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IP Math

<p>The box-and-whisker plot represents a set of data with $80$ data points.</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/1769497950121-106nfxli3zxd.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk0OTc5NTAxMjEtMTA2bmZ4bGkzenhkLnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3ODU3MTA4ODcsImV4cCI6MTc4NTcxMTE4N30.pYgSqzRWFfkNrkHmIUiRNxxVfFhcgHzaYM19QN3Godc" style="max-width: min(400px, 100%); height: auto; display: block;" alt="Box-and-whisker plot showing data values at 7, 48, 65, 78, and 85." class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the greatest and the least possible number of data points with the value of $78$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the greatest and the least possible number of data points with values at least $78$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the greatest and the least possible number of data points with the value of $85$.</span></div></li></ol>

The box-and-whisker plot represents a set of data with $80$ data points.

Find the greatest and the least possible number of data points with the value of $78$.

Find the greatest and the least possible number of data points with values at least $78$.

Find the greatest and the least possible number of data points with the value of $85$.

IP2U15 HOT Q4

Statistics

<p>The number of minutes taken by four boys to complete a question is $x - y$, $8$, $10$ and $x + y$. The mean time taken is $9$ minutes.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$x = 9$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that the standard deviation is $5$ minutes, find the value of $y$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Four girls answered the same question and the mean time taken is also $9$ minutes but the standard deviation is $2$ minutes. Compare and comment on these two sets of data.</span></div></li></ol>

The number of minutes taken by four boys to complete a question is $x - y$, $8$, $10$ and $x + y$. The mean time taken is $9$ minutes.

Given that the standard deviation is $5$ minutes, find the value of $y$.

Four girls answered the same question and the mean time taken is also $9$ minutes but the standard deviation is $2$ minutes. Compare and comment on these two sets of data.

IP2U15 HOT Q3

<p>$16$ pupils sat for a test. The sum of their marks is $504$. The sum of the squares of their marks is $16833$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Calculate</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the mean,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the standard deviation, of these test marks.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">One of the marks was $43$ but was incorrectly recorded as&nbsp;$25$. Calculate the correct mean and standard deviation of these test marks.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain, stating your reasons clearly,</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">why the mean has increased,</span></div></li><li><div class="list-item-paragraph">why the standard deviation has increased.</div></li></ol></li></ol>

$16$ pupils sat for a test. The sum of their marks is $504$. The sum of the squares of their marks is $16833$.

One of the marks was $43$ but was incorrectly recorded as $25$. Calculate the correct mean and standard deviation of these test marks.

IP2U15 HOT Q2

<p><strong>Variance</strong></p>
<p>Variance measures the spread or dispersion of a set of data points around their mean (average). For a group of discrete data, the variance ($\sigma^2$) is calculated as the average of the squared differences between each data point and the mean.</p>
<p style="text-align: center">\[\sigma^2 = \frac{\sum(x - \bar{x})^2}{n}\]</p>
<p><strong>Standard Deviation</strong></p>
<p>The standard deviation is the square root of the variance. It gives a measure of the average deviation from the mean, expressed in the same units as the data.</p>
<p style="text-align: center">\[\sigma = \sqrt{\frac{\sum(x - \bar{x})^2}{n}}\]</p>
<p>Show in an algebraic way that, an alternative expression for standard deviation of a set of discrete data is given by $\sigma = \sqrt{\frac{\sum x^2}{n} - \bar{x}^2}$.</p>

Variance measures the spread or dispersion of a set of data points around their mean (average). For a group of discrete data, the variance ($\sigma^2$) is calculated as the average of the squared differences between each data point and the mean.

\[\sigma^2 = \frac{\sum(x - \bar{x})^2}{n}\]

The standard deviation is the square root of the variance. It gives a measure of the average deviation from the mean, expressed in the same units as the data.

\[\sigma = \sqrt{\frac{\sum(x - \bar{x})^2}{n}}\]

Show in an algebraic way that, an alternative expression for standard deviation of a set of discrete data is given by $\sigma = \sqrt{\frac{\sum x^2}{n} - \bar{x}^2}$.

