Point Estimation Exercises - Revision history

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‎Exercises

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‎Exercises

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Suppose that $Y\thicksim N\left( 6,2\right)$ , and that $\bar{Y}$ is the sample mean of a (simple) random sample of size
2013-09-19T21:29:19Z

Created page with " = Exercises = <ol> <li>Suppose that <math>Y\thicksim N\left( 6,2\right) </math>, and that <math>\bar{Y}</math> is the sample mean of a (simple) random sample of size <ma..."

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= Exercises =

<ol>
<li>Suppose that <math>Y\thicksim N\left( 6,2\right) </math>, and that <math>\bar{Y}</math> is the sample mean of a (simple) random sample of size <math>n</math>. Find:
<ol>
<li><math>\Pr \left( Y>8\right) ;</math></li>
<li><math>\Pr \left( \bar{Y}>8\right) \;</math>when <math>n=1;</math></li>
<li><math>\Pr \left( \bar{Y}>8\right) \;</math>when <math>n=2;</math></li>
<li><math>\Pr \left( \bar{Y}>8\right) \;</math>when <math>n=5;</math></li></ol>

Sketch, on the same axes, the sampling distribution of <math>\bar{Y}</math> for <math>n=1,2,5</math>.</li>
<li>In a certain population, 60% of all adults own a car. If a simple random sample of 100 adults is taken, what is the probability that at least 70% of the sample will be car owners? (Optional: use EXCEL to find the exact probability.)</li>
<li>When set correctly, a machine produces hamburgers of mean weight <math>100g</math> each and standard deviation <math>5g</math> each. The weight of hamburgers is known to be normally distributed. The hamburgers are sold in packets of four.
<ol>
<li>What is the sampling distribution of the total weight of hamburgers in a packet? In stating this sampling distribution, state carefully what results you using and any assumptions you have to make.</li>
<li>A customer claims that packets of hamburgers are underweight. A trading standards officer is sent to investigate. He selects one packet of four hamburgers and finds that the weight of hamburgers in it is <math>390g</math>. What is the probability of a packet weighing as little as <math>390g</math> if the machine is set correctly? Do you consider that this finding constitutes evidence that the machine has been set to deliver underweight hamburgers?</li></ol>
</li>
<li>A discrete random variable, <math>Y</math>, has the following probability distribution:
<table border="1">

<tr class="header">
<th align="center"><math>y</math></th>
<th align="center"><math>0</math></th>
<th align="center"><math>1</math></th>
<th align="center"><math>2</math></th>
</tr>

<tr class="odd">
<td align="center"><math>p\left( y\right) </math></td>
<td align="center"><math>0.3</math></td>
<td align="center"><math>0.4</math></td>
<td align="center"><math>0.3</math></td>
</tr>

</table>

<ol>
<li>What are <math>E\left[ Y\right] </math> and <math>y_{\min }</math>, where <math>y_{\min }</math> is the smallest possible value of <math>Y?</math></li>
<li>Simple random samples of two observations are to be drawn with replacement from this population. Write down all possible samples, and the probability of each sample. Use this to obtain the sampling distribution of each of the following statistics:
<ol>
<li>the sample mean, <math>\bar{Y};</math></li>
<li>the minimum of the two observations, <math>M</math>.</li></ol>
</li>
<li>Calculate <math>E\left[ \bar{Y}\right] </math> and <math>E\left[ M\right] </math>. State whether each is an unbiased estimator of the corresponding population parameter.</li></ol>
</li>
<li>A random sample of size three is drawn from the distribution of a Bernoulli random variable <math>X</math>, where
<math>\Pr \left( X=0\right) =0.3,\;\;\;\Pr \left( X=1\right) =0.7.</math>
<ol>
<li>Enumerate all the possible samples, and find their probabilities of being drawn. You should have eight possible samples.</li>
<li>Find the sampling distribution of the random variable <math>T</math>, the total number of ones in each sample.</li>
<li>Check that the probability distribution of <math>T</math> is the Binomial distribution for <math>n=3</math> and <math>\pi =0.7</math>, by calculating
<math>\Pr \left( T=t\right) =\binom{3}{t}\left( 0.7\right) ^{t}\left( 0.3\right)^{3-t}</math>
for <math>t=0,1,2,3</math>.</li>
<li>Find the probability distribution of <math>P</math>, the sample proportion of ones. How is this probability distribution related to that of <math>T?</math></li>
<li>Is <math>P</math> an unbiased estimator of <math>\Pr \left( X=1\right) ?</math></li></ol>
</li>
<li>A simple random sample of three observations is taken from a population with mean <math>\mu </math> and variance <math>\sigma ^{2}</math>. The three sample random variables are denoted <math>Y_{1},Y_{2},Y_{3}</math>. A sample statistic is being sought to estimate <math>\mu </math>. The statistics being considered are
<ol>
<li><ol>
<li><math>A_{1}=\dfrac{1}{3}\left( Y_{1}+Y_{2}+Y_{3}\right) ;</math></li>
<li><math>A_{2}=\dfrac{1}{2}\left( Y_{1}+Y_{2}\right) ;</math></li>
<li><math>A_{3}=\dfrac{1}{2}\left( Y_{1}+Y_{2}+Y_{3}\right) ;</math></li>
<li><math>A_{4}=0.75Y_{1}+0.75Y_{2}-0.5Y_{3}</math>.</li></ol>
</li>
<li>Which of these statistics yields an unbiased estimator of <math>\mu ?</math></li>
<li>Of those that are unbiased, which is the most efficient?</li>
<li>Of those that are unbiased, find the efficiency with respect to <math>A_{1}. </math></li></ol>
</li></ol>

