Lecture 5 Handout.pdf - ECON 400 Estimation of Means and Proportions Parameters are the numerical measures of a population that describe probability

Lecture 5 Handout.pdf - ECON 400 Estimation of Means and...

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ECON 400 September 17, 2019
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Estimation of Means and Proportions Parameters are the numerical measures of a population that describe probability distributions. We have seen many of these such as p , n , r , N , λ, θ etc. Sample statistics such as ¯ x , s can also be used to make inference about their population counterparts, μ, σ . The parameters p , λ, θ that we have studied are population parameters. Their sample counterparts are ˆ p , ˆ λ, ˆ θ . Statistics September 17, 2019 2 / 48
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Estimation of Means and Proportions The sample statistics are used to estimate their corresponding population parameters. However, this is not a perfect measure, of course. We also need to decide what the best sample statistic is for each population parameter, and we need some way to decide between them. In some cases for example, the sample statistics of the sample average, median, or mode may be he best choice to estimate μ . Statistics September 17, 2019 3 / 48
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Estimation of Means and Proportions No one sample statistic will always be closest to the population parameter (the truth). Let’s consider an example where we know the population parameter, the truth. Consider flipping a coin 4 times. Say we are interested in the number of heads. We know from our knowledge of probability theory that the expected number of heads is two of the four tosses. However our results will vary with finite tosses. Either the mean, median, or mode may be closest. This is easy to discount as we know the population parameter. What if we were discussing the amount of time a person in a survey says they spend on the phone each day. We certainly don’t know what the result would be if we surveyed every possible person, the population. Then our choice is not so obvious. Statistics September 17, 2019 4 / 48
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Sampling Distribution Sample statistics are variables, with their own probability distributions. They are known as Sampling Distributions . We can see the behavior of a sample statistic by taking many samples, and producing a histogram of the result, approximating the probability density function of ¯ x /the sample median/the sample mode/etc. Consider ¯ x from a distribution with mean μ = 10 and σ = 2. If we take many samples from this distribution and calculate the means, we can plot the sample means into a histogram to see how close ¯ x came to the population parameter μ = 10 Statistics September 17, 2019 5 / 48
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Sampling Distribution The following histograms were generated by telling the computer to draw a sample of 50 from a distribution with mean 10 and standard deviation 2, 1000 times. The 1000 sample means, medians, and modes were used to construct the following histograms. Tell me which sample statistic you think performed better. Statistics September 17, 2019 6 / 48
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Estimation of Means and Proportions Statistics September 17, 2019 7 / 48
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Estimation of Means and Proportions The mean and median both performed well. The average of each of the mean, median and modes were: 10.0082, 10.0046, and 5.5204.
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