Variance and Standard Deviation

<p>One hundred students were asked how many hand phones they have in their home. The results of the survey are summarized in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">No. of hand phones ($x$)</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">No. of students ($f$)</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$fx$</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$1$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$15$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$c$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$2$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$25$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$50$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$3$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$b$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$d$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$4$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$a$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$84$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$12$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$60$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$6$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$30$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Total</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$100$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$e$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the values of $a$, $b$, $c$, $d$ and $e$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">the mean and standard deviation of the number of hand phones in their home.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">There are $5$ red cards, $6$ yellow cards and $4$ blue cards in a box. Annie draws a card from the box at random. Write down the probability that she will not get a blue card.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Brandon draws $2$ cards at random, one after the other, without replacement. Find the probability that the two cards are of different colour.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Annie and Brandon take turn to draw one card at a time, without replacement, until either player gets a blue card. Assuming that Annie starts first, find the probability that she will get a blue card exactly on her second try.</span></div></li></ol>

One hundred students were asked how many hand phones they have in their home. The results of the survey are summarized in the table below.

There are $5$ red cards, $6$ yellow cards and $4$ blue cards in a box. Annie draws a card from the box at random. Write down the probability that she will not get a blue card.

Brandon draws $2$ cards at random, one after the other, without replacement. Find the probability that the two cards are of different colour.

Annie and Brandon take turn to draw one card at a time, without replacement, until either player gets a blue card. Assuming that Annie starts first, find the probability that she will get a blue card exactly on her second try.

Mean and Standard Deviation for Grouped Data

<p>The height of students at a school is shown as a cumulative frequency graph below.</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/1769331137120-oubtwaa4dlf.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzExMzcxMjAtb3VidHdhYTRkbGYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDg4NywiZXhwIjoxNzg1NzExMTg3fQ.BndNvG_vplRMwY6ffHjCqQzc2DdCDyBDYhJSEySDmZA" style="max-width: min(400px, 100%); height: auto; display: block;" alt="Cumulative frequency graph showing Number of students vs Height (cm)" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use the graph to estimate</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the median,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the inter-quartile range.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">If&nbsp;$20$% of the students are taller than $h$ cm, find the value of $h$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Copy and complete the grouped frequency table of students' heights.</span></div></li></ol>
<div class="content-block content-block--table"><div class="table-container"><table><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Heights (cm)</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">No. of students</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$120 \le x &lt; 130$</td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$130 \le x &lt; 140$</td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$140 \le x &lt; 150$</td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$150 \le x &lt; 160$</td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$160 \le x &lt; 170$</td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$170 \le x &lt; 180$</td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr></tbody></table></div></div>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Using your grouped frequency table, calculate an estimate of the standard deviation.</div></li><li><div class="list-item-paragraph">The students at the school took measurements of their weights using a faulty weighing machine. The box and whisker graph below shows the students' weights (in kg) measured by the faulty weight machine.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1769331223446-mab2gukchu8.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzEyMjM0NDYtbWFiMmd1a2NodTgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDg4NywiZXhwIjoxNzg1NzExMTg3fQ.GpR2LDha7PkuwP0dJx1TZF4Gdv7yZ2Ng-50qAuwTNzA" alt="List item image" class="aligncenter" style="max-width: 100%; height: auto;" /></div><ol class="i-list"><li><div class="list-item-paragraph">If every student's true weight was actually $5$ kg lower, describe briefly how the box and whisker graph will differ from the given graph.</div></li><li><div class="list-item-paragraph">State the correct value of the median weight.</div></li></ol></li></ol>

The height of students at a school is shown as a cumulative frequency graph below.

If $20$% of the students are taller than $h$ cm, find the value of $h$.

Copy and complete the grouped frequency table of students' heights.

Using your grouped frequency table, calculate an estimate of the standard deviation.

The students at the school took measurements of their weights using a faulty weighing machine. The box and whisker graph below shows the students' weights (in kg) measured by the faulty weight machine.

