334 lines
10 KiB
Plaintext
334 lines
10 KiB
Plaintext
<section xml:id="sec-Confidence-Intervals" xmlns:xi="http://www.w3.org/2001/XInclude">
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<title>Confidence Intervals</title>
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<p>
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When estimating the value of some parameter, the maximum likelihood estimation provides the single estimate with highest possible likelihood.
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But we might prefer to have a range of values, with some quantitative measure of how certain we are that the true parameter value could be in the range.
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</p>
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<p>
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Suppose some measurement has a mean <m>\mu</m> and variance <m>\sigma^2</m> across the entire population.
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Independent measurements <m>X_1, X_2, \dotsc, X_n</m> and calculate the sample mean
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<md>
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<mrow> A_n = \frac{X_1 + \dotsb + X_n}{n}. </mrow>
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</md>
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Then, according to <xref ref="thm-CLT"/>, <m>A_n \approx \Norm\left(\mu, \frac{\sigma^2}{n}\right)</m> as long as we take a sufficient number of measurements.
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However, we don't have access to the true values of <m>\mu, \sigma^2</m>.
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</p>
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<p>
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We can make an estimate <m>s^2</m> of <m>\sigma^2</m> from the collected data, so we know how spread out the normal distribution should be.
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</p>
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<definition xml:id="def-sample-variance">
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<statement>
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<p>
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Let <m>X_1, X_2, \dotsc, X_n</m> be independent samples taken from a population.
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Let <m>A_n = \frac{X_1 + \dotsb + X_n}{n}</m> be the sample mean.
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The <term>sample variance</term> is:
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<md>
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<mrow> s^2 = \frac{\sum \left(X_i - A_n\right)^2}{n - 1} = \frac{\left(\sum X_i^2\right) - n \left(A_n\right)^2}{n - 1}. </mrow>
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</md>
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The expression:
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<md>
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<mrow> \sqrt{\frac{s^2}{n}} = \frac{s}{\sqrt{n}} </mrow>
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</md>
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is called the <term>standard error of the mean</term>.
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</p>
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</statement>
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</definition>
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<p>
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Note that <m>s^2</m> is an estimate of the variance of each individual measurement <m>X_i</m> taken from the population.
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Therefore, <m>\frac{s^2}{n}</m> is an estimate of the variance of the mean <m>A_n</m>.
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Hence "standard error of the mean" is an estimate of the standard deviation of the mean.
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</p>
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<p>
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Next, we want to pick a distance <m>c</m> so that our measured sample mean <m>A_n</m> is within <m>c</m> of the true population mean <m>\mu</m> with a high probability.
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We can pick some value <m>\alpha \in (0, 1)</m>, and let <m>c_{\alpha} = \Phi^{-1}\left(1 - \frac{\alpha}{2}\right)</m>.
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Then the interval:
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<md>
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<mrow> \left[A_n - c_{\alpha} \left(\frac{s}{\sqrt{n}}\right), A_n + c_{\alpha} \left(\frac{s}{\sqrt{n}}\right)\right] </mrow>
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</md>
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is called a <term>confidence interval</term> with confidence level <m>1 - \alpha</m>.
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</p>
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<!-- TODO: explanation of c_{\alpha} -->
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<theorem xml:id="thm-CI">
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<statement>
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<p>
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Let <m>X_1, X_2, \dotsc, X_n</m> be independent samples taken from a population.
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Suppose <m>A_n</m> is the sample mean and <m>s^2</m> the sample variance.
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Then the 95% confidence limits are:
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<md>
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<mrow> \mu_{\ell} \amp = A_n - 1.96 \left(\frac{s}{\sqrt{n}}\right) </mrow>
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<mrow> \mu_{h} \amp = A_n + 1.96 \left(\frac{s}{\sqrt{n}}\right) </mrow>
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</md>
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</p>
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</statement>
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</theorem>
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<example>
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<statement>
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<p>
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Suppose measurements are taken of the heights of several plants in a garden plot.
