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Estimating unknown parameters

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Advanced High School Statistics

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Slides developed by Mine Çetinkaya-Rundel of OpenIntro, modified by Leah Dorazio for use with AHSS.

The slides may be copied, edited, and/or shared via the CC BY-SA license

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Parameter estimation

  • We are often interested in population parameters.
  • Since complete populations are difficult (or impossible) to collect data on, we use sample statistics as point estimates for the unknown population parameters of interest.
  • Sample statistics vary from sample to sample.
  • Quantifying how sample statistics vary provides a way to estimate the margin of error associated with our point estimate.

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Parameter estimation (cont.)

Before we get to quantifying the variability among samples, let's try to understand how and why point estimates vary from sample to sample.

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Suppose we randomly sample 1000 people and we find the proportion of them that plan to vote Democratic in the next election. Then, we repeat this three more times. Would you expect the proportions to be the exactly same, similar, or very different?

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Parameter estimation (cont.)

Before we get to quantifying the variability among samples, let's try to understand how and why point estimates vary from sample to sample.

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Suppose we randomly sample 1000 people and we find the proportion of them that plan to vote Democratic in the next election. Then, we repeat this three more times. Would you expect the proportions to be the exactly same, similar, or very different?

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Similar, but not exactly the same. Why? Because of sampling variability.

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Here, we are taking a sample and using the sample proportion p̂ as our estimate for the true (unknown) population proportion p.

Similarly, we could use the sample average x̄ to estimate the true population average µ .

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Variability in the estimate p̂

In the last chapter we saw that the standard deviation of p̂ is given by

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So to calculate how much our estimate p̂ is likely to vary from sample to sample, we would like to use this formula. However, what is the problem??

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Variability in the estimate p̂

In the last chapter we saw that the standard deviation of p̂ is given by

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So to calculate how much our estimate p̂ is likely to vary from sample to sample, we would like to use this formula. However, what is the problem??

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p is unknown! That is the quantity we are trying to estimate.

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So what should we do?

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Variability in the estimate p̂

In the last chapter we saw that the standard deviation of p̂ is given by

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​

​

So to calculate how much our estimate p̂ is likely to vary from sample to sample, we would like to use this formula. However, what is the problem??

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p is unknown! That is the quantity we are trying to estimate.

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So what should we do?

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Use p̂ in the formula since it is our best estimate for p. This gives us the SE of p̂.

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