A Math

A Math (IGCSE)

E Math

H1 Math

H2 Math

IB HL Math

IB SL Math

IP Math

<ol class="abc-list"><li><div class="list-item-paragraph">The masses of a randomly chosen bolt and a randomly chosen nut are denoted by $M$ grams and $W$ grams respectively. $M$ and $W$ are independent random variables with the distributions $\mathrm{N}(272, 8^2)$ and $\mathrm{N}(98, 5^2)$ respectively.</div><ol class="i-list"><li><div class="list-item-paragraph">Find the range of values of $a$ for which $P(a &lt; M &lt; 275) &gt; 0.2024$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Calculate the probability that twice the mass of a randomly chosen bolt differs from the total mass of $5$ randomly chosen nuts by less than $80$ grams.<span class="inline-marks">[3]</span></div></li></ol></li><li><div class="list-item-paragraph">Bolts are manufactured to fit into holes in steel plates. The bolts have diameters, in cm, that follow the distribution $\mathrm{N}(2.65, 0.03^2)$ and the diameters of the holes, in cm, follow the distribution $\mathrm{N}(2.72, 0.02^2)$. A manufacturer sells boxes of twenty pairs of these bolts and steel plates, where each pair consists of one randomly selected bolt and one randomly selected steel plate. A pair is acceptable only if the diameter of the hole in the steel plate is at least $0.02$ cm larger, but no more than $0.15$ cm larger, than the diameter of the bolt. Use a normal distribution to estimate the probability that the average number of acceptable pairs in $50$ boxes is more than $18$.<span class="inline-marks">[4]</span></div></li></ol>

Central Limit Theorem

2024 RI P2 Q9

Sampling

<p>Aran is a telemarketer working in a large city. Each day, he speaks to $50$ potential customers and on average, he successfully makes a sale $6\%$ of the time. The number of successful sales each day is denoted by the random variable $T$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">State, in context, two assumptions needed for&nbsp;$T$ to be well modelled by a binomial distribution.<span class="inline-marks">[2]</span></div></li></ol>
<p>Assume now that&nbsp;$T$ has a binomial distribution.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Find the probability that Aran makes at least $5$ successful sales on a randomly chosen day.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Aran receives a bonus if he makes at least $5$ successful sales a day. Find the probability that he receives a bonus on $2$ days out of a randomly chosen $5$-day work week.<span class="inline-marks">[2]</span></div></li></ol>
<p>Aran is part of a team of $100$ similar telemarketers.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Estimate the probability that on a randomly chosen day, the team makes more than $3.5$ successful sales on average.<span class="inline-marks">[3]</span></div></li></ol>

Aran is a telemarketer working in a large city. Each day, he speaks to $50$ potential customers and on average, he successfully makes a sale $6\%$ of the time. The number of successful sales each day is denoted by the random variable $T$.

Assume now that $T$ has a binomial distribution.

Aran is part of a team of $100$ similar telemarketers.

2024 CJC P2 Q7 

<p>Box $A$ contains five cards numbered $1$, $1$, $2$, $2$ and $3$ and Box $B$ contains three cards numbered $4$, $4$ and $5$. Cards numbered $1$ and $5$ are red, whereas cards numbered $2$, $3$ and $4$ are blue.</p>
<p>A card is drawn from each of the two boxes. If both cards are the same colour then the score will be the product of the numbers on the two cards. If both cards are different in colour then the score will be the sum of the numbers on the two cards.</p>
<p>Let&nbsp;$X$ be the score obtained. The probability distribution of $X$ is given below:</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 400px;"><colgroup><col style="width: 80px;"><col style="width: 80px;"><col style="width: 80px;"><col style="width: 80px;"><col style="width: 80px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">$x$</th><th style="text-align: center; --cell-justify: center">$5$</th><th style="text-align: center; --cell-justify: center">$7$</th><th style="text-align: center; --cell-justify: center">$8$</th><th style="text-align: center; --cell-justify: center">$12$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$P(X = x)$</td><td style="text-align: center; --cell-justify: center">$h$</td><td style="text-align: center; --cell-justify: center">$k$</td><td style="text-align: center; --cell-justify: center">$\frac{1}{3}$</td><td style="text-align: center; --cell-justify: center">$k$</td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;$h = \frac{2}{5}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find $k$, $ \mathrm{E}(X)$ and $ \mathrm{Var}(X)$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Find the probability that the mean score of $60$ independent observations of $X$ lies between $6.5$ and $7.5$.<span class="inline-marks">[3]</span></div></li></ol>
<p>In a game, a player pays $\$20$ and draws a card each from Box $A$ and Box $B$. She receives, in dollars, an amount equal to three times the score obtained. Find the variance of her winnings per game.</p>

