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E Math (IGCSE)

H1 Math

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IB AA HL Math

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N(A) E Math

PSLE

PSLE Foundation Mathematics

<p><strong>In this question you should assume that&nbsp;$T$, $X$ and $Y$ follow independent normal distributions. You should also state the parameters of any distribution you use.</strong></p>
<p>A customer service call centre categorises incoming calls for follow-up resolution of the issues. The time taken to categorise an incoming call, $T$ minutes, follows a normal distribution $\mathrm{N}(2, 0.2^2)$. Incoming calls are categorised as routine or complex calls. The time taken to resolve a routine call, $X$ minutes, follows a normal distribution $\mathrm{N}(5, k)$ and the time taken to resolve a complex call, $Y$ minutes, follows a normal distribution $\mathrm{N}(20, 6^2)$. Any incoming call is first categorised as either routine or complex and is immediately followed up with the resolution of the issue. The duration of the call is taken to be the sum of the times taken to categorise the call and to resolve the issue.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Sketch the distribution of $T$ for&nbsp;$1.2 \le t \le 2.8$.<span class="inline-marks">[2]</span></div></li></ol>
<p>For a randomly chosen incoming call that is categorised as routine and then resolved, there is a probability of 0.254 that the duration of the call is more than 8 minutes.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Show that&nbsp;$k = 2.24$ correct to 2 decimal places.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">On a weekday morning where there are 20 incoming calls, $n$ calls are categorised as complex calls. Given that there is a probability of at most 0.01 that the mean time taken to resolve the $n$ complex calls exceeds 24 mins, write down an inequality involving $\sqrt{n}$ and hence find the possible values of $n$.<span class="inline-marks">[3]</span></div></li></ol>
<p>There is a review of the resolution processes such that the time taken to resolve a complex call is now reduced by 20%.</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Find the probability that the time taken to resolve 2 complex calls is more than twice the time taken to resolve a routine call.<span class="inline-marks">[3]</span></div></li></ol>

**In this question you should assume that $T$, $X$ and $Y$ follow independent normal distributions. You should also state the parameters of any distribution you use.**

A customer service call centre categorises incoming calls for follow-up resolution of the issues. The time taken to categorise an incoming call, $T$ minutes, follows a normal distribution $\mathrm{N}(2, 0.2^2)$. Incoming calls are categorised as routine or complex calls. The time taken to resolve a routine call, $X$ minutes, follows a normal distribution $\mathrm{N}(5, k)$ and the time taken to resolve a complex call, $Y$ minutes, follows a normal distribution $\mathrm{N}(20, 6^2)$. Any incoming call is first categorised as either routine or complex and is immediately followed up with the resolution of the issue. The duration of the call is taken to be the sum of the times taken to categorise the call and to resolve the issue.

For a randomly chosen incoming call that is categorised as routine and then resolved, there is a probability of 0.254 that the duration of the call is more than 8 minutes.

There is a review of the resolution processes such that the time taken to resolve a complex call is now reduced by 20%.

Normal Distribution: Model and Parameters

Solving Normal Probability Inequalities

Linear Transformations of a Normal Variable

Determining Sample Size from Probability Conditions

Normal Models for Sample Means and Averages

Finding Unknown Normal Mean and Standard Deviation

Linear Combinations of Independent Normal Variables

2026 RI TP Q12

Continuous Random Variables

Sampling

<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.

Estimating Parameters from Summarised and Coded Data

Central Limit Theorem

Sample Mean and Its Sampling Distribution

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 $n$.

<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 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? 


Assumptions for Sampling

<p>If a large number of samples of size $n$ are taken form a distribution which has a value of $2.5$ for its mean and variance, and approximately $5\%$ of the sample means are less than $2.025$, find $n$.</p>

If a large number of samples of size $n$ are taken form a distribution which has a value of $2.5$ for its mean and variance, and approximately $5\%$ of the sample means are less than $2.025$, find $n$.

Sampling From A Non-Normal Population (Central Limit Theorem)  

<p>If a large number of samples of size $n$ are taken form a distribution which has a value of $5.2$ for its mean and variance, and approximately $23.6\%$ of the sample means are less than $5$, find $n$.</p>

If a large number of samples of size $n$ are taken form a distribution which has a value of $5.2$ for its mean and variance, and approximately $23.6\%$ of the sample means are less than $5$, find $n$.

