Point Estimation - Revision history

Rb: /* Estimating the population proportion */

2014-09-11T10:42:52Z

‎Estimating the population proportion

Rb: /* Estimating the population proportion */

2014-09-11T10:41:15Z

‎Estimating the population proportion

Rb: /* Point Estimation */

2014-09-11T10:39:52Z

‎Point Estimation

Rb: /* Principle of Estimation */

2014-09-11T10:39:31Z

‎Principle of Estimation

Rb: /* Principle of Estimation */

2014-09-11T10:37:11Z

‎Principle of Estimation

Rb: /* Example: Normal approximation II */

2014-09-11T10:33:13Z

‎Example: Normal approximation II

Rb: /* Example: Normal approximation II */

2014-09-11T10:32:46Z

‎Example: Normal approximation II

Rb: /* Example: Normal approximation II */

2014-09-11T10:28:15Z

‎Example: Normal approximation II

Rb: /* Footnotes */

2013-10-01T15:29:54Z

‎Footnotes

Rb at 15:29, 1 October 2013

2013-10-01T15:29:39Z

@@ Line 233: / Line 233: @@
 <math>0,\dfrac{1}{n},\dfrac{2}{n},...,\dfrac{n-1}{n},1</math>
-and is the sample proportion <math>P</math> of successes. Recall that in an earlier [Statistics_SamplingDistributions#The_Binomial_case|Section], the total number of successes in <math>n</math> Bernoulli trials, <math>T</math>, was shown to have a Binomial distribution, and that <math>\bar{X}</math> was represented as
+and is the sample proportion <math>P</math> of successes. Recall that in an earlier [[Statistics_SamplingDistributions#The_Binomial_case|Section]], the total number of successes in <math>n</math> Bernoulli trials, <math>T</math>, was shown to have a Binomial distribution, and that <math>\bar{X}</math> was represented as
 <math>\bar{X}=\dfrac{T}{n}.</math>
@@ Line 247: / Line 247: @@
 Here, <math>t</math> is the total number of successes in a given sample.
-So, we know (from this [Statistics_SamplingDistributions#The_Binomial_case|Section]) that
+So, we know (from this [[Statistics_SamplingDistributions#The_Binomial_case|Section]]) that
 <math>\begin{aligned}
@@ Line 255: / Line 255: @@
 so that <math>P</math> is an unbiased estimator of <math>\pi</math>.
-From the results in [Statistics_SamplingDistributions#The_Binomial_case|Section], we know that the sampling distribution of <math>P</math> is related to that of the Binomial random variable <math>T</math>. Suppose that we draw a random sample of size <math>50</math> from the distribution of a Bernoulli random variable with success probability <math>\pi =0.6</math>, say. What is
+From the results in [[Statistics_SamplingDistributions#The_Binomial_case|Section]], we know that the sampling distribution of <math>P</math> is related to that of the Binomial random variable <math>T</math>. Suppose that we draw a random sample of size <math>50</math> from the distribution of a Bernoulli random variable with success probability <math>\pi =0.6</math>, say. What is
 <math>\Pr \left( P\leqslant 0.4\right),</math>
@@ Line 282: / Line 282: @@
 These days it is increasingly difficult to imagine that we will not have a computer at hand should we be in a position to have to make a calculation like this <ref>The obvious example being a University exam!
-</ref>. When introducing the [Probability_Norm#The_Normal_distribution_as_an_approximation_to_the_Binomial_distribution|Normal distribution] we argued that it can be used to approximate a binomial distribution. The argument was, at the time, based on a visual case. We now know that we can argue that since the sample proportion <math>P</math> is a sample mean, and since the [Statistics_SamplingDistributions#Sampling_from_Non-Normal_distributions|Central Limit Theorem] can be applied to the sample mean. Hence we have
+</ref>. When introducing the [[Probability_Norm#The_Normal_distribution_as_an_approximation_to_the_Binomial_distribution|Normal distribution]] we argued that it can be used to approximate a binomial distribution. The argument was, at the time, based on a visual case. We now know that we can argue that since the sample proportion <math>P</math> is a sample mean, and since the [[Statistics_SamplingDistributions#Sampling_from_Non-Normal_distributions|Central Limit Theorem]] can be applied to the sample mean. Hence we have
 <math>\dfrac{P-\pi }{\sqrt{\pi \left( 1-\pi \right) /n}}\thicksim N\left(0,1\right) \;\;\;\text{approximately}</math>

