Confidence Intervals - Revision history

Rb: /* Sampling Variability */

2020-04-20T15:56:41Z

‎Sampling Variability

Rb at 15:43, 1 October 2013

2013-10-01T15:43:31Z

Rb at 21:00, 21 September 2013

2013-09-21T21:00:29Z

Rb: Created page with " = Point and Interval Estimation = In Section, it was noted that an estimate of a population parameter is a single number. Bearing in mind the fact th..."

2013-09-21T20:50:06Z

Created page with " = Point and Interval Estimation = In Section, it was noted that an estimate of a population parameter is a single number. Bearing in mind the fact th..."

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 == Sampling Variability ==
-Consider the simplest case of sampling from a normal distribution, <math>X\thicksim N\left( \mu ,\sigma ^{2}\right) </math> with the intention of estimating <math>\mu </math>. The obvious estimator is the sample mean, <math>\bar{X},</math> with sampling distribution <math>\bar{X}\thicksim N\left( \mu ,\sigma^{2}/n\right)</math> (see this [[Statistics_SamplingDistributions#The_Sampling_Distribution|section]] for details). Here, <math>\mu </math> is unknown: one issue will be whether the population parameter <math>\sigma ^{2}</math> is also unknown. From the general principle that population parameters are unknown, the answer should be “yes”, but it will be convenient to (initially) assume, for simplicity, that <math>\sigma ^{2}</math> is actually known.
+Consider the simplest case of sampling from a normal distribution, <math>X\sim N\left( \mu ,\sigma ^{2}\right) </math> with the intention of estimating <math>\mu </math>. The obvious estimator is the sample mean, <math>\bar{X},</math> with sampling distribution <math>\bar{X}\sim N\left( \mu ,\sigma^{2}/n\right)</math> (see this [[Statistics_SamplingDistributions#The_Sampling_Distribution|section]] for details). Here, <math>\mu </math> is unknown: one issue will be whether the population parameter <math>\sigma ^{2}</math> is also unknown. From the general principle that population parameters are unknown, the answer should be “yes”, but it will be convenient to (initially) assume, for simplicity, that <math>\sigma ^{2}</math> is actually known.
 The variance of the distribution of <math>\bar{X}</math> measures the dispersion of this distribution, and thus gives some idea of the range of values of <math>\bar{X}</math> that might be obtained in drawing different samples of size <math>n</math>. The question is, though, what is an extreme or untypical value of <math>\bar{X}</math>? The conventional way to define this is by using a multiple of the '''standard error '''of <math>\bar{X},</math> <math>SE\left( \bar{X}\right)</math>, as a measure of sampling variability. Then, extreme values do not belong to the interval