<p>A number of random samples of size $n$ are taken from a normal population with mean $83$ and variance $21$ and the sample means are calculated. If at least $95$% of the sample means exceed $80$, find the least possible value of $n$.</p>

A number of random samples of size $n$ are taken from a normal population with mean $83$ and variance $21$ and the sample means are calculated. If at least $95$% of the sample means exceed $80$, find the least possible value of $n$.

Using GC to find the possible value of n

<p>A large number of random samples of size $n$ are taken from a normal population with mean $30$ and variance $36$ and the sample means are calculated. If at most $95$% of the sample means exceed $27$, find the largest possible value of $n$.</p>

A large number of random samples of size $n$ are taken from a normal population with mean $30$ and variance $36$ and the sample means are calculated. If at most $95$% of the sample means exceed $27$, find the largest possible value of $n$.

<p>Each month the amount of electricity, $X$ measured in kilowatt-hours (kWh), used by a household in a particular city may be assumed to follow a normal distribution with mean $950$ and standard deviation $\sigma $ . The charge for electricity used per month is fixed at $\$0.22$ per kWh.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Given that $65\%$ of the households uses less than $960$ kWh of electricity in a month, find the value of $\sigma $, correct to $1$ decimal place.</div><p style="text-align: right;">[2]</p></li></ol>
<p>For the rest of the question, $\sigma $ is the value found in part <strong>(i)</strong>.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the probability that the difference in the amount of electricity used among $2$ randomly chosen households in a particular month is not more than $30$ kWh.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">In the month of August, the mayor of the city decides to provide $50\%$ and $30\%$ subsidies for the electricity bills of households in the North and South districts of the city respectively. Find the probability that the total electricity bill of $2$ randomly chosen North district households and $1$ South district household is less than $\$360$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">In December, a random sample of n households is chosen to study the mean monthly electricity usage per household in the city. Find the least value of n if the probability of the sample mean being less than $955$ kWh is at least $0.9$.</div><p style="text-align: right;">[3]</p></li></ol>

Each month the amount of electricity, $X$ measured in kilowatt-hours (kWh), used by a household in a particular city may be assumed to follow a normal distribution with mean $950$ and standard deviation $\sigma $ . The charge for electricity used per month is fixed at $\$0.22$ per kWh.

For the rest of the question, $\sigma $ is the value found in part **(i)**.

2017 RVHS P2 Q10

<p><strong>In this question you should, where applicable, state clearly all the distributions that you use, together with the values of the appropriate parameters.</strong></p>
<p>A supermarket produces its own packs of white sesame seeds and fennel seeds for sale under its house brand. The masses (in grams) of a randomly chosen pack of white sesame seeds and that of a randomly chosen pack of fennel seeds, denoted by $W$ and $F$ respectively, have independent normal distributions. The means and standard deviations of these distributions are shown in the following table</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: left; --cell-justify: flex-start">Mass of a pack of</th><th style="text-align: left; --cell-justify: flex-start">Mean (g)</th><th style="text-align: left; --cell-justify: flex-start">Standard deviation (g)</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">white sesame seeds $({W})$</td><td style="text-align: left; --cell-justify: flex-start">$250$</td><td style="text-align: left; --cell-justify: flex-start">${\sigma}$</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">fennel seeds $({F})$</td><td style="text-align: left; --cell-justify: flex-start">$300$</td><td style="text-align: left; --cell-justify: flex-start">$2.5$</td></tr></tbody></table></div></div>
<p>It is given that&nbsp;$5\%$ of the packs of white sesame seeds have a mass of less than $245$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760683631808-ixscivqms" data-list-item-path="item-1760683631808-ixscivqms"><div class="list-item-paragraph">Show that the value of $\sigma $ is $\mathrm{3}\mathrm{.0398}$, correct to $5$ significant figures.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760683632580-3yz2d73ha" data-list-item-path="item-1760683632580-3yz2d73ha"><div class="list-item-paragraph">Find the probability that the total mass of six randomly chosen packs of white sesame seeds is less than five times the mass of a randomly chosen pack of fennel seeds by not more than $20$ grams.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760683634757-wnzo2h8xn" data-list-item-path="item-1760683634757-wnzo2h8xn"><div class="list-item-paragraph">$n$ packs of white sesame seeds and $n$ packs of fennel seeds are chosen at random. Find the smallest value of $n$ such that the probability that the mean mass of these $2n$ packs being at least $278$grams is less than $0.015$.<span class="inline-marks">[4]</span></div></li></ol>

**In this question you should, where applicable, state clearly all the distributions that you use, together with the values of the appropriate parameters.**

A supermarket produces its own packs of white sesame seeds and fennel seeds for sale under its house brand. The masses (in grams) of a randomly chosen pack of white sesame seeds and that of a randomly chosen pack of fennel seeds, denoted by $W$ and $F$ respectively, have independent normal distributions. The means and standard deviations of these distributions are shown in the following table

It is given that $5\%$ of the packs of white sesame seeds have a mass of less than $245$.