= Footnotes =

@@ Line 6: / Line 6: @@
 <ol>
-<li><p><math>[L1,L2]</math>Suppose that <math>Y\thicksim N\left( 6,2\right) </math>, and that <math>\bar{Y}</math> is the sample mean of a (simple) random sample of size <math>n</math>. Find:</p>
+<li><p><math>[L1,L2]</math>Suppose that <math>Y\sim N\left( 6,2\right) </math>, and that <math>\bar{Y}</math> is the sample mean of a (simple) random sample of size <math>n</math>. Find:</p>
 <ol>
 <li><p><math>\Pr \left( Y>8\right)</math>; {0.0793}</p></li>

@@ Line 41: / Line 41: @@
 <ol>
-<li><p><math>[L1,L2]</math> What are <math>E\left[ Y\right] </math> and <math>y_{\min }</math>, where <math>y_{\min }</math> is the smallest possible value of <math>Y?</math></p></li>
+<li><p><math>[L2]</math> What are <math>E\left[ Y\right] </math> and <math>y_{\min }</math>, where <math>y_{\min }</math> is the smallest possible value of <math>Y?</math></p></li>
 <li><p><math>[L1,L2]</math> Simple random samples of two observations are to be drawn with replacement from this population. Write down all possible samples, and the probability of each sample. {e.g. (<math>P(y_1=0, y_2=2)=0.09</math>} Use this to obtain the sampling distribution of each of the following statistics:</p>
 <ol>

