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<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767944223552-75yqyislj5x.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDQyMjM1NTItNzV5cXlpc2xqNXgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTkwOSwiZXhwIjoxNzkyNjk2MjA5fQ.j6H0C1NZOej-ArKVi5xkuhwEiFcbE9Eu-0ypAoDr7DI" style="max-width: min(500px, 100%); height: auto; display: block;" alt="Cumulative frequency graph for Mathematics and Science examinations" class="aligncenter custom-width" /></figure></div>
<p>The cumulative frequency graph shows the distribution of marks for&nbsp;\(400\) students in a Mathematics and Science examination.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the median mark for the Mathematics examination.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the interquartile range for the Science examination.</span></div></li><li><div class="list-item-content"><span class="list-item-text">If the distinctions for Mathematics were allocated to the top \(25\%\) of the students, find the mark to achieve a distinction.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">One student is chosen at random. You may assume that the marks gained in the two examinations are independent. Expressing your answer as a fraction in its lowest terms, calculate the probability that the student gains</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">less than \(20\) marks for both examinations,</span></div></li><li><div class="list-item-content"><span class="list-item-text">more than \(60\) marks on one examination, but not on the other.</span></div></li></ol></li></ol>

The cumulative frequency graph shows the distribution of marks for \(400\) students in a Mathematics and Science examination.

One student is chosen at random. You may assume that the marks gained in the two examinations are independent. Expressing your answer as a fraction in its lowest terms, calculate the probability that the student gains

Percentiles and Quartiles from Cumulative Frequency Curves

Probabilities of Successive Events

Problems Involving Two Cumulative Frequency Curve

Regional Academic Standards: A Comparative Statistical Study

Statistics

Probability

<p>The Mathematics Preliminary Examination results for students across schools \(X\) and \(Y\) were recorded. Schools \(X\) and \(Y\) have&nbsp;\(200\) and \(150\) math students respectively.</p>
<p>The following cumulative frequency curves show the distribution of the marks.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767943223594-jdsp45ot1uc.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDMyMjM1OTQtamRzcDQ1b3QxdWMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTkwOSwiZXhwIjoxNzkyNjk2MjA5fQ.SMdFq6ZFkuFadVLraoz8FA7QX5CY9Arg_TeAJlevaZg" style="max-width: min(500px, 100%); height: auto; display: block;" alt="Cumulative frequency curve for student's marks in Math Prelim Examination in School X and Y" class="aligncenter custom-width" /></figure><div class="image-caption" style="text-align: center;">Cumulative frequency curve for student's marks in Math Prelim Examination in School X and Y</div></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the median mark for both schools \(X\) and \(Y\).</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Calculate the interquartile range for both schools \(X\) and \(Y\).</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">A teacher analysed the data above and commented: "School \(X\) performs better than school \(Y\) because the cumulative frequency curve of \(X\) is always above \(Y\)." Use your results in <strong>(a)</strong> and <strong>(b)</strong> to argue if you agree with the teacher's analysis.</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">A distinction grade is given for students who scored \(80\) marks or more.<br><br>Which school has a greater percentage distinction for the Math preliminary examination?</div></li></ol>

The Mathematics Preliminary Examination results for students across schools \(X\) and \(Y\) were recorded. Schools \(X\) and \(Y\) have \(200\) and \(150\) math students respectively.

The following cumulative frequency curves show the distribution of the marks.

Find the median mark for both schools \(X\) and \(Y\).

Calculate the interquartile range for both schools \(X\) and \(Y\).

A distinction grade is given for students who scored \(80\) marks or more.

Which school has a greater percentage distinction for the Math preliminary examination?

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