2023 HCI P2 Q10

<p>The random variable $X$ is normally distributed with mean $\mu $ and variance ${{\sigma }^{2}}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760683461527-ea96b8ekp" data-list-item-path="item-1760683461527-ea96b8ekp"><div class="list-item-paragraph">Find $\mathrm{P}\left( X&gt;\mu +0.05\sigma \right)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760683462143-lzfp5v0xe" data-list-item-path="item-1760683462143-lzfp5v0xe"><div class="list-item-paragraph">The mean of a random sample of $n$ independent observations of $X$ is denoted by $\bar{X}$.</div><div class="list-item-paragraph">Find the least value of $n$ such that the probability that&nbsp;$\bar{X}$ differs from $\mu $ by at least $0.05\sigma $ is less than $0.4$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760683463553-6e6h0ye4h" data-list-item-path="item-1760683463553-6e6h0ye4h"><div class="list-item-paragraph">In another random sample of $m$ independent observations of $X$, the probability that at least $4$ of these observations exceeds $\mu +0.05\sigma $ each is at most $\mathrm{0}\mathrm{.95}$. Find the greatest value of $m$.<span class="inline-marks">[3]</span></div></li></ol>

The random variable $X$ is normally distributed with mean $\mu $ and variance ${{\sigma }^{2}}$.

Maximum Number of Binomial Trials

Standardising Normal Random Variables

2023 NJC P2 Q6

Discrete Random Variables (DRV)

<p>The owner of a fruit stall, Mr Lee, bought a large batch of durians from a durian farm.</p>
<p>The weights (in kg) of a sample of six durians from this batch are</p>
<p style="text-align: center">$1.0$,        $1.1$,       $1.3$,       $1.7$,       $a$,       $b$.</p>
<p>It is known that the unbiased estimates for the population mean and variance based on this sample are $1.25$kg and $0.095$kg$^{2}$ respectively.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the values of $a$ and $b$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Assuming that the weight of durians from Robert’s farm follows a normal distribution with mean $1.60$kg and standard deviation $0.3$kg, carry out an appropriate test at the $5%$ level to find out whether there is significant evidence that this batch of durians is not taken from Robert’s farm.</div><p style="text-align: right;">[4]</p></li></ol>
<p>The mean and standard deviation of the weight of papayas from Robert’s farm are $1.2$kg and $0.45$kg respectively. Mr Lee bought a batch of papayas from Robert’s farm. Find the least number of papayas in the batch such that the probability of the average weight exceeding $1.3$kg is less than $0.05$.</p><p style="text-align: right;">[3]</p>

The owner of a fruit stall, Mr Lee, bought a large batch of durians from a durian farm.

The weights (in kg) of a sample of six durians from this batch are

$1.0$,        $1.1$,       $1.3$,       $1.7$,       $a$,       $b$.

It is known that the unbiased estimates for the population mean and variance based on this sample are $1.25$kg and $0.095$kg$^{2}$ respectively.

The mean and standard deviation of the weight of papayas from Robert’s farm are $1.2$kg and $0.45$kg respectively. Mr Lee bought a batch of papayas from Robert’s farm. Find the least number of papayas in the batch such that the probability of the average weight exceeding $1.3$kg is less than $0.05$.