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The measurements are given below.
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(Note: for the sake of a managable calculation, only five measurements are given.
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However, for <xref ref="thm-CLT"/> to apply, the number of measurements would need to be greater.)
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</p>
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<table>
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<title>Plant Height Measurements</title>
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<tabular halign="center">
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<row bottom="minor">
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<cell>Plant <m>i</m></cell>
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<cell>Height <m>H_i</m> (in)</cell>
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</row>
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<row>
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<cell><m>1</m></cell>
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<cell><m>9.2</m></cell>
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</row>
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<row>
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<cell><m>2</m></cell>
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<cell><m>9.8</m></cell>
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</row>
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<row>
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<cell><m>3</m></cell>
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<cell><m>10.1</m></cell>
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</row>
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<row>
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<cell><m>4</m></cell>
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<cell><m>9.5</m></cell>
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</row>
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<row>
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<cell><m>5</m></cell>
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<cell><m>10.4</m></cell>
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</row>
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</tabular>
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</table>
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<p>
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Then the sample mean is:
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<md>
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<mrow> A_5 = \frac{H_1 + \dotsb + H_5}{5} \amp = \frac{9.2 + \dotsb + 10.4}{5} = 9.8. </mrow>
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</md>
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The sum of the squares is:
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<md>
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<mrow> \sum H_i^2 \amp = H_1^2 + \dotsb + H_5^2 = 9.2^2 + \dotsb + 10.4^2 = 481.1 </mrow>
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</md>
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So the sample variance is:
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<md>
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<mrow> s^2 = \frac{\left(\sum H_i\right)^2 - 5 (A_5)^2}{5 - 1} = \frac{481.1 - 5(9.8)^2}{4} \approx 0.225 </mrow>
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</md>
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Now, the standard error of the mean is:
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<md>
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<mrow> \frac{s}{\sqrt{n}} = \sqrt{\frac{0.225}{5}} \approx 0.212 </mrow>
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</md>
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Finally, we can give 95% confidence limits:
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<md>
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<mrow> \mu_{\ell} \amp = 9.8 - 1.96(0.212) \approx 9.38 </mrow>
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<mrow> \mu_{h} \amp = 9.8 + 1.96(0.212) \approx 10.22 </mrow>
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</md>
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</p>
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</statement>
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</example>
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<p>
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If the individual measurements <m>X_1, \dotsc, X_n</m> are indicator random variables, then the mean <m>A_n</m> represents the proportion which take on the value 1.
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For example, if the <m>X_i</m>'s indicate heads on coin flips, then <m>A_n</m> represents the proportion which come up heads.
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In this setting, we have a nice shortcut formula (<xref ref="example-binomial-variance"/>) for the variance of <m>S_n = X_1 + \dotsc + X_n</m>.
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We can make use of this formula to get the variance of <m>A_n</m> without going through the sample variance calculation above:
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<md>
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<mrow> \Var(S_n) \amp = np(1 - p) </mrow>
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<mrow> \Var(A_n) \amp = \Var\left(\frac{S_n}{n}\right) </mrow>
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<mrow> \amp = \frac{1}{n^2} \Var\left(S_n\right) </mrow>
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<mrow> \amp = \frac{1}{n^2} (np(1 - p)) </mrow>
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<mrow> \amp = \frac{p(1 - p)}{n}. </mrow>
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</md>
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Since we won't have access to the true value of <m>p</m>, we can use the maximum likelihood estimate <m>\widehat{p}</m>.
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</p>
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<example>
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<statement>
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<p>
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Suppose in a sample of 100 people, 40 have a particular trait.
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The maximum likelihood estimate for the probability <m>p</m> of a random person having the trait is <m>\widehat{p} = \frac{40}{100} = 0.4</m>.
