<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@vertical+block@ps10" data-block-type="vertical" data-runtime-version="1" data-runtime-class="LmsRuntime" data-has-score="False" data-request-token="c3593e6e105311efa60b0e90929de005" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="False" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">Problem set 10</h2>
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<p>You can download this pdf file from <a href="/assets/courseware/v1/4fb15b1faa1e825eed001977cb07871b/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/pdf_mit18_05_s22_ps10.pdf">here</a></p><iframe src="/assets/courseware/v1/4fb15b1faa1e825eed001977cb07871b/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/pdf_mit18_05_s22_ps10.pdf" width="100%" height="1000px"> </iframe>
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<h2 class="hd hd-2 unit-title">Pset 10 problem checker</h2>
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Problem 1
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps10-1-check-problem-progress"></div>
<div class="problem">
<div>
<p>
To check your answers put them in the appropriate box and click the &#8216;Check' button. The checker can do arithmetic. So, 3^2, [mathjaxinline]5/3[/mathjaxinline] and [mathjaxinline]3*4[/mathjaxinline] are okay. </p>
<p>
Unless otherwise stated the checker expects numerical answers to 3 significant digits. </p>
<p>
For formulas you can use R notation. Remember to use a * for multiplication, i.e. [mathjaxinline]3*x[/mathjaxinline] <b class="bf">NOT </b> [mathjaxinline]3x[/mathjaxinline]. </p>
<p>
The checker knows many of the standard functions, e.g. use exp(x) and ln(x) for the exponential and natural log functions. </p>
<p>
The formula checker expects a specific letter for the variables, so be sure to use the ones specified. </p>
<p><b class="bf">1(a)</b> Give the 95% confidence interval. Enter it in standard interval form, e.g <tt class="tt">[1.23, 4.56]</tt>. Give your answers to 3 significant digits. <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps10-1-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
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<div class="unanswered inline">
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<span class="status unanswered" id="status_ps10-1-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps10-1-check_2_1" class="answer"/>
<div id="display_ps10-1-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps10-1-check_2_1_dynamath" name="input_ps10-1-check_2_1_dynamath"/>
</div>
</div></div> </p>
<p>
Give the 70% confidence interval. Enter it in standard interval form. Give the answers to 3 significant digits. <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps10-1-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps10-1-check_3_1" id="input_ps10-1-check_3_1" aria-describedby="status_ps10-1-check_3_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_ps10-1-check_3_1"/>
<span class="status unanswered" id="status_ps10-1-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps10-1-check_3_1" class="answer"/>
<div id="display_ps10-1-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps10-1-check_3_1_dynamath" name="input_ps10-1-check_3_1_dynamath"/>
</div>
</div></div> </p>
<p>
<b class="bf">1(b)</b>
<em>No answer checker given.</em>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 1" />
<div class="submit-attempt-container">
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<span class="submit-label">Submit</span>
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<div class="submission-feedback" id="submission_feedback_ps10-1-check">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
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Problem 2
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps10-2-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">2(a)</b> Give the 95% [mathjaxinline]z[/mathjaxinline]-confidence interval for the mean. Enter it in standard interval form. Give the answers to 3 significant digits. <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps10-2-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps10-2-check_2_1" id="input_ps10-2-check_2_1" aria-describedby="status_ps10-2-check_2_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_ps10-2-check_2_1"/>
<span class="status unanswered" id="status_ps10-2-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps10-2-check_2_1" class="answer"/>
<div id="display_ps10-2-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps10-2-check_2_1_dynamath" name="input_ps10-2-check_2_1_dynamath"/>
</div>
</div></div> </p>
<p><b class="bf">2(b)</b> Give the 98% [mathjaxinline]z[/mathjaxinline]-confidence interval for the mean. Enter it in standard interval form. Give the answers to 3 significant digits. <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps10-2-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps10-2-check_3_1" id="input_ps10-2-check_3_1" aria-describedby="status_ps10-2-check_3_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_ps10-2-check_3_1"/>
<span class="status unanswered" id="status_ps10-2-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps10-2-check_3_1" class="answer"/>
<div id="display_ps10-2-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps10-2-check_3_1_dynamath" name="input_ps10-2-check_3_1_dynamath"/>
</div>
</div></div> </p>
<p><b class="bf">2(c)</b> Give the 95% [mathjaxinline]t[/mathjaxinline]-confidence interval for the mean. Enter it in standard interval form. Give the answers to 3 significant digits. <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ps10-2-check_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps10-2-check_4_1" id="input_ps10-2-check_4_1" aria-describedby="status_ps10-2-check_4_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_ps10-2-check_4_1"/>
<span class="status unanswered" id="status_ps10-2-check_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps10-2-check_4_1" class="answer"/>
<div id="display_ps10-2-check_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps10-2-check_4_1_dynamath" name="input_ps10-2-check_4_1_dynamath"/>
</div>
</div></div> </p>
<p>
Give the 98% [mathjaxinline]t[/mathjaxinline]-confidence interval for the mean. Enter it in standard interval form. Give the answers to 3 significant digits. <div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_ps10-2-check_5_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps10-2-check_5_1" id="input_ps10-2-check_5_1" aria-describedby="status_ps10-2-check_5_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_ps10-2-check_5_1"/>
<span class="status unanswered" id="status_ps10-2-check_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps10-2-check_5_1" class="answer"/>
<div id="display_ps10-2-check_5_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps10-2-check_5_1_dynamath" name="input_ps10-2-check_5_1_dynamath"/>
</div>
</div></div> </p>
</div>
<div class="action">
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<div class="submission-feedback" id="submission_feedback_ps10-2-check">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
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Problem 3