Unbiased Estimates of Population Parameters

Mean Tests for Normal Populations with Known Variance

Hypothesis Tests Using Critical Regions and $p$-Values

Two-Tailed Hypothesis Tests

2012 AJC MYCT P2 Q4

Hypothesis Testing

<p>A company manufactures screen protectors for handphones. On average, $17\%$ of the screen protectors are cracked. The screen protectors are packaged into boxes of $20$. It should be assumed that the number of cracked screen protectors in a box of $20$ screen protectors follows a binomial distribution.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that the probability of having no more than $3$ cracked screen protectors in a randomly chosen box is $0.55041$, correct to $5$ significant figures.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Find the most probable number of cracked screen protectors in a randomly chosen box.<span class="inline-marks">[1]</span></div></li></ol>
<p>Every month, the company exports a large shipment of n cartons of screen protectors. In each carton, there are $8$ boxes of screen protectors. Every carton is assumed to be filled completely.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Using a suitable approximation, find the least value of $n$ such that in a randomly chosen month, there is at least a $99\%$ chance that the average number of boxes per carton having no more than $3$ cracked screen protectors is at most $5$.<span class="inline-marks">[4]</span></div></li></ol>
<p>To determine whether cartons are ready for local or overseas sales, cartons are randomly selected for inspection. In a carton,</p>
<p style="margin-left: 3rem">• if there are at least $4$ boxes that contain no more than $3$ cracked screen protectors each, the carton is ready for local sales;</p>
<p style="margin-left: 3rem">• if there are at least $6$ boxes that contain no more than $3$ cracked screen protectors each, the carton is ready for overseas sales.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">A carton that is not ready for overseas sales is randomly selected. Find the probability that the carton is ready for local sales.<span class="inline-marks">[3]</span></div></li></ol>

A company manufactures screen protectors for handphones. On average, $17\%$ of the screen protectors are cracked. The screen protectors are packaged into boxes of $20$. It should be assumed that the number of cracked screen protectors in a box of $20$ screen protectors follows a binomial distribution.

Every month, the company exports a large shipment of n cartons of screen protectors. In each carton, there are $8$ boxes of screen protectors. Every carton is assumed to be filled completely.

To determine whether cartons are ready for local or overseas sales, cartons are randomly selected for inspection. In a carton,

• if there are at least $4$ boxes that contain no more than $3$ cracked screen protectors each, the carton is ready for local sales;

• if there are at least $6$ boxes that contain no more than $3$ cracked screen protectors each, the carton is ready for overseas sales.

Mode of a Binomial Distribution

Conditional Probability

Binomial Probabilities Using Technology

Repeated Binomial Experiments

2022 EJC P2 Q6

Probability

<p>A random sample of $100$ packets of a breakfast cereal was examined and the mass, $m$ grams, of the contents recorded. The results revealed that $\sum{m=3010}$ and $\sum{{{m}^{2}}=91\,280}$. Obtain an unbiased estimate of the population variance.</p><p style="text-align: right;">[2]</p>
<p>By taking the unbiased estimate obtained above to be the population variance, find the size of the random sample required so that there is a probability of at least $0.98$ that the sample mean will be within $0.5$g of the true mean.</p><p style="text-align: right;">[5]</p>

A random sample of $100$ packets of a breakfast cereal was examined and the mass, $m$ grams, of the contents recorded. The results revealed that $\sum{m=3010}$ and $\sum{{{m}^{2}}=91\,280}$. Obtain an unbiased estimate of the population variance.

By taking the unbiased estimate obtained above to be the population variance, find the size of the random sample required so that there is a probability of at least $0.98$ that the sample mean will be within $0.5$g of the true mean.

ASRJC Sampling Tutorial Q5

<p>A computer can give independent observations of a random variable $X$ with probability distribution given by $\mathrm{P}\left( X=0 \right)=\frac{3}{4}$ and $\mathrm{P}\left( X=2 \right)=\frac{1}{4}$. It is programmed to output a value for the random variable $Y$ defined by $Y={{X}_{1}}+{{X}_{2}}$, where ${{X}_{1}}$ and ${{X}_{2}}$ are independent observations of $X$. Tabulate the probability distribution of $Y$, show that $E\left( Y \right)=1$.</p><p style="text-align: right;">[3]</p>
<p>The random variable $T$ is defined by $T={{Y}^{2}}$. Find $\mathrm{E}\left( T \right)$ and show that $\mathrm{Var}\left( T \right)=\frac{63}{4}$.</p><p style="text-align: right;">[3]</p>
<p>The computer is programmed to produce a large number $n$ of independent values of $T$ and to calculate the mean $M$ of these values. Find the smallest value of $n$ such that $\mathrm{P}\left( M<3 \right)>0.99$.</p><p style="text-align: right;">[4]</p>

A computer can give independent observations of a random variable $X$ with probability distribution given by $\mathrm{P}\left( X=0 \right)=\frac{3}{4}$ and $\mathrm{P}\left( X=2 \right)=\frac{1}{4}$. It is programmed to output a value for the random variable $Y$ defined by $Y={{X}_{1}}+{{X}_{2}}$, where ${{X}_{1}}$ and ${{X}_{2}}$ are independent observations of $X$. Tabulate the probability distribution of $Y$, show that $E\left( Y \right)=1$.

The random variable $T$ is defined by $T={{Y}^{2}}$. Find $\mathrm{E}\left( T \right)$ and show that $\mathrm{Var}\left( T \right)=\frac{63}{4}$.