<p>The weights of $200$ secondary two pupils at Ace Secondary School are represented by the cumulative frequency curve shown below.</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/1769330982039-i6xy3zqu22q.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzA5ODIwMzktaTZ4eTN6cXUyMnEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDg4NywiZXhwIjoxNzg1NzExMTg3fQ.iWKfW91B7m8C3UxzwJ7Uwspbnlw9xHO7xt8qNpQnu6w" style="max-width: min(400px, 100%); height: auto; display: block;" alt="Cumulative Frequency Curve for Weights of Pupils" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use the graph to find</span></div><ol class="i-list"><li><div class="list-item-paragraph">the median weight of the pupils,</div></li><li><div class="list-item-paragraph">the interquartile range,</div></li><li><div class="list-item-paragraph">the probability that at least one of two pupils chosen at random weighs at least $55$ kg.</div></li></ol></li><li><div class="list-item-paragraph">The weights of the pupils are represented in the box-and-whiskers plot shown below. Write down the values of $a$, $b$, $c$, $d$ and $e$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1769331007631-b9z4edd5xs5.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzEwMDc2MzEtYjl6NGVkZDV4czUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDg4NywiZXhwIjoxNzg1NzExMTg3fQ.vvog7DAbMyLqTdchkst7a6CojAOH2mlqehPqqDzq3io" alt="List item image" class="aligncenter" style="max-width: 100%; height: auto;" /></div></li><li><div class="list-item-paragraph">The weights of another $200$ secondary two pupils at Best Secondary School are represented in another box-and-whiskers plot shown below. Compare briefly the weights of the secondary two pupils from the two schools.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1769331018689-izkd9lsqqf.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzEwMTg2ODktaXprZDlsc3FxZi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzEwODg3LCJleHAiOjE3ODU3MTExODd9.274BUYP4Zrx3otlB77wdYn-LtS66010gXV6gBzWWRHA" alt="List item image" class="aligncenter" style="max-width: 100%; height: auto;" /></div></li></ol>

The weights of $200$ secondary two pupils at Ace Secondary School are represented by the cumulative frequency curve shown below.

The weights of the pupils are represented in the box-and-whiskers plot shown below. Write down the values of $a$, $b$, $c$, $d$ and $e$.

<p>The mass $x$ grams of each of $90$ oranges of variety $A$ were recorded. The data obtained was illustrated as shown in the table.</p>
<div class="content-block content-block--table"><div class="table-container"><table><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Mass $x$ grams</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">No. of oranges</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$60 &lt; x \le 70$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$1$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$70 &lt; x \le 80$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$3$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$80 &lt; x \le 90$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$9$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$90 &lt; x \le 100$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$28$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$100 &lt; x \le 110$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$37$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$110 &lt; x \le 120$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$8$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$120 &lt; x \le 130$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$4$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the mean and standard deviation for the above data.</span></div></li><li><div class="list-item-content"><span class="list-item-text">In an another variety, variety $B$, of the $90$ oranges, the lightest orange has the mass $55$ grams and the heaviest orange has the mass $138$ grams. The median, lower quartile and upper quartile are $102$ grams, $93$ grams and $109$ grams respectively. Draw the box-whisker diagram for this data.</span></div></li><li><div class="list-item-content"><span class="list-item-text">If you were to buy some oranges, which variety would you buy? Explain your answer.</span></div></li></ol>

The mass $x$ grams of each of $90$ oranges of variety $A$ were recorded. The data obtained was illustrated as shown in the table.

Find the mean and standard deviation for the above data.

In an another variety, variety $B$, of the $90$ oranges, the lightest orange has the mass $55$ grams and the heaviest orange has the mass $138$ grams. The median, lower quartile and upper quartile are $102$ grams, $93$ grams and $109$ grams respectively. Draw the box-whisker diagram for this data.

If you were to buy some oranges, which variety would you buy? Explain your answer.

<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/1769330783936-6uq3u26qw5f.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzA3ODM5MzYtNnVxM3UyNnF3NWYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDg4NywiZXhwIjoxNzg1NzExMTg3fQ.h2FT2zZq3jC8rik_N6ymwHvbHQ-WNQvuolV5holrNaU" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The cumulative frequency curve above represents daily wages of $60$ male adults in factory $X$. Use the graph to estimate, as accurately as possible,</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the median daily wage,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the interquartile range,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the $30^{\mathrm{th}}$ percentile,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the number of male adults whose daily wage is less than or equal to $\$45$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the fraction of the $60$ male adults whose daily wage is more than $\$25$.</span></div></li></ol>

The cumulative frequency curve above represents daily wages of $60$ male adults in factory $X$. Use the graph to estimate, as accurately as possible,

the number of male adults whose daily wage is less than or equal to $\$45$,

the fraction of the $60$ male adults whose daily wage is more than $\$25$.

