Sample Means. Parameters The mean and standard deviation of a population are parameters. Mu represents the population mean. Sigma represents the population.

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Presentation transcript:

Sample Means

Parameters The mean and standard deviation of a population are parameters. Mu represents the population mean. Sigma represents the population standard deviation. Example: IQ test scores are normally distributed with a mean (mu) of 100 and a standard deviation (sigma) of 15.

Statistics Suppose that x-bar is the mean of an SRS of size n drawn from a large population having mean mu and standard deviation sigma. The mean of the sampling distribution of x-bar is and its standard deviation is adjusted by the square root of the sample size:

Notes on and The sample mean x-bar is an unbiased estimator of the population mean mu. The values of x-bar are less spread out for larger samples. The standard deviation decreases at a rate of. Thus, you must take a sample four times as large to cut the standard deviation in half. The population must be at least ten times the sample size to use the formula sigma divided by radical n. These rules apply no matter what shape the population has.

Sampling Distribution of a Sample Mean If you draw an SRS of any size from a normally distributed population, the sampling distribution of the sample means will also be normally distributed.

Example

Central Limit Theorem If a sample of size n is large enough, a population with mean mu and standard deviation sigma will give a sampling distribution of means which is approximately normal. This is true no matter the shape of the population distribution. Basic rule: if n is greater than or equal to 30, then the Central Limit Theorem applies.

Example

Law of Large Numbers Revisited If you draw observations from any population with a finite mean mu, the mean x-bar of the observed values gets closer and closer to mu as the number of trials approaches the maximum. In the long-run, the more samples of a given size which are taken, the closer the mean of the sampling distribution approaches the true mean of the population and stays there.

Homework Worksheet