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So the standard error will be:
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<md>
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<mrow> \sqrt{\frac{(0.4)(1 - 0.4)}{100}} = \sqrt{\frac{0.24}{100}} \approx 0.049. </mrow>
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</md>
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Therefore, 95% confidence limits for <m>p</m> are:
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<md>
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<mrow> p_{\ell} \amp = 0.4 - 1.96(0.049) \approx 0.304 </mrow>
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<mrow> p_{h} \amp = 0.4 + 1.96(0.049) \approx 0.496 </mrow>
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</md>
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</p>
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</statement>
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</example>
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<exercises xml:id="exercises-Confidence-Intervals">
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<exercise>
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<statement>
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<p>
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Suppose we flip a coin 100 times and count 60 heads.
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Let <m>p</m> be the (unknown) probability that the coin comes up heads on a flip.
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Give an approximate 95% confidence interval for the value of <m>p</m>.
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</p>
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</statement>
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<answer>
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<p>
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<m>[0.504, 0.696]</m>.
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</p>
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</answer>
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</exercise>
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<exercise>
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<introduction>
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<p>
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Suppose in a sample of 100 people, 12 are left-handed.
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</p>
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</introduction>
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<task>
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<statement>
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<p>
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Give a 95% confidence interval for the proportion <m>p</m> of left-handed people.
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</p>
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</statement>
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<answer>
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<p>
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<m>[0.056, 0.184]</m>.
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</p>
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</answer>
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</task>
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<task>
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<statement>
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<p>
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Give a 90% confidence interval.
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</p>
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</statement>
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<answer>
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<p>
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<m>[0.066, 0.174]</m>.
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</p>
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</answer>
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</task>
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</exercise>
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<exercise>
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<statement>
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<p>
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The weights of five mice are measured and recorded below.
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Give a 95% confidence interval for the sample mean weight of mice.
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(Pretend 5 measurements is large enough for the CLT to apply.)
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</p>
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<table>
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<title></title>
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<tabular halign="center">
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<row header="yes" bottom="minor">
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<cell>mouse <m>i</m></cell>
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<cell>weight <m>W_i</m> (g)</cell>
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</row>
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<row>
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<cell>1</cell>
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<cell>26</cell>
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</row>
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<row>
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<cell>2</cell>
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<cell>32</cell>
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</row>
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<row>
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<cell>3</cell>
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<cell>33</cell>
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</row>
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<row>
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<cell>4</cell>
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<cell>20</cell>
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</row>
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<row>
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<cell>5</cell>
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<cell>29</cell>
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</row>
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</tabular>
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</table>
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</statement>
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<answer>
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<p>
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<!-- avg weight = 28 --> <!-- sum of squares = 4030 --> <!-- s^2 = 27.5 --> <!-- \sqrt{s^2/n} \approx 2.35 --> <m>[23.39, 32.61]</m>.
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</p>
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</answer>
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</exercise>
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<exercise>
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<statement>
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<p>
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The heights of five plants are measured and recorded below.
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Give a 95% confidence interval around the sample mean for the heights of the plants.
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(Pretend 5 measurements is large enough for the CLT to apply.)
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</p>
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<table>
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<title></title>
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<tabular halign="center">
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<row header="yes" bottom="minor">
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<cell>plant <m>i</m></cell>
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<cell>height <m>H_i</m> (in)</cell>
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</row>
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<row>
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<cell>1</cell>
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<cell>15</cell>
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</row>
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<row>
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<cell>2</cell>
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<cell>14</cell>
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</row>
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<row>
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<cell>3</cell>
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<cell>18</cell>
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</row>
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<row>
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<cell>4</cell>
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<cell>21</cell>
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</row>
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<row>
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<cell>5</cell>
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<cell>17</cell>
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</row>
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</tabular>
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</table>
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</statement>
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<answer>
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<p>
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<!-- avg weight = 17 --> <!-- sum of squares = 1475 --> <!-- s^2 = 7.5 --> <!-- \sqrt{s^2/n} \approx 1.22 --> <m>[14.61, 19.39]</m>.
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</p>
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</answer>
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</exercise>
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</exercises>
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</section> |