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 <math>\bar{x}\pm k SE\left( \bar{X}\right) ,</math>
-with the value <math>k</math> chosen suitably. Then, the factor <math>\pm k SE \left( \bar{X}\right) </math> is the measure of sampling variability around <math>\bar{x}</math>. Clearly, the parameter <math>SE\left( \bar{X}\right) </math> has to be known in order for the measure of `sampling variability to be computed. This is why we make the initial assumption of a known <math>\sigma</math>, such that we can easily calculate <math>SE\left( \bar{X}\right) = \sqrt(\sigma^2 / n)</math>. As you can see this measure partially reflects the inherent variability in the population, as represented by the population variance <math>\sigma ^{2}</math>. <ref>Usually <math>\sigma^2</math> is, as <math>\mu</math>, unknown and it has to be estimated by using the sample variance <math>s^{2}</math> instead. It is via the use of <math>s^{2}</math> that the measure of sampling variability is being calculated from a single sample. This will be discussed below.
+with the value <math>k</math> chosen suitably. Then, the factor <math>\pm k SE \left( \bar{X}\right) </math> is the measure of sampling variability around <math>\bar{x}</math>. Clearly, the parameter <math>SE\left( \bar{X}\right) </math> has to be known in order for the measure of ‘sampling variability to be computed. This is why we make the initial assumption of a known <math>\sigma</math>, such that we can easily calculate <math>SE\left( \bar{X}\right) = \sqrt(\sigma^2 / n)</math>. As you can see this measure partially reflects the inherent variability in the population, as represented by the population variance <math>\sigma ^{2}</math>. <ref>Usually <math>\sigma^2</math> is, as <math>\mu</math>, unknown and it has to be estimated by using the sample variance <math>s^{2}</math> instead. It is via the use of <math>s^{2}</math> that the measure of sampling variability is being calculated from a single sample. This will be discussed below.
 </ref>
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 This standard error has to be known in order to calculate the interval estimate. But, it has frequently been emphasised that population parameters are in general unknown. So, '''assuming''' that <math>\sigma ^{2}</math> '''is known''' has to be seen as an unrealistic but simplifying assumption. This assumption allows us to see the principles behind the construction of an interval estimator or confidence interval without other complications. We shall now have to investigate the consequences of relaxing this assumption.
 = Unknown <math>\sigma</math> and the <math>t</math> distribution =
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 Sometimes the estimator of <math>SE\left( \bar{X}\right) </math> is denoted <math>\widehat{SE}\left( \bar{X}\right) </math>, with estimate <math>\widehat{se}\left( \bar{X}\right) </math>, but these are a bit clumsy to use in general.
-= Using <math>\widehat{SE}\left( \bar{X}\right)</math> =
+== Using <math>\widehat{SE}\left( \bar{X}\right)</math> ==
 So, instead of using
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 This seems like a simple solution, but unfortunately it is not an innocuous solution. This is because the distribution of the random variable <math>T</math> combines two sources of randomness, <math>\bar{X}</math> and <math>S^{2}</math>. As a result, the distribution of <math>T</math> is '''NOT''' <math>N\left( 0,1\right) </math>.
-The distribution of <math>T</math> was discovered by a statistician called W.S. Gossett who worked at the Guinness Brewery in Dublin, and wrote under the pen name ‘Student’. The distribution of <math>T</math> is called ''Student''`''s'' <math>t</math> ''distribution'', or more commonly, just the <math>t</math> ''distribution''. This distribution depends on a parameter, just like other distributions: here, the parameter is called the ''degrees of freedom''. Before discussing the properties of this distribution, we summarise the distribution statement:
+The distribution of <math>T</math> was discovered by a statistician called W.S. Gossett who worked at the Guinness Brewery in Dublin, and wrote under the pen name ‘Student’. The distribution of <math>T</math> is called ''Student''‘''s'' <math>t</math> ''distribution'', or more commonly, just the <math>t</math> ''distribution''. This distribution depends on a parameter, just like other distributions: here, the parameter is called the ''degrees of freedom''. Before discussing the properties of this distribution, we summarise the distribution statement:
 <ul>
 <li><p>in random sampling from <math>N\left( \mu ,\sigma ^{2}\right) </math>,</p>
-<p><math>T=\dfrac{\bar{X}-\mu }{\widehat{SE}\left( \bar{X}\right) }%
+<p><math>T=\dfrac{\bar{X}-\mu }{\widehat{SE}\left( \bar{X}\right) }\thicksim t_{n-1},</math></p></li>
 <li><p>the <math>t</math> distribution with <math>n-1</math> degrees of freedom.</p></li></ul>
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 <td align="center">Appendix</td>
 <td align="center">''Other Texts''</td>
-<td align="center"><source enclose="none">tinv(0.05,df)</source></td>
+<td align="center"><code>tinv(0.05,df)</code></td>
 </tr>
 <tr class="even">
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 This normal-based confidence interval is narrower than the <math>t</math> based one: this is the consequence of the extra dispersion of the <math>t</math> distribution compared with the <math>N\left( 0,1\right) </math> distribution. The underlying reason for the increased dispersion is, of course, the fact that we do not know the value of <math>\sigma^2</math>. Although we have a sampel estimate <math>s^2</math> we need to acknowledge that there is sampling variation with respect to this estimate on top of the sampling variation we have for <math>\bar{x}</math>.
 = Realationships between Normal, <math>t</math> and <math>\chi ^{2}</math> distributions =
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 The usual sampling distribution of the sample mean <math>\bar{X}</math> also assumes sampling from a normal distribution. In an earlier [[Statistics_SamplingDistributions#Sampling_from_Non-Normal_distributions| Section]], the effect of sampling from a non-normal distribution was discussed. Provided that one draws a random sample from a population with mean <math>\mu </math> and variance <math>\sigma ^{2}</math>, the Central Limit Theorem assures us that
-<math>\dfrac{\bar{X}-\mu }{\sigma /\sqrt{n}}\thicksim N\left( 0,1\right) \;\;\;\;\;%
+<math>\dfrac{\bar{X}-\mu }{\sigma /\sqrt{n}}\thicksim N\left( 0,1\right) \;\;\;\;\;\text{approximately,}</math>
 or, equivalently,
-<math>\bar{X}\thicksim N\left( \mu ,\dfrac{\sigma ^{2}}{n}\right) \;\;\;\;\;\text{%
+<math>\bar{X}\thicksim N\left( \mu ,\dfrac{\sigma ^{2}}{n}\right) \;\;\;\;\;\text{approximately.}</math>
 There is the presumption that the quality of the approximation improves as <math>n\rightarrow \infty </math>, that is, as the sample size increases. The larger the <math>n</math>, the better.
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 For such an apparently large sample size, this is quite a wide confidence interval. Better precision of estimation would require a larger sample size.
 = Footnotes =