@@ Line 229: / Line 229: @@
 = Estimating the population proportion =
-In an earlier [Statistics_SamplingDistributions#The_Binomial_case|Section] we showed how a Bernoulli random variable <math>X</math>, taking on the value <math>0</math> with probability <math>1-\pi </math> and the value <math>1</math> with probability <math>\pi</math>, could represent a population of zeros and ones, with <math>1</math> representing the fact that a household in the population owned a Tablet computer, and a <math>0</math> that it doesn’t. Here, <math>\pi </math> is the unknown population proportion of Tablet computer ownership. Above it was shown that the sample mean <math>\bar{X}</math> of a random sample of size <math>n</math> from a Bernoulli distribution takes on values
+In an earlier [[Statistics_SamplingDistributions#The_Binomial_case|Section]] we showed how a Bernoulli random variable <math>X</math>, taking on the value <math>0</math> with probability <math>1-\pi </math> and the value <math>1</math> with probability <math>\pi</math>, could represent a population of zeros and ones, with <math>1</math> representing the fact that a household in the population owned a Tablet computer, and a <math>0</math> that it doesn’t. Here, <math>\pi </math> is the unknown population proportion of Tablet computer ownership. Above it was shown that the sample mean <math>\bar{X}</math> of a random sample of size <math>n</math> from a Bernoulli distribution takes on values
 <math>0,\dfrac{1}{n},\dfrac{2}{n},...,\dfrac{n-1}{n},1</math>

@@ Line 96: / Line 96: @@
 == Point Estimation ==
-The adjective “point” seems to play no role in the discussion. It serves mainly to distinguish the ideas from another concept of estimation called ''interval estimation'', which is discussed in [Confidence_Intervals|another section]. The distinction is simply that a '''point estimate '''is a single number, whilst an '''interval estimate''' is an interval of numbers. As will be discussed in the [Confidence_Intervals|confidence interval section] the latter has the significant advantage that it gives a clear indication that there is n certainty about the value of the unknown population parameter.
+The adjective “point” seems to play no role in the discussion. It serves mainly to distinguish the ideas from another concept of estimation called ''interval estimation'', which is discussed in [[Confidence_Intervals|another section]]. The distinction is simply that a '''point estimate '''is a single number, whilst an '''interval estimate''' is an interval of numbers. As will be discussed in the [[Confidence_Intervals|confidence interval section]] the latter has the significant advantage that it gives a clear indication that there is n certainty about the value of the unknown population parameter.
 = Properties of Estimators =

@@ Line 88: / Line 88: @@
 Most of the quantities in this table have appeared in the discussion of [[Descriptive| descriptive statistics]]. The (sample) correlation coefficient <math>R</math> is shown explicitly as a function of the sample covariance and the sample variances to emphasise that the estimator of the population correlation coefficient <math>\rho </math> is derived from other estimators. Out of the quantities in this table, we shall be concerned almost exclusively with the behaviour of the sample mean <math>\bar{X}</math>. The sample variance <math>S^{2}</math> will also appear in a minor role, but none of the other quantities will concern us further.
-One other small detail to explain in this table is the absence of an expression for the estimator or estimate of the population proportion. This is because the sample mean is the required quantity, when sampling from a population of zeros and ones or equivalently from the distribution of a Bernoulli random variable - see [Statistics_SamplingDistributions#The_Binomial_case| this Section]. The possible sample values for <math>\bar{x}</math> are
+One other small detail to explain in this table is the absence of an expression for the estimator or estimate of the population proportion. This is because the sample mean is the required quantity, when sampling from a population of zeros and ones or equivalently from the distribution of a Bernoulli random variable - see [[Statistics_SamplingDistributions#The_Binomial_case| this Section]]. The possible sample values for <math>\bar{x}</math> are
 <math>0,\dfrac{1}{n},\dfrac{2}{n},...,\dfrac{n-1}{n},1:</math>