@@ Line 3: / Line 3: @@
 = Exercises =
-Worked solutions to these questions can be found here: [http://youtu.be/TCHG3mP3q1g?hd=1 Q1], [http://youtu.be/iXwkvtZpjm8?hd=1 Q2], [http://youtu.be/mQymVsrxmPU?hd=1 Q3], [http://youtu.be/x4lzkCx3bNw?hd=1 Q4]
+Worked solutions to these questions can be found here: [http://youtu.be/TCHG3mP3q1g?hd=1 Q1], [http://youtu.be/iXwkvtZpjm8?hd=1 Q2], [http://youtu.be/mQymVsrxmPU?hd=1 Q3], [http://youtu.be/x4lzkCx3bNw?hd=1 Q4], [http://youtu.be/PqZko0Yi8Fc?hd=1 Q5] and [http://youtu.be/8-uQdtCZdlA?hd=1 Q6]
 <ol>
@@ Line 52: / Line 52: @@
 <p><math>\Pr \left( X=0\right) =0.3,\;\;\;\Pr \left( X=1\right) =0.7.</math></p>
 <ol>
-<li><p>Enumerate all the possible samples, and find their probabilities of being drawn. You should have eight possible samples.</p></li>
+<li><p><math>[L1,L2]</math> Enumerate all the possible samples, and find their probabilities of being drawn. You should have eight possible samples. {e.g. <math>P(1,0,1)=0.3*0.7^2)</math>}</p></li>
-<li><p>Find the sampling distribution of the random variable <math>T</math>, the total number of ones in each sample.</p></li>
+<li><p><math>[L1,L2]</math> Find the sampling distribution of the random variable <math>T</math>, the total number of ones in each sample. {e.g. <math>P(T=1)=0.189)</math>}</p></li>
-<li><p>Check that the probability distribution of <math>T</math> is the Binomial distribution for <math>n=3</math> and <math>\pi =0.7</math>, by calculating</p>
+<li><p><math>[L2]</math> Check that the probability distribution of <math>T</math> is the Binomial distribution for <math>n=3</math> and <math>\pi =0.7</math>, by calculating</p>
 <p><math>\Pr \left( T=t\right) =\binom{3}{t}\left( 0.7\right) ^{t}\left( 0.3\right)^{3-t}</math></p>
 <p>for <math>t=0,1,2,3</math>.</p></li>
-<li><p>Find the probability distribution of <math>P</math>, the sample proportion of ones. How is this probability distribution related to that of <math>T?</math></p></li>
+<li><p><math>[L1,L2]</math> Find the probability distribution of <math>P</math>, the sample proportion of ones. How is this probability distribution related to that of <math>T?</math> {e.g. <math>Pr(P=2/3)=0.441)</math>}</p></li>
-<li><p>Is <math>P</math> an unbiased estimator of <math>\Pr \left( X=1\right) ?</math></p></li></ol>
+<li><p><math>[L2]</math> Is <math>P</math> an unbiased estimator of <math>\Pr \left( X=1\right) ?</math> {yes}</p></li></ol>
 </li>
-<li><p>A simple random sample of three observations is taken from a population with mean <math>\mu </math> and variance <math>\sigma ^{2}</math>. The three sample random variables are denoted <math>Y_{1},Y_{2},Y_{3}</math>. A sample statistic is being sought to estimate <math>\mu </math>. The statistics being considered are</p>
+<li><p><math>[L2]</math> A simple random sample of three observations is taken from a population with mean <math>\mu </math> and variance <math>\sigma ^{2}</math>. The three sample random variables are denoted <math>Y_{1},Y_{2},Y_{3}</math>. A sample statistic is being sought to estimate <math>\mu </math>. The statistics being considered are</p>
 <ol>
 <li><ol>
@@ Line 68: / Line 68: @@
 <li><p><math>A_{4}=0.75Y_{1}+0.75Y_{2}-0.5Y_{3}</math>.</p></li></ol>
 </li>
-<li><p>Which of these statistics yields an unbiased estimator of <math>\mu ?</math></p></li>
+<li><p>Which of these statistics yields an unbiased estimator of <math>\mu ?</math> {<math>A_1,A_2</math> and <math>A_4</math>}</p></li>
-<li><p>Of those that are unbiased, which is the most efficient?</p></li>
+<li><p>Of those that are unbiased, which is the most efficient? {<math>Var(A_1)=Var(Y)/3</math>}</p></li>
 <li><p>Of those that are unbiased, find the efficiency with respect to <math>A_{1}. </math></p></li></ol>
 </li></ol>
 = Footnotes =

