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Sampling Distributions and Forward Inference

Sampling Distributions and Forward Inference. By simulating the process of drawing random samples of size N from a population with a specific mean and variance, we can learn (a) how much error we can expect on average and

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Sampling Distributions and Forward Inference

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  1. Sampling Distributions and Forward Inference • By simulating the process of drawing random samples of size N from a population with a specific mean and variance, we can learn • (a) how much error we can expect on average and • (b) how much variation there will be on average in the errors observed • Sampling distribution: the distribution of a sample statistic (e.g., a mean) when sampled under known sampling conditions from a known population.

  2. n = 2 mean of sample means = 10 SD of sample means = 4.16 n = 5 mean of sample means = 10 SD of sample means = 2.41 n = 15 mean of sample means = 10 SD of sample means = 0.87

  3. The mean as an unbiased statistic • Note that the distributions of sample means were normal and centered at the mean of the population. • Thus, we “expect” or predict any sample mean to equal the population mean. Why? The average (i.e., typical) sample mean is equal to the mean of the population. • In this sense, the mean is considered an “unbiased” statistic. To the degree that a sample mean differs from a population mean, it is just as likely to be too high or too low.

  4. The variance as a biased statistic • The value of the sample variance we “expect,” however, is not equal to the population variance. • Specifically, a variance observed in a sample drawn from a population will tend to be smaller than the population variance, especially when sample sizes are small.

  5. Standard Error of the Mean • The amount of sampling error we expect is an average error--the standard deviation of the sampling distribution. This represents, on average, how much of an error we should expect. • The standard deviation of a sampling distribution is often called a standard error.

  6. Standard Error of the Mean • When dealing with means, we call this the standard error of the mean. • Its calculation is simple: To find the SD of the sampling distribution of means, all you need to know is the standard deviation of scores in the population, and your sample size. [in forward inference]

  7. Standard Error of the Mean • The equation implies that sampling error decreases as sample size increases. • This is important because it implies that if we want to make sampling error as small as possible, we want to use as large of a sample size as we can manage.

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