Sample Distributions

# Sample Distributions - Sample Distributions The Basic Idea...

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The Basic Idea of Sampling Distributions | The Dispersion in Sampling Distributions | Problem Situations: the Sampling Distribution of the Sample Mean | Problem Situations: The Sampling Distribution of the Sample Proportion | In Conclusion The Basic Idea of Sampling Distributions Last week, we looked at four problem situations in which probability plays a central role in using data to solve problems: Contingency Tables, Expected Value, The Binomial Probability Distribution, and The Normal Probability Distribution. This week, we will narrow our focus to the Normal Probability Distribution, extending its usefulness tremendously by applying it to a very special case: Sampling Distributions. We will look at two problem situations within Sampling Distributions: the Sampling Distribution of the Sample Mean, and the Sampling Distribution of the Sample Proportion. Now the idea of sampling distributions is very simple, but, as we will see, it is powerful and very useful. Here’s the basic idea. Imagine that we are interested in the average income of Keller students; we take a random sample of n = 16 students; and we calculate the mean income reported by the students (after assuring them of confidentiality, and providing a valid reason for our request so that they respond to it and provide an honest answer). The mean that we calculate gives us an idea about the mean income of all Keller students. This is good. But there’s a problem: if we take another sample of n = 16 Keller students, and we compute the mean income for them, the mean will be somewhat different. In fact, imagine taking a great number of random samples of size n = 16, and computing the mean for each of them. The good news is that they are likely to be roughly the same, but the bad news is that they will vary one from another. Which one is right? What is the “true” mean income of the population of Keller students? Seeing the Pattern in Sampling Distributions . The beauty of probability distributions, as we saw last week, is that while any individual random sample’s mean is unknown (before we calculate it), probability distributions have a pattern. And the pattern that emerges, when we take our large number of sample means and we create their distribution, is a normal probability distribution. This distribution, like all distributions, has a mean and a standard deviation. The Central Tendency in Sampling Distributions . Now it turns out, very usefully, that the mean of this new distribution of sample means is equal to the population mean. We represent this as , where represents the mean of the distribution of sample means, and represents the mean of the original population. So in our example, if we look at our distribution of sample means of Keller students’ incomes, the mean of this distribution will point at, and be very close to, the mean income for the population of all Keller students. The Dispersion in Sampling

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## This note was uploaded on 05/23/2010 for the course BUS 1300 taught by Professor Nunya during the Spring '10 term at E. Washington.

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Sample Distributions - Sample Distributions The Basic Idea...

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