Box $A$ contains five cards numbered $1$, $1$, $2$, $2$ and $3$ and Box $B$ contains three cards numbered $4$, $4$ and $5$. Cards numbered $1$ and $5$ are red, whereas cards numbered $2$, $3$ and $4$ are blue.

A card is drawn from each of the two boxes. If both cards are the same colour then the score will be the product of the numbers on the two cards. If both cards are different in colour then the score will be the sum of the numbers on the two cards.

Let $X$ be the score obtained. The probability distribution of $X$ is given below:

In a game, a player pays $\$20$ and draws a card each from Box $A$ and Box $B$. She receives, in dollars, an amount equal to three times the score obtained. Find the variance of her winnings per game.

2018 TJC P2 Q8

Discrete Random Variables (DRV)

<p>The random variable $X$ has the distribution $X \sim \mathrm{N}(\mu, \sigma^2)$. It is given that $X_1$ and $X_2$ are $2$ independent observations of $X$ with $\mathrm{P}(4 &lt; X &lt; 6) = 0.5$ and $\mathrm{P}(X_1 + X_2 &gt; 10) = 0.5$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$\mu = 5$ and state, with a reason, the value of $\mathrm{P}(2X &gt; 10)$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the value of $\sigma$, giving your answer to $5$ significant figures.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">A random sample of $n$ observations of $X$ is obtained. Given that there is a probability of more than $95\%$ such that the sample mean is within $0.1$ units of $\mu$, find the least value of $n$.</span><span class="inline-marks">[4]</span></div></li></ol>

The random variable $X$ has the distribution $X \sim \mathrm{N}(\mu, \sigma^2)$. It is given that $X_1$ and $X_2$ are $2$ independent observations of $X$ with $\mathrm{P}(4 < X < 6) = 0.5$ and $\mathrm{P}(X_1 + X_2 > 10) = 0.5$.

Show that $\mu = 5$ and state, with a reason, the value of $\mathrm{P}(2X > 10)$.

Find the value of $\sigma$, giving your answer to $5$ significant figures.

A random sample of $n$ observations of $X$ is obtained. Given that there is a probability of more than $95\%$ such that the sample mean is within $0.1$ units of $\mu$, find the least value of $n$.

<p>A private-hire vehicle firm, Euber, employs drivers to operate a fleet of cars. The mileage covered by an Euber driver on a weekday is denoted by $M$ km. A random sample of $60$ Euber drivers is taken and their mileage can be summarised by</p>
<p style="text-align: center">\[\sum(m-250) = 540 \text{ and}\hspace{0.5em} \sum(m-259)^2 = 8176.\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Calculate unbiased estimates of the population mean and variance.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>Assume that the population mean and variance follow the answers calculated in part <strong>(i)</strong>.</p>
<p>Random samples of $n$ Euber drivers are taken, where&nbsp;$n$ is large.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Over a long period of time, it is found that&nbsp;$25\%$ of the samples have a mean mileage of at least $260$ km. Find the value of $n$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain whether it is necessary to assume that&nbsp;$M$ follows a normal distribution in order for your calculations in part <strong>(ii)</strong> to hold.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">It is now given that&nbsp;$M$ is normally distributed with population variance $5^2$. Estimate the sample size needed so that Euber can be $95\%$ certain that the sample mean mileage will differ from the true mean by less than $500$m.</span><span class="inline-marks">[2]</span></div></li></ol>

A private-hire vehicle firm, Euber, employs drivers to operate a fleet of cars. The mileage covered by an Euber driver on a weekday is denoted by $M$ km. A random sample of $60$ Euber drivers is taken and their mileage can be summarised by

\[\sum(m-250) = 540 \text{ and } \sum(m-259)^2 = 8176.\]

Calculate unbiased estimates of the population mean and variance.