<p>The cumulative frequency curve below illustrates the marks obtained by $120$ students in a Science test for Class $A$.</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/1769330518184-k7awj2ogt0h.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzA1MTgxODQtazdhd2oyb2d0MGgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDg4NywiZXhwIjoxNzg1NzExMTg3fQ.15WhuJ-TfR6ltsF00WLHyGEspSriongQuskL1kESr-o" style="max-width: min(400px, 100%); height: auto; display: block;" alt="Cumulative frequency curve for Science test marks" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use the graph to find,</span></div><ol class="i-list"><li><div class="list-item-paragraph">the $80^{th}$ percentile.</div></li><li><div class="list-item-paragraph">the interquartile range.</div><div class="list-item-paragraph">In this test, only $15$% of the students scored a distinction.</div></li><li><div class="list-item-paragraph">Find the least marks for a student to be considered for distinction.</div><div class="list-item-paragraph">Two student are chosen at random.</div></li><li><div class="list-item-paragraph">Find the probability that both the students scored above $80$ marks.</div></li></ol></li><li><div class="list-item-paragraph">The cumulative frequency curve can be represented using the following box-and-whisker diagram.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1769330604036-b08esn5p68u.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzA2MDQwMzYtYjA4ZXNuNXA2OHUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDg4NywiZXhwIjoxNzg1NzExMTg3fQ.N4FbJu0z_Ys1CRz9MftNZZ_D32cMuwVpf1MaCgFLQhQ" alt="List item image" class="aligncenter" style="max-width: 100%; height: auto;" /></div><ol class="i-list"><li><div class="list-item-paragraph">State the value of $x_{3}$.</div></li><li><div class="list-item-paragraph">State the value of $x_{5}-x_{1}$. What does $x_{5}-x_{1}$ represent?</div></li></ol></li><li><div class="list-item-paragraph">The box-and-whisker diagram below illustrates the marks obtained by $120$ students in a Science test for Class $B$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1769499239663-n5u3oytoth.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk0OTkyMzk2NjMtbjV1M295dG90aC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzEwODg3LCJleHAiOjE3ODU3MTExODd9.HrMtCFqPzv5ayd8m-97CRrQfmFQeyjGNe6mcOrkRtQE" alt="List item image" class="aligncenter custom-width" style="max-width: min(400px, 100%); height: auto;" /></div><div class="list-item-paragraph">Compare the test results for the two classes in two different ways.</div></li><li><div class="list-item-paragraph">The results of the science test of Class $C$ of $120$ students have the same median but a smaller standard deviation. Describe how the cumulative frequency curve will differ from the given curve.</div></li></ol>

The cumulative frequency curve below illustrates the marks obtained by $120$ students in a Science test for Class $A$.

<p>A teacher recorded the individual time taken by $200$ students to complete a cross-country race. The results are presented in the form of a cumulative frequency curve.</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/1769330452944-u09rnsjaqz.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzA0NTI5NDQtdTA5cm5zamFxei5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzEwODg3LCJleHAiOjE3ODU3MTExODd9.EaWEiqXLmdDTXwCjpPNdjVAyvcZ9DYQCUg7S1vX9egs" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use the curve to estimate</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the median time taken to run the race,</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the interquartile range of the distribution,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the percentage of students who took longer than $66$ min to finish the race,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the probability of choosing a student who took $45$ min or less time to finish the race.</span><span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">The frequency table relates to the same cross-country race results.</span></div></li></ol>
<div class="content-block content-block--table"><div class="table-container"><table><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Frequency</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Time ($t$ min)</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$13$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$30 &lt; t \le 40$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$r$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$40 &lt; t \le 50$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$90$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$50 &lt; t \le 60$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$s$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$60 &lt; t \le 70$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$9$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$70 &lt; t \le 80$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$4$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$80 &lt; t \le 90$</td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the value of $r$ and the value of $s$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Showing your method clearly, estimate the mean time required to run the race.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Estimate the standard deviation of the distribution.</span><span class="inline-marks">[2]</span></div></li></ol>

A teacher recorded the individual time taken by $200$ students to complete a cross-country race. The results are presented in the form of a cumulative frequency curve.

The frequency table relates to the same cross-country race results.