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 <ul>
 <li><p>a 95% confidence interval for <math>\mu </math> is given by the random interval or interval estimator</p>
-<p><math>\left[ \bar{X}-1.96SE\left( \bar{X}\right) ,\;\bar{X}+1.96\limfunc{%
+<p><math>\left[ \bar{X}-1.96SE\left( \bar{X}\right) ,\;\bar{X}+1.96 SE\left( \bar{X}\right) \right]</math></p></li></ul>
 == The Interval Estimate ==
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 This seems like a simple solution, but unfortunately it is not an innocuous solution. This is because the distribution of the random variable <math>T</math> combines two sources of randomness, <math>\bar{X}</math> and <math>S^{2}</math>. As a result, the distribution of <math>T</math> is '''NOT''' <math>N\left( 0,1\right) </math>.
-The distribution of <math>T</math> was discovered by a statistician called W.S. Gossett who worked at the Guinness Brewery in Dublin, and wrote under the pen name ‘Student’. The distribution of <math>T</math> is called <math>\emph{Student}`\emph{s}</math> <math>t</math> ''distribution'', or more commonly, just the <math>t</math> ''distribution''. This distribution depends on a parameter, just like other distributions: here, the parameter is called the ''degrees of freedom''. Before discussing the properties of this distribution, we summarise the distribution statement:
+The distribution of <math>T</math> was discovered by a statistician called W.S. Gossett who worked at the Guinness Brewery in Dublin, and wrote under the pen name ‘Student’. The distribution of <math>T</math> is called ''Student''`''s'' <math>t</math> ''distribution'', or more commonly, just the <math>t</math> ''distribution''. This distribution depends on a parameter, just like other distributions: here, the parameter is called the ''degrees of freedom''. Before discussing the properties of this distribution, we summarise the distribution statement:
 <ul>
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 The presence of <math>n-1</math> degrees of freedom can be explained in a number of ways. One explanation is based on the expression for the estimator <math>S^{2}</math>:
-<math>S^{2}=\dfrac{1}{n-1}\dsum\limits_{i=1}^{n}\left( X_{i}-\bar{X}\right) ^{2}.</math>
+<math>S^{2}=\dfrac{1}{n-1}\sum\limits_{i=1}^{n}\left( X_{i}-\bar{X}\right) ^{2}.</math>
 Here, the divisor <math>n-1</math> in this expression leads to the degrees of freedom for <math>T</math>. This is actually the main justification for using a divisor <math>n-1</math> in a sample variance <math>S^{2}</math> rather than <math>n</math>, although using <math>n-1</math> also leads to an unbiased estimator.
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 So, the variance is only defined for <math>\nu \geqslant 2</math>, and
-<math>\limfunc{var}\left[ T\right] \geqslant 1,</math>
+<math>var\left[ T\right] \geqslant 1,</math>
 which explains the extra dispersion of the <math>t</math> distribution relative to <math>N\left( 0,1\right) </math>.
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 One way of doing this is to compare some “typical” probabilities. The difficulty with this is that one cannot produce a table of <math>t</math> distribution probabilities
-<math>\Pr \left( T\leqslant t\right) \;\;\;\text{for\ \ \ }T\thicksim t_{\nu }</math>
+<math>\Pr \left( T\leqslant t\right) \;\;\;\text{for }T\thicksim t_{\nu }</math>
 to compare with those for
-<math>\Pr \left( Z\leqslant z\right) \;\;\;\text{for\ \ \ }Z\thicksim N\left(0,1\right) :</math>
+<math>\Pr \left( Z\leqslant z\right) \;\;\;\text{for }Z\thicksim N\left(0,1\right) :</math>
 there would have to be a table for each value of <math>\nu</math>. If you need a precise probabilities form the t-distribution you need to use a software like EXCEL. The following table gives some numerical values:
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 Since
-<math>E\left[ P\right] =\pi ,\;\;\;\;\limfunc{var}\left[ P\right] =\dfrac{\pi \left( 1-\pi \right) }{n},</math>
+<math>E\left[ P\right] =\pi ,\;\;\;\;var\left[ P\right] =\dfrac{\pi \left( 1-\pi \right) }{n},</math>
-<math>\dfrac{P-\pi }{\sqrt{\limfunc{var}\left[ P\right] }}\thicksim N\left(0,1\right) \;\;\;\;\;\text{approximately.}</math>
+<math>\dfrac{P-\pi }{\sqrt{var\left[ P\right] }}\thicksim N\left(0,1\right) \;\;\;\;\;\text{approximately.}</math>
 So, one could hope to use this to provide an approximate confidence interval for <math>\pi </math>. There is a minor complication here in that <math>var\left[ P\right] </math> depends on the unknown parameter <math>\pi </math>, but, there is an obvious estimator (<math>P)</math> available which could used to provide an estimator of <math>var\left[ P\right] </math>.
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 = Footnotes =
   <references />