Assume that the population mean and variance follow the answers calculated in part **(i)**.

Random samples of $n$ Euber drivers are taken, where $n$ is large.

Over a long period of time, it is found that $25\%$ of the samples have a mean mileage of at least $260$ km. Find the value of $n$.

Explain whether it is necessary to assume that $M$ follows a normal distribution in order for your calculations in part **(ii)** to hold.

It is now given that $M$ is normally distributed with population variance $5^2$. Estimate the sample size needed so that Euber can be $95\%$ certain that the sample mean mileage will differ from the true mean by less than $500$m.

Sample Mean

2019 EJC BT2 P2 Q8 [Modified]

<p>A large number of samples of size $n$ are taken from a normal population with mean $74$ and standard deviation $6$. It was found that&nbsp;$28.2\%$ of the samples have sample mean that exceeds $75$. Estimate the value of $n$.</p>

A large number of samples of size $n$ are taken from a normal population with mean $74$ and standard deviation $6$. It was found that $28.2\%$ of the samples have sample mean that exceeds $75$. Estimate the value of $n$.

<p>An IP school has&nbsp;$2000$ students – there are $200$ students in each level from Secondary 1-4, $600$ in JC1 and $600$ in JC2. The Student Council is interested in finding $x$ hours, the amount of time each student spends in a week doing homework. The president of the Student Council thus decides to survey $50$ students. The data gathered is summarised as follows:</p>
<p style="text-align: center">$\sum{x}=600$;	$\sum{{{x}^{2}}}=7300$</p>
<ol class="i-list"><li><div class="list-item-paragraph">Calculate the unbiased estimates of the population mean and variance.	</div></li><li><div class="list-item-paragraph">Given that&nbsp;$\mu $=12, find the probability that the sample mean exceeds $12.1$.</div></li></ol>

An IP school has $2000$ students – there are $200$ students in each level from Secondary 1-4, $600$ in JC1 and $600$ in JC2. The Student Council is interested in finding $x$ hours, the amount of time each student spends in a week doing homework. The president of the Student Council thus decides to survey $50$ students. The data gathered is summarised as follows:

2012 NJC Q10 [Modified]

<ol class="i-list"><li><div class="list-item-paragraph">Factory $A$ produces nuts whose mass may be assumed to be normally distributed with mean $\mu $ grams and standard 	deviation $\sigma $ grams. A random sample of $50$ nuts is taken. It is given that the probability that the mean mass is less than $247$ grams is $0.018079$, correct to $5$ significant figures. It is also given that the probability that the total mass exceeds $12600$ grams is $0.78397$, correct to $5$ significant figures. Find the values of $\mu $ and $\sigma $, giving your answers to the nearest grams. <span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">(For this question, you should state clearly the values of the parameters of any normal distribution you use.) <br>Factory $B$ produces bolts and nuts. The masses, in grams, of bolts and nuts produced are modelled as having independent normal distributions with means and standard deviations as shown in the table: <br></div><div class="table-container"><table style="table-layout: fixed; width: 450px;"><colgroup><col style="width: 50px;"><col style="width: 200px;"><col style="width: 200px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p><strong>Mean Mass (in grams)</strong></p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p><strong>Standard Deviation (in grams)</strong></p></th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p><strong>Bolts</strong></p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$745$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$7.3$</p></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p><strong>Nuts</strong></p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$250$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$5$</p></td></tr></tbody></table></div><ol class="abc-list"><li><div class="list-item-paragraph">Find the probability that the mass of a randomly chosen bolt differs from $3$ times the mass of a randomly chosen nut by at least $40$ grams. 	<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">This factory introduces a new process which is able to reduce the mass of each nut by $10\%$. Find the probability that the total mass, after the introduction of this process, of $10$ randomly chosen nuts is less than $2.24$ kg. 	<span class="inline-marks">[3]</span></div></li></ol></li></ol>