Find the value of $r$ and the value of $s$.

Showing your method clearly, estimate the mean time required to run the race.

Estimate the standard deviation of the distribution.

<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/1769330191844-v3siqj5rqg.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzAxOTE4NDQtdjNzaXFqNXJxZy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzEwODg3LCJleHAiOjE3ODU3MTExODd9.DZomLYgw7Arnq9NTnYbEvnv0pNMI3uNaGHDNPclncBE" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The diagram above is the cumulative frequency curve for the daily time spent surfing Facebook of $250$ students.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use the graph to find</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the median time spent surfing Facebook,</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the number of students who spend more than $20$ minutes a day surfing Facebook,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the $90^{th}$ percentile,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the interquartile range.</span><span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">A second group of students have a higher median and a larger standard deviation. Describe how the cumulative frequency graph will differ from this given curve.</span><span class="inline-marks">[2]</span></div></li></ol>

The diagram above is the cumulative frequency curve for the daily time spent surfing Facebook of $250$ students.

A second group of students have a higher median and a larger standard deviation. Describe how the cumulative frequency graph will differ from this given curve.

<p>The frequency table below summarizes the volume of apple juice dispensed from vending machine $A$ for&nbsp;$50$ trials.</p>
<div class="content-block content-block--table"><div class="table-container"><table><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Volume from $A$ ($x$ ml)</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Frequency</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$246 &lt; x \le 248$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$4$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$248 &lt; x \le 250$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$7$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$250 &lt; x \le 252$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$28$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$252 &lt; x \le 254$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$6$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$254 &lt; x \le 256$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$5$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">For vending machine $A$, calculate</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the mean,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the standard deviation.</span><span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Another vending machine $B$ is used to dispense apple juice. The mean and standard deviation of the volumes of apple juice dispensed by vending machine $B$ is $251.36$ ml and $2.22$ ml respectively. The nominal volume of each cup of apple juice dispensed from these vending machines is $250$ ml. If you run a vending machine business, which machine would you buy? Explain your choice.</span><span class="inline-marks">[2]</span></div></li><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">What is the range of volume dispensed by machine $A$?</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Why is the standard deviation preferred over the range as a measure of the spread of data?</span><span class="inline-marks">[1]</span></div></li></ol></li></ol>

The frequency table below summarizes the volume of apple juice dispensed from vending machine $A$ for $50$ trials.

Another vending machine $B$ is used to dispense apple juice. The mean and standard deviation of the volumes of apple juice dispensed by vending machine $B$ is $251.36$ ml and $2.22$ ml respectively. The nominal volume of each cup of apple juice dispensed from these vending machines is $250$ ml. If you run a vending machine business, which machine would you buy? Explain your choice.

<p>A survey is conducted to study the amount of daily pocket money (in dollars) of a group of students in a class. The results of the survey, in terms of $x$, is as given in the data set&nbsp;$\{x-3,\ x+2,\ x+7,\ x-6,\ x+4,\ x,\ x+5\}$, where&nbsp;$x &gt; 0$.</p>
<p>Write down an expression for its</p>
<ol class="abc-list"><li><div class="list-item-paragraph">range,</div></li><li><div class="list-item-paragraph">lower quartile.</div></li></ol>

A survey is conducted to study the amount of daily pocket money (in dollars) of a group of students in a class. The results of the survey, in terms of $x$, is as given in the data set $\{x-3,\ x+2,\ x+7,\ x-6,\ x+4,\ x,\ x+5\}$, where $x > 0$.

<p>The waiting time in minutes for&nbsp;$50$ patients at two hospitals were given as follows.</p>
<p>Hospital $A$:</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr style="height: 43px;"><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">Time (minutes)</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$20 &lt; t \le 24$</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$24 &lt; t \le 28$</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$28 &lt; t \le 32$</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$32 &lt; t \le 36$</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$36 &lt; t \le 40$</th></tr></thead><tbody><tr style="height: 43px;"><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">Number of patients</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$4$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$14$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$23$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$6$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$3$</td></tr></tbody></table></div></div>
<p>Hospital $B$:</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 200px;"><colgroup><col style="width: 200px;"></colgroup><thead><tr><th style="text-align: left; --cell-justify: flex-start">Mean (minutes) = $29.2$</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">Standard Deviation (minutes) = $3.5$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">For Hospital $A$, calculate an estimate of</span></div><ol class="i-list"><li data-list-item-id="item-a-i" data-list-item-path="item-a/item-a-i"><div class="list-item-content"><span class="list-item-text">the mean waiting time,</span></div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-content"><span class="list-item-text">the standard deviation, correct to $3$ significant figures.</span></div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">Compare briefly, the waiting time for the two hospitals.</span></div></li></ol>

The waiting time in minutes for $50$ patients at two hospitals were given as follows.