@@ Line 2: / Line 2: @@
 = Exercises =
 <ol>
-<li><p>Suppose that <math>Y\thicksim N\left( 6,2\right) </math>, and that <math>\bar{Y}</math> is the sample mean of a (simple) random sample of size <math>n</math>. Find:</p>
+<li><p><math>[L1,L2]</math>Suppose that <math>Y\thicksim N\left( 6,2\right) </math>, and that <math>\bar{Y}</math> is the sample mean of a (simple) random sample of size <math>n</math>. Find:</p>
 <ol>
-<li><p><math>\Pr \left( Y>8\right) ;</math></p></li>
+<li><p><math>\Pr \left( Y>8\right)</math>; {0.0793}</p></li>
-<li><p><math>\Pr \left( \bar{Y}>8\right) \;</math>when <math>n=1;</math></p></li>
+<li><p><math>\Pr \left( \bar{Y}>8\right) \;</math>when <math>n=1;</math> {0.0793}</p></li>
-<li><p><math>\Pr \left( \bar{Y}>8\right) \;</math>when <math>n=2;</math></p></li>
+<li><p><math>\Pr \left( \bar{Y}>8\right) \;</math>when <math>n=2;</math> {0.0228}</p></li>
-<li><p><math>\Pr \left( \bar{Y}>8\right) \;</math>when <math>n=5;</math></p></li></ol>
+<li><p><math>\Pr \left( \bar{Y}>8\right) \;</math>when <math>n=5;</math> {0.0000}</p></li></ol>
 <p>Sketch, on the same axes, the sampling distribution of <math>\bar{Y}</math> for <math>n=1,2,5</math>.</p></li>
-<li><p>In a certain population, 60% of all adults own a car. If a simple random sample of 100 adults is taken, what is the probability that at least 70% of the sample will be car owners? (Optional: use EXCEL to find the exact probability.)</p></li>
+<li><p><math>[L1,L2]</math> In a certain population, 60% of all adults own a car. If a simple random sample of 100 adults is taken, what is the probability that at least 70% of the sample will be car owners? (Optional: use EXCEL to find the exact probability.) {0.0207 and 0.0262 are both approximations}</p></li>
-<li><p>When set correctly, a machine produces hamburgers of mean weight <math>100g</math> each and standard deviation <math>5g</math> each. The weight of hamburgers is known to be normally distributed. The hamburgers are sold in packets of four.</p>
+<li><p><math>[L1,L2]</math>When set correctly, a machine produces hamburgers of mean weight <math>100g</math> each and standard deviation <math>5g</math> each. The weight of hamburgers is known to be normally distributed. The hamburgers are sold in packets of four.</p>
 <ol>
-<li><p>What is the sampling distribution of the total weight of hamburgers in a packet? In stating this sampling distribution, state carefully what results you using and any assumptions you have to make.</p></li>
+<li><p>What is the sampling distribution of the total weight of hamburgers in a packet? In stating this sampling distribution, state carefully what results you using and any assumptions you have to make. {N(400,400), independence}</p></li>
-<li><p>A customer claims that packets of hamburgers are underweight. A trading standards officer is sent to investigate. He selects one packet of four hamburgers and finds that the weight of hamburgers in it is <math>390g</math>. What is the probability of a packet weighing as little as <math>390g</math> if the machine is set correctly? Do you consider that this finding constitutes evidence that the machine has been set to deliver underweight hamburgers?</p></li></ol>
+<li><p>A customer claims that packets of hamburgers are underweight. A trading standards officer is sent to investigate. He selects one packet of four hamburgers and finds that the weight of hamburgers in it is <math>390g</math>. What is the probability of a packet weighing as little as <math>390g</math> if the machine is set correctly? Do you consider that this finding constitutes evidence that the machine has been set to deliver underweight hamburgers? {0.3085}</p></li></ol>
 </li>
 <li><p>A discrete random variable, <math>Y</math>, has the following probability distribution:</p>
@@ Line 39: / Line 41: @@
 <ol>
-<li><p>What are <math>E\left[ Y\right] </math> and <math>y_{\min }</math>, where <math>y_{\min }</math> is the smallest possible value of <math>Y?</math></p></li>
+<li><p><math>[L1,L2]</math> What are <math>E\left[ Y\right] </math> and <math>y_{\min }</math>, where <math>y_{\min }</math> is the smallest possible value of <math>Y?</math></p></li>
-<li><p>Simple random samples of two observations are to be drawn with replacement from this population. Write down all possible samples, and the probability of each sample. Use this to obtain the sampling distribution of each of the following statistics:</p>
+<li><p><math>[L1,L2]</math> Simple random samples of two observations are to be drawn with replacement from this population. Write down all possible samples, and the probability of each sample. {e.g. (<math>P(y_1=0, y_2=2)=0.09</math>} Use this to obtain the sampling distribution of each of the following statistics:</p>
 <ol>
-<li><p>the sample mean, <math>\bar{Y};</math></p></li>
+<li><p>the sample mean, <math>\bar{Y};</math> {e.g. <math>P(\bar{Y}=0.5)=0.24)</math>}</p></li>
-<li><p>the minimum of the two observations, <math>M</math>.</p></li></ol>
+<li><p>the minimum of the two observations, <math>M</math>. {<math>P(M=1)=0.4</math>}</p></li></ol>
 </li>
-<li><p>Calculate <math>E\left[ \bar{Y}\right] </math> and <math>E\left[ M\right] </math>. State whether each is an unbiased estimator of the corresponding population parameter.</p></li></ol>
+<li><p><math>[L2]</math> Calculate <math>E\left[ \bar{Y}\right] </math> and <math>E\left[ M\right] </math>. State whether each is an unbiased estimator of the corresponding population parameter. {<math>\bar{Y}</math> yes, <math>M</math> no}</p></li></ol>
 </li>
 <li><p>A random sample of size three is drawn from the distribution of a Bernoulli random variable <math>X</math>, where</p>