2017 MJC P2 Q10

<ol class="abc-list"><li><div class="list-item-paragraph">A farm in the west of Singapore grows turnips for sale to the local market.</div><ol class="i-list"><li><div class="list-item-paragraph">$5$ turnips are randomly chosen. Find the probability that exactly one turnip weighs less than the lower 		quartile weight and exactly two turnips weigh more than the median weight.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">The mass of a randomly chosen turnip has mean $40$ g and standard deviation of $3$ g. If the probability that the mean mass of a large sample of $n$ turnips is greater than $39.6$ g exceeds $0.95$, find the least value of $n$.	<span class="inline-marks">[3]</span></div></li></ol></li><li><div class="list-item-paragraph">A random variable $X$ has the distribution $X\sim N\left( 40,{{3}^{2}} \right)$. The random variable $Y$ is related to $X$ by the formula 	$Y=aX-\frac{1}{b}$, where $a$ and $b$ are constants and $a&gt;0$. Given that $\mathrm{P}(Y&lt;85)=\mathrm{P}(Y&gt;155)=0.075$, find the values of $\mathrm{E}(Y)$ and $\mathrm{Var}(Y)$, and hence find the values of $a$ and $b$.<span class="inline-marks">[5]</span></div></li></ol>

2015 SRJC P2 Q11

<p>The voltage of a battery is a random variable with the distribution $\text{N}\hspace{0.1em}\left( \mu ,\ {{\sigma }^{2}} \right)$. The mean voltage of $4$ randomly selected batteries is denoted by $\overline{X}$. It is given that $\text{P}\hspace{0.1em}\left( \overline{X}&lt;22.2 \right)=\text{P}\hspace{0.1em}\left( \overline{X}&gt;26.2 \right)=0.0128$. State the value of $\mu $ and find the value of $\sigma $.</p>

The voltage of a battery is a random variable with the distribution $\text{N}\left( \mu ,\ {{\sigma }^{2}} \right)$. The mean voltage of $4$ randomly selected batteries is denoted by $\overline{X}$. It is given that $\text{P}\left( \overline{X}<22.2 \right)=\text{P}\left( \overline{X}>26.2 \right)=0.0128$. State the value of $\mu $ and find the value of $\sigma $.

2015 YJC P2 Q6

<p>The masses, in kilograms, for a randomly chosen pumpkin and a randomly chosen watermelon, are normally distributed with means $4.2$ and $8.5$, and standard deviations $1.6$ and $2.2$ respectively.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the range of values of $k$ such that the average mass of five randomly chosen watermelons exceeds $k$ kilograms 	occurs at most $10\%$ of the time. 	<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the probability that the total mass of five randomly chosen pumpkins differs from three times the average mass 	of five randomly chosen watermelons by more than $5$ kg. <span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">A fruit store has&nbsp;$70$ batches, each consisting of $n$ pumpkins. It is known that, on average, $3$ in $20$ pumpkins are 	rotten. Given that in the $70$ batches, the probability of having an average of at least $4$ rotten pumpkins per batch is 	more than $0.7$. Determine the minimum value of $n$ . <span class="inline-marks">[4]</span></div></li></ol>

The masses, in kilograms, for a randomly chosen pumpkin and a randomly chosen watermelon, are normally distributed with means $4.2$ and $8.5$, and standard deviations $1.6$ and $2.2$ respectively.

2014 MJC P2 Q10	

<p style="margin-top: 0.25rem; margin-bottom: 0.5rem">The exhaustion time of an alkaline battery is defined as the time it takes to discharge completely under controlled experimental conditions. Research has shown that the exhaustion time of AA-size alkaline batteries manufactured by the market leaders, Durazer and Energi-cell, have the following population parameters:</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 500px;"><colgroup><col style="width: 100px;"><col style="width: 200px;"><col style="width: 200px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p><strong>Mean (in hours)</strong></p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p><strong>Standard deviation (in hours)</strong></p></th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p><strong>Durazer</strong></p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$19.87$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$0.24$</p></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p><strong>Energi-cell</strong></p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$20.16$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$0.19$</p></td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-paragraph">$50$ Durazer and $80$ Energi-cell batteries are tested. Determine the distribution of the sample mean exhaustion time 	for each brand. <span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence find the probability that the difference in the mean exhaustion times of the two brands of batteries is less than 	$0.2$ hours. <span class="inline-marks">[3]</span></div></li></ol>