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps10-3-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">3(a)</b> Give point estimate for the variance from the data. (Give to 2 significant digits) <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps10-3-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps10-3-check_2_1" id="input_ps10-3-check_2_1" aria-describedby="status_ps10-3-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps10-3-check_2_1"/>
<span class="status unanswered" id="status_ps10-3-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps10-3-check_2_1" class="answer"/>
<div id="display_ps10-3-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps10-3-check_2_1_dynamath" name="input_ps10-3-check_2_1_dynamath"/>
</div>
</div></div> </p>
<p>
Give the 95% confidence interval for the variance. Enter it in standard interval form. Give the answers to 2 significant digits. <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps10-3-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps10-3-check_3_1" id="input_ps10-3-check_3_1" aria-describedby="status_ps10-3-check_3_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_ps10-3-check_3_1"/>
<span class="status unanswered" id="status_ps10-3-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps10-3-check_3_1" class="answer"/>
<div id="display_ps10-3-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps10-3-check_3_1_dynamath" name="input_ps10-3-check_3_1_dynamath"/>
</div>
</div></div> </p>
<p>
<b class="bf">3(b)</b>
<em>No answer checker given.</em>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 3" />
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<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_ps10-3-check" >
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<div class="submission-feedback" id="submission_feedback_ps10-3-check">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
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<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps10-4-check" data-block-type="problem" data-runtime-version="1" data-runtime-class="LmsRuntime" data-has-score="True" data-request-token="c3593e6e105311efa60b0e90929de005" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
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<div id="problem_ps10-4-check" class="problems-wrapper" role="group"
aria-labelledby="ps10-4-check-problem-title"
data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps10-4-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps10-4-check/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps10-4-check-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps10-4-check-problem-progress" tabindex="-1">
Problem 4
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps10-4-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">4(a)</b> Give the 95% confidence interval for the variance. Enter it in standard interval form. Give the answers to 3 significant digits. <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps10-4-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps10-4-check_2_1" id="input_ps10-4-check_2_1" aria-describedby="status_ps10-4-check_2_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_ps10-4-check_2_1"/>
<span class="status unanswered" id="status_ps10-4-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps10-4-check_2_1" class="answer"/>
<div id="display_ps10-4-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps10-4-check_2_1_dynamath" name="input_ps10-4-check_2_1_dynamath"/>
</div>
</div></div> </p>
<p>
<b class="bf">4(b)</b>
<em>No answer checker given.</em>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 4" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_ps10-4-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps10-4-check">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
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<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps10-5-check" data-block-type="problem" data-runtime-version="1" data-runtime-class="LmsRuntime" data-has-score="True" data-request-token="c3593e6e105311efa60b0e90929de005" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
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<div id="problem_ps10-5-check" class="problems-wrapper" role="group"
aria-labelledby="ps10-5-check-problem-title"
data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps10-5-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps10-5-check/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps10-5-check-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps10-5-check-problem-progress" tabindex="-1">
Problem 5
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps10-5-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">5(a)</b> Approximately what fraction of the time will the chosen [mathjaxinline]\mu[/mathjaxinline] be in the confidence interval? <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps10-5-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps10-5-check_2_1" id="input_ps10-5-check_2_1" aria-describedby="status_ps10-5-check_2_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_ps10-5-check_2_1"/>
<span class="status unanswered" id="status_ps10-5-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps10-5-check_2_1" class="answer"/>
<div id="display_ps10-5-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps10-5-check_2_1_dynamath" name="input_ps10-5-check_2_1_dynamath"/>
</div>
</div></div> </p>
<p><b class="bf">5b(i)</b> Give the prior probability [mathjaxinline]\mu[/mathjaxinline] is in the given confidence interval (3 significant digits) <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps10-5-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps10-5-check_3_1" id="input_ps10-5-check_3_1" aria-describedby="status_ps10-5-check_3_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_ps10-5-check_3_1"/>
<span class="status unanswered" id="status_ps10-5-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps10-5-check_3_1" class="answer"/>
<div id="display_ps10-5-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps10-5-check_3_1_dynamath" name="input_ps10-5-check_3_1_dynamath"/>
</div>
</div></div> </p>
<p><b class="bf">5b(ii)</b> Give the posterior probability [mathjaxinline]\mu[/mathjaxinline] is in the given confidence interval (3 significant digits) <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ps10-5-check_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps10-5-check_4_1" id="input_ps10-5-check_4_1" aria-describedby="status_ps10-5-check_4_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_ps10-5-check_4_1"/>
<span class="status unanswered" id="status_ps10-5-check_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps10-5-check_4_1" class="answer"/>
<div id="display_ps10-5-check_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps10-5-check_4_1_dynamath" name="input_ps10-5-check_4_1_dynamath"/>
</div>
</div></div> </p>
<p>
<b class="bf">5(c)</b>
<em>No answer checker given.</em>
</p>
</div>
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<input type="hidden" name="problem_id" value="Problem 5" />
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