For Hospital $A$, calculate an estimate of

Compare briefly, the waiting time for the two hospitals.

Technical Report: Variance Analysis of Biological Samples

<div class="content-block content-block--table"><div class="table-container"><table><thead><tr><th style="text-align: center; --cell-justify: center">Mass ($m$) in grams</th><th style="text-align: center; --cell-justify: center">Frequency</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$40 &lt; m \le 50$</td><td style="text-align: center; --cell-justify: center">$15$</td></tr><tr><td style="text-align: center; --cell-justify: center">$50 &lt; m \le 60$</td><td style="text-align: center; --cell-justify: center">$20$</td></tr><tr><td style="text-align: center; --cell-justify: center">$60 &lt; m \le 70$</td><td style="text-align: center; --cell-justify: center">$5$</td></tr></tbody></table></div></div>
<p>The table shows the distribution of the mass of a number of eggs.</p>
<p>Calculate an estimate of</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the mean mass of the eggs,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the standard deviation of the mass of the eggs.</span></div></li></ol>

The table shows the distribution of the mass of a number of eggs.

the standard deviation of the mass of the eggs.

Technical Report: Variance Analysis of Mass Distribution

<p>The thirty students in a class each took a Biology test. Their scores are shown in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table><colgroup><col style="width: 120px;"><col style="width: 60px;"><col style="width: 60px;"><col style="width: 60px;"><col style="width: 60px;"><col style="width: 60px;"><col style="width: 60px;"></colgroup><thead><tr style="height: 43px;"><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">Score</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$4$</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$5$</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$6$</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$7$</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$8$</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$10$</th></tr></thead><tbody><tr style="height: 43px;"><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">Frequency</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$2$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$5$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$11$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$8$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$3$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$1$</td></tr></tbody></table></div></div>
<p>Calculate</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the mean score,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the standard deviation.</span></div></li></ol>

The thirty students in a class each took a Biology test. Their scores are shown in the table below.

Technical Report: Variance Analysis of Student Achievement

<p>The test results of a class of students are shown below.</p>
<div class="content-block content-block--table"><div class="table-container"><table><thead><tr><th style="text-align: center; --cell-justify: center">Marks</th><th style="text-align: center; --cell-justify: center">$1-5$</th><th style="text-align: center; --cell-justify: center">$6-10$</th><th style="text-align: center; --cell-justify: center">$11-15$</th><th style="text-align: center; --cell-justify: center">$16-20$</th><th style="text-align: center; --cell-justify: center">$21-25$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Frequency</td><td style="text-align: center; --cell-justify: center">$0$</td><td style="text-align: center; --cell-justify: center">$3$</td><td style="text-align: center; --cell-justify: center">$27$</td><td style="text-align: center; --cell-justify: center">$8$</td><td style="text-align: center; --cell-justify: center">$2$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Calculate an estimate of</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the mean marks,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the standard deviation of the marks.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">If the mean and standard deviation for the entire school’s results in the same test are $13.5$ marks and $2.1$ marks respectively. Compare briefly, the class results against the entire school’s results.</span></div></li></ol>

The test results of a class of students are shown below.

If the mean and standard deviation for the entire school’s results in the same test are $13.5$ marks and $2.1$ marks respectively. Compare briefly, the class results against the entire school’s results.