@@ Line 86: / Line 86: @@
 </table>
-Most of the quantities in this table have appeared in the discussion of [Descriptive| descriptive statistics]. The (sample) correlation coefficient <math>R</math> is shown explicitly as a function of the sample covariance and the sample variances to emphasise that the estimator of the population correlation coefficient <math>\rho </math> is derived from other estimators. Out of the quantities in this table, we shall be concerned almost exclusively with the behaviour of the sample mean <math>\bar{X}</math>. The sample variance <math>S^{2}</math> will also appear in a minor role, but none of the other quantities will concern us further.
+Most of the quantities in this table have appeared in the discussion of [[Descriptive| descriptive statistics]]. The (sample) correlation coefficient <math>R</math> is shown explicitly as a function of the sample covariance and the sample variances to emphasise that the estimator of the population correlation coefficient <math>\rho </math> is derived from other estimators. Out of the quantities in this table, we shall be concerned almost exclusively with the behaviour of the sample mean <math>\bar{X}</math>. The sample variance <math>S^{2}</math> will also appear in a minor role, but none of the other quantities will concern us further.
 One other small detail to explain in this table is the absence of an expression for the estimator or estimate of the population proportion. This is because the sample mean is the required quantity, when sampling from a population of zeros and ones or equivalently from the distribution of a Bernoulli random variable - see [Statistics_SamplingDistributions#The_Binomial_case| this Section]. The possible sample values for <math>\bar{x}</math> are

@@ Line 330: / Line 330: @@
 <math>\Pr \left( Y=y\right) =0</math>
-for any value <math>y</math> - see the [[Probability_Norm#The_Normal_distribution Section]] on the Normal Distribution. Approximating <math>\Pr \left( T=t\right) </math> in this way is clearly not a good idea.
+for any value <math>y</math> - see the [[Probability_Norm#The_Normal_distribution|Section]] on the Normal Distribution. Approximating <math>\Pr \left( T=t\right) </math> in this way is clearly not a good idea.
 The solution is to approximate <math>\Pr \left( T=t\right) </math> by an area under the density of <math>Y</math> as

@@ Line 409: / Line 409: @@
 * This clip goes through the basics of estimators and tackles the issue of unbiasedness [http://youtu.be/6GhSiM0frIk].
 =Footnotes=
   <references />

@@ Line 144: / Line 144: @@
 == Biased Sampling Procedures ==
-All of these results are based on drawing a random sample from the population. One would expect that a sampling procedure not based on the principles behind random sampling (see this [[Populations_Samples_SampleDistributions#Samples_and_Sampling|Section]]) would automatically generate biased estimators of population parameters. This is an important issue for the design of official sample surveys and market research surveys. The consequences of inappropriate design are illustrated in the following example.
+All of these results are based on drawing a random sample from the population. One would expect that a sampling procedure not based on the principles behind random sampling (see this [Populations_Samples_SampleDistributions#Samples_and_Sampling|Section]) would automatically generate biased estimators of population parameters. This is an important issue for the design of official sample surveys and market research surveys. The consequences of inappropriate design are illustrated in the following example.
 To estimate the mean income of households in the UK, a researcher uses a sample of households selected at random from the telephone directory. The sample mean income will be taken as an estimate of the population mean income. To illustrate the possible consequences of this sample design, suppose that it is known that the population of households with telephones have a mean income of £20000, whilst households without telephones have mean income £10000. Out of the total population, 80% of households have a telephone.
@@ Line 205: / Line 205: @@
 \Pr ( -\varepsilon <\bar{X}-\mu <\varepsilon ) &=&\Pr (-\frac{\varepsilon }{\sigma /\sqrt{n}}< \frac{( \bar{X}-\mu )}{\sigma /\sqrt{n}}<\frac{\varepsilon }{\sigma /\sqrt{n}})\\
 &=& \Pr (-\frac{\varepsilon }{\sigma /\sqrt{n}}< Z <\frac{\varepsilon }{\sigma /\sqrt{n}}).\end{aligned}</math>
 Why did we do that? Because now we are looking at the probability for <math>\dfrac{\left\vert \bar{X}-\mu \right\vert }{\sigma /\sqrt{n}}</math> and as seen above we exactly know how this is distributed, it follows a standard normal distribution, <math>Z \thicksim N(0,1)</math>. Hence we can use standard normal distribution Tables to calculate these type of probabilities.
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 This correction is called the ''continuity correction''. As it turns out, this correction becomes less important as <math>n \rightarrow \infty</math> and it is, therefore, not always used when we are dealing with large <math>n</math>?
 =Footnotes=
   <references />