
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When the sample size is large, we can use the statistics of a normal distributions to describe the sample data. Interesting note: In this case the sample distribution of the sample mean is normal even if the parent distribution is not normal! (Central Limit Theorem). Suppose we are interested in estimating the height of all freshman in college. We take a random sample of 100 and find the sample mean height is = x 67.5 inches and the sample standard deviation is S = 4 inches. Q. What would be our estimate of the population mean height?. Well instead of giving a single value (67.5 inches), it is more appropriate to give a range, e.g., “it is probably between 66 inches and 69 inches.” Such an estimate is an interval estimate. The size of the interval depends on: 1. The sample size, n . Clearly the larger the estimate, the greater the precision. 2. The variability in the population ( σ ), which can be estimated by the variability in the data (S) 3. The level of confidence we wish to have We cannot be 100% confident about the measurement. Say we choose a 95% confidence interval. This is a typical value and can be interpreted as follows: On 95% of the occasions the population mean falls within the interval that is estimated; 5% of the time it does not.
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This note was uploaded on 04/17/2010 for the course CHE 213 taught by Professor Staff during the Winter '08 term at Oregon State.

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Handout3 - Week 3 Statistics of Sampling from...

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