Analytical Report: Comparative Class Performance Statistics

<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 250px;"><colgroup><col style="width: 250px;"></colgroup><tbody><tr style="height: 43px;"><td rowspan="2" style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center"><p style="text-align: center;"><strong>Formula to find mean of a set of data</strong></p><p style="text-align: center;">Grouped Data: $\frac{\sum fx}{\sum f}$</p></td></tr><tr style="height: 40px;"></tr></tbody></table></div></div>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 250px;"><colgroup><col style="width: 250px;"></colgroup><tbody><tr style="height: 43px;"><td rowspan="2" style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center"><p style="text-align: center;"><strong>Formula to find standard deviation</strong></p><p style="text-align: center;">$\sqrt{\frac{\sum x^2}{n} - \bar{x}^2} \text{ or}\hspace{0.5em} \sqrt{\frac{\sum (x - \bar{x})^2}{n}}$</p></td></tr><tr style="height: 40px;"></tr></tbody></table></div></div>
<p>The frequency table shows the distribution of the number of times a class of $30$ students access Facebook in a day.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 600px;"><colgroup><col style="width: 200px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p>Number of access to Facebook</p><p>(daily)</p></th><th style="text-align: center; --cell-justify: center">$0 &lt; t \le 2$</th><th style="text-align: center; --cell-justify: center">$2 &lt; t \le 4$</th><th style="text-align: center; --cell-justify: center">$4 &lt; t \le 6$</th><th style="text-align: center; --cell-justify: center">$6 &lt; t \le 8$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Number of students</td><td style="text-align: center; --cell-justify: center">$4$</td><td style="text-align: center; --cell-justify: center">$6$</td><td style="text-align: center; --cell-justify: center">$13$</td><td style="text-align: center; --cell-justify: center">$7$</td></tr></tbody></table></div></div>
<p>Calculate the mean and standard deviation of this data.</p>

<p style="text-align: center;"><strong>Formula to find mean of a set of data</strong></p><p style="text-align: center;">Grouped Data: $\frac{\sum fx}{\sum f}$</p>

<p style="text-align: center;"><strong>Formula to find standard deviation</strong></p><p style="text-align: center;">$\sqrt{\frac{\sum x^2}{n} - \bar{x}^2} \text{ or } \sqrt{\frac{\sum (x - \bar{x})^2}{n}}$</p>

The frequency table shows the distribution of the number of times a class of $30$ students access Facebook in a day.

<p>Number of access to Facebook</p><p>(daily)</p>

Calculate the mean and standard deviation of this data.

Mean and Standard Deviation of Grouped Data

<p>Two brands of light bulbs were tested for their reliability. $10$ light bulbs of each brand were randomly selected and they were switched on for a number of days until they blew out. The number of days that each bulb lasted was then noted. The results are given below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 605px;"><colgroup><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"></colgroup><thead><tr style="height: 43px;"><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">Brand</th><th colspan="10" style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">Number of days bulb lasted</th></tr></thead><tbody><tr style="height: 43px;"><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$A$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$101$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$100$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$99$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$105$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$102$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$101$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$98$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$100$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$97$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$97$</td></tr></tbody></table></div></div>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 605px;"><colgroup><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"></colgroup><thead><tr style="height: 43px;"><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">Brand</th><th colspan="10" style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">Number of days bulb lasted</th></tr></thead><tbody><tr style="height: 43px;"><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$B$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$95$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$101$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$100$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$103$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$92$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$107$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$106$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$94$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$98$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$104$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">For Brand $A$, find</span></div><ol class="i-list"><li data-list-item-id="item-a-i" data-list-item-path="item-a/item-a-i"><div class="list-item-content"><span class="list-item-text">the mean number of days the light bulbs lasted,</span></div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-content"><span class="list-item-text">the standard deviation of the number of days, correct to $3$ significant figures.</span></div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">For Brand $B$, find</span></div><ol class="i-list"><li data-list-item-id="item-b-i" data-list-item-path="item-b/item-b-i"><div class="list-item-content"><span class="list-item-text">the mean number of days the light bulbs lasted,</span></div></li><li data-list-item-id="item-b-ii" data-list-item-path="item-b/item-b-ii"><div class="list-item-content"><span class="list-item-text">the standard deviation of the number of days, correct to $3$ significant figures.</span></div></li></ol></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-content"><span class="list-item-text">Which brand of light bulb is more reliable? Give a reason for your answer.</span></div></li></ol>

Two brands of light bulbs were tested for their reliability. $10$ light bulbs of each brand were randomly selected and they were switched on for a number of days until they blew out. The number of days that each bulb lasted was then noted. The results are given below.

Which brand of light bulb is more reliable? Give a reason for your answer.

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