The exhaustion time of an alkaline battery is defined as the time it takes to discharge completely under controlled experimental conditions. Research has shown that the exhaustion time of AA-size alkaline batteries manufactured by the market leaders, Durazer and Energi-cell, have the following population parameters:

<p><strong>Standard deviation (in hours)</strong></p>

<p>The life, in hours, of a randomly chosen light bulb produced by a manufacturer is normally distributed with mean $1100$ and standard deviation $70$. How large a sample is required such that the probability that the mean life in the sample shall exceed $1120$ is not more than $5\%$? <br></p>
<p>Do you need to use Central Limit Theorem in your working? <br></p>

The life, in hours, of a randomly chosen light bulb produced by a manufacturer is normally distributed with mean $1100$ and standard deviation $70$. How large a sample is required such that the probability that the mean life in the sample shall exceed $1120$ is not more than $5\%$? 


Do you need to use Central Limit Theorem in your working? 


<p>The amount of rainfall $X$ in a certain city per day has mean $1.3$ metres and variance $2.8$ square metres. The amount of rainfall for a sample of $60$ days is collected.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Calculate the mean and variance of $\overline{X}$.</div></li><li><div class="list-item-paragraph">Find the probability that the mean amount of rainfall exceeds $0.8$ metres.</div></li><li><div class="list-item-paragraph">State two assumptions about the sample used in obtaining this estimate in <strong>(ii)</strong>.</div></li></ol>

The amount of rainfall $X$ in a certain city per day has mean $1.3$ metres and variance $2.8$ square metres. The amount of rainfall for a sample of $60$ days is collected.

<p>A sample, of size $73$, has a standard deviation of $4.622$. Estimate the population variance. </p>

A sample, of size $73$, has a standard deviation of $4.622$. Estimate the population variance. 

Finding The Population Variance Given Sample Variance

<p>A manufacturer of candles claimed that it produced birthday candles with a mean burning time of $6$ minutes. A random sample of $150$ birthday candles was tested and the burning times, $X$ minutes, were summarized by $\sum{(x-5)=120}$ and $\sum{{{(x-5)}^{2}}=638}$ . Calculate the unbiased estimates for the mean $\mu $ and variance${{\sigma }^{2}}$ .</p>

A manufacturer of candles claimed that it produced birthday candles with a mean burning time of $6$ minutes. A random sample of $150$ birthday candles was tested and the burning times, $X$ minutes, were summarized by $\sum{(x-5)=120}$ and $\sum{{{(x-5)}^{2}}=638}$ . Calculate the unbiased estimates for the mean $\mu $ and variance${{\sigma }^{2}}$ .

2005 SAJC P2 Q28

<p style="margin-bottom: 0.5rem">The thirty students in a class each took a Biology test. The marks obtained are shown in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 400px;"><colgroup><col style="width: 100px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Marks</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$4$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$5$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$6$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$7$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$8$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$10$</p></th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Frequency</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$2$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$5$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$11$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$8$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$3$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$1$</p></td></tr></tbody></table></div></div>
<p>Calculate the unbiased estimates of the population mean and variance.  </p>

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

Calculate the unbiased estimates of the population mean and variance.  