The computer is programmed to produce a large number $n$ of independent values of $T$ and to calculate the mean $M$ of these values. Find the smallest value of $n$ such that $\mathrm{P}\left( M<3 \right)>0.99$.

Constructing and Using Discrete Probability Distributions

Expectation and Variance of Discrete Random Variables

Linear Combinations of Random Variables

N1998 P2 Q7

<p>The waiting times, in minutes, for orders of Chili Crab and Pepper Crab at a popular hawker stall are modelled as having independent normal distributions with means and standard deviations as shown in the following table:</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 450px;"><colgroup><col style="width: 150px;"><col style="width: 150px;"><col style="width: 150px;"></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">Mean Waiting Time</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Standard Deviation</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Chili Crab</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">${\mu}$</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">Pepper Crab</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$10$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$2$</td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-paragraph">If the probability that one randomly chosen order of Pepper Crab takes a longer waiting time than one randomly chosen order of Chili Crab is $0.290$, show that&nbsp;$\mu =12.0$, correct to $3$ significant figures.<span class="inline-marks">[3]</span></div></li></ol>
<p>Customers who drive park their cars at a nearby car park at $\$0.05$ per minute. The walking time between the car park and the hawker stall is negligible. The time taken by a randomly chosen customer to finish his meal is normally distributed with mean $20$ minutes and standard deviation $4$ minutes.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the probability that a randomly chosen customer who parked at the car park, bought one order of Pepper Crab, finished his meal and immediately drove off, spent more than $\$2$ on car park charges.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">A random sample of $n$ customers who each make one order of Pepper Crab is taken. Given that the probability that their mean waiting time exceeds $10.5$ minutes is at most $0.0385$, find the smallest possible value of $n$. Explain briefly if any assumption is needed for the above calculation.<span class="inline-marks">[4]</span></div></li></ol>

The waiting times, in minutes, for orders of Chili Crab and Pepper Crab at a popular hawker stall are modelled as having independent normal distributions with means and standard deviations as shown in the following table:

Customers who drive park their cars at a nearby car park at $\$0.05$ per minute. The walking time between the car park and the hawker stall is negligible. The time taken by a randomly chosen customer to finish his meal is normally distributed with mean $20$ minutes and standard deviation $4$ minutes.

2013 CJC P2 Q12

<p>A sample of $n$ observations was taken from a population of mean $\mu$ and variance 10. Find the least sample size required such that the probability of the sample mean lying between $\mu -0.5$ and $\mu +0.5$ is more than 0.89. State an assumption that you have to make in order to proceed with this calculation.</p><p style="text-align: right;">[6]</p>

A sample of $n$ observations was taken from a population of mean $\mu$ and variance 10. Find the least sample size required such that the probability of the sample mean lying between $\mu -0.5$ and $\mu +0.5$ is more than 0.89. State an assumption that you have to make in order to proceed with this calculation.

2012 CJC P2 Q11

<p>In this question you should state the parameters of any distributions you use.</p>
<p>A teacher, while analysing the performance of a large cohort for a recent quiz, finds that the score of each student from the entire cohort follows the distribution $\mathrm{N}\left( 50,{{\sigma }^{2}} \right)$. He also finds that $95\%$ of the students scored between $36$ and $64$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the value of ${{\sigma }^{2}}$.</div><p style="text-align: right;">[2]</p></li></ol>
<p>You may assume $\sigma =7$ for the remaining parts of the question.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Three students are selected at random from the cohort. Find the probability that</div><ol class="i-list"><li><div class="list-item-paragraph">the sum of the scores of the first two students is less than $1.5$ times the score of the third student,</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">the lowest score of the three students is more than $53$.</div><p style="text-align: right;">[2]</p></li></ol></li><li><div class="list-item-paragraph">Find the least integer $n$ such that there is at least a $95\%$ chance that the average score of $n$ randomly chosen students is at least $48$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">$15$ students are selected at random. Find the probability that less than half of them scored at most $55$ for the quiz.</div><p style="text-align: right;">[3]</p></li></ol>

In this question you should state the parameters of any distributions you use.

A teacher, while analysing the performance of a large cohort for a recent quiz, finds that the score of each student from the entire cohort follows the distribution $\mathrm{N}\left( 50,{{\sigma }^{2}} \right)$. He also finds that $95\%$ of the students scored between $36$ and $64$.

You may assume $\sigma =7$ for the remaining parts of the question.

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