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Sampling Distributions

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hosea
22 diapositives · 381 Views · Sep 10, 2014

This guide explores the concepts of sampling distributions, statistics, and the Central Limit Theorem. It clarifies key parameters: mean (μ), standard deviation (σ), and sample proportions (p). Learn how to calculate sample means and proportions from different samples and understand their distributions. Discover the importance of sample size and how the normal distribution approximates sample means when the sample size is large enough (n ≥ 30). Engage with practical examples using small populations to illustrate theoretical concepts effectively.

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Sampling Distributions
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  1. Sampling Distributions The “What If?” game

  2. Parameters • Values that describe a characteristic of the POPULATION • Most of the time, there is no way for us to really know what this number is • μ = mean • σ = standard deviation • p (or π) = proportion • α = y-int. of LSRL • β = slope of LSRL Of the POPULATION

  3. Statistics • Value computed from a sample • = mean • s = standard deviation • = proportion • a = y-int of LSRL • b = slope of LSRL Of the SAMPLE

  4. Distribution • All the values a variable can take, and the number of times that it takes each value • A distribution is just a picture of the data

  5. Put them together

  6. Sampling Distribution • The distribution of possible values of a statistic, from all the possible samples of the same size, from the same population

  7. Sampling Distribution • I take a sample from a population, calculate a statistic. • What if I could take every possible sample of that size from that population, and calculate the same statistic every time, and then plot all of these values • The picture (distribution) of all those statistics from all the samples (sampling)

  8. Sampling Distributions • We are going to be concerned with the distributions of sample proportions, , and the distributions of sample means, • Since we often don’t know the true proportion, p, or the true mean, μ, the only information we have to base decisions on is the statistics and

  9. Sample Proportions • = # in the sample that have this characteristic Sample size

  10. Assumptions - Proportions • If we assume that our sample is not too big, less than 10% of the population so we can have independence • And • If we assumer our sample is big enough, where np > 10 and n(1 – p) > 10 • Then we can use a normal curve to approximate the sampling distribution

  11. If we’re going to use a normal model, we need: • Mean • Standard deviation

  12. Suppose we have a population of six people: Alice, Ben, Charles, Denise, Edward, & Frank We are interested in the proportion of females. This is called the parameter of interest. What is the proportion of females? Draw samples of two from this population. How many different samples are possible? 6C2 =15

  13. Find the 15 different samples that are possible & find the sample proportion of the number of females in each sample Alice & Ben .5 Alice & Charles .5 Alice & Denise 1 Alice & Edward .5 Alice & Frank .5 Ben & Charles 0 Ben & Denise .5 Ben & Edward 0 Ben & Frank 0 Charles & Denise .5 Charles & Edward 0 Charles & Frank 0 Denise & Edward .5 Denise & Frank .5 Edward & Frank 0 Find the mean & standard deviation of all p-hats.

  14. Once you have your distribution of all the sample proportions in the whole wide world, from this size sample from the population…

  15. The mean of all the sample proportions (statistics) in the whole wide world, all the p-hats, is equal to the value of the proportion for the whole population (parameter) These are found on the formula chart!

  16. Sample Means • = Add all the individual values Divide by how many there are

  17. Assumptions - Means • We want to be able to use a normal model • Central Limit Theorem – When n is sufficiently large, the sampling distribution of is well approximated by a normal curve, even when the population distribution itself is not normal

  18. Assumptions - Means • So, what is “sufficiently large”? • n ≥ 30

  19. Consider the population of 5 fish in my pond – the length of fish (in inches): 2, 7, 10, 11, 14 What is the mean and standard deviation of this population?

  20. Let’s take samples of size 2 (n = 2) from this population: How many samples of size 2 are possible? 5C2 = 10 Find all 10 of these samples and record the sample means. What is the mean and standard deviation of the sample means? sx = 2.4919 mx = 8.8

  21. Repeat this procedure with sample size n= 3 What is the mean and standard deviation of the sample means? Find all 10 of these samples and record the sample means. sx = 1.6613 mx = 8.8

  22. The mean of all the sample means (statistics) in the whole wide world, all the x-bars, is equal to the value of the meanfor the whole population (parameter) mx= m sx= These are found on the formula chart!