Statistical Estimates of Biology Test Marks

<p class="paragraph-header">Distribution of $\bar{X}$ </p>
<ol class="ABC-list"><li><div class="list-item-paragraph"><strong>When&nbsp;$X$ follows a Normal Distribution with Known Population and Variance</strong></div></li></ol>
<p>If a random sample of size $n$ is taken from a <strong>normal distribution</strong> with mean, $E(X)=\mu $ and variance, $Var(X)={{\sigma }^{2}}$, then the sample mean, $\bar{X}$ , follows a <strong>normal distribution </strong> with mean $\mu $and variance $\frac{{{\sigma }^{2}}}{n}$ . </p>
<p style="text-align: center">$X\sim N\left( \mu ,{{\sigma }^{2}} \right)\Rightarrow \bar{X}\sim N\left( \mu ,\frac{{{\sigma }^{2}}}{n} \right)$  and ${{X}_{1}}+{{X}_{2}}+...+{{X}_{n}}\sim N(n\mu ,n{{\sigma }^{2}})$</p>
<ol class="ABC-list" start="2"><li><div class="list-item-paragraph"><strong>When&nbsp;$X$ follows Any Distribution with Known Population and Variance</strong></div></li></ol>
<p class="paragraph-header">Central Limit Theorem, CLT</p>
<p>If a random sample of size $n$, where&nbsp;$n$ is large ($n&gt;30$), is taken from a <strong>any distribution</strong> with mean, $E(X)=\mu $ and variance, $Var(X)={{\sigma }^{2}}$, then by CLT, </p>
<p style="text-align: center">${{X}_{1}}+{{X}_{2}}+...+{{X}_{n}}\sim N(n\mu ,n{{\sigma }^{2}})$ approximately,</p>
<p style="text-align: center">Sample mean of $X$, $\overline{X}\sim N\left( \mu ,\frac{{{\sigma }^{2}}}{n} \right)$ approximately. </p>
<p>Note, if&nbsp;$X$ is already normally distributed, then there is <strong>no </strong> need to use <strong>Central Limit Theorem</strong>. </p>

If a random sample of size $n$ is taken from a **normal distribution** with mean, $E(X)=\mu $ and variance, $Var(X)={{\sigma }^{2}}$, then the sample mean, $\bar{X}$ , follows a **normal distribution ** with mean $\mu $and variance $\frac{{{\sigma }^{2}}}{n}$ . 

$X\sim N\left( \mu ,{{\sigma }^{2}} \right)\Rightarrow \bar{X}\sim N\left( \mu ,\frac{{{\sigma }^{2}}}{n} \right)$  and ${{X}_{1}}+{{X}_{2}}+...+{{X}_{n}}\sim N(n\mu ,n{{\sigma }^{2}})$

If a random sample of size $n$, where $n$ is large ($n>30$), is taken from a **any distribution** with mean, $E(X)=\mu $ and variance, $Var(X)={{\sigma }^{2}}$, then by CLT, 

${{X}_{1}}+{{X}_{2}}+...+{{X}_{n}}\sim N(n\mu ,n{{\sigma }^{2}})$ approximately,

Sample mean of $X$, $\overline{X}\sim N\left( \mu ,\frac{{{\sigma }^{2}}}{n} \right)$ approximately. 

Note, if $X$ is already normally distributed, then there is **no ** need to use **Central Limit Theorem**. 

The Sampling Distribution of the Sample Mean $\bar{X}$ 

Statistical Inference and Unbiased Estimation

<p>An aptitude test for deep-sea divers produces scores which are normally distributed on a scale from $0$ to $100$. A random sample of $160$ divers were assessed, and each of their individual test scores, $x$, was recorded. The results are summarised by</p>
<p style="text-align: center">$\sum x = 9120, \qquad\qquad\qquad \sum (x-\bar{x})^{2} = 35775$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance.</div></li><li><div class="list-item-paragraph">Assuming that the unbiased estimate of the population mean obtained above is equal to the population mean, find the value of $c$ such that there is a probability of $0.05$ that a sample of $160$ scores has a sample mean that differs from its mean score by more than $c$.</div></li><li><div class="list-item-paragraph">Comment on the validity of the calculations if the scores had not been known to be normally distributed.</div></li></ol>

An aptitude test for deep-sea divers produces scores which are normally distributed on a scale from $0$ to $100$. A random sample of $160$ divers were assessed, and each of their individual test scores, $x$, was recorded. The results are summarised by

$\sum x = 9120, \qquad\qquad\qquad \sum (x-\bar{x})^{2} = 35775$.

Math Question Bank