A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

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

<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>

Normal Probabilities and Quantiles

Central Limit Theorem

2024 RI P2 Q9

<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

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 $n$.

<p>The weights of a randomly chosen $10$-month old male baby, $X$ kg, and a randomly chosen female baby, $Y$ kg, are modelled as having independent normal distributions with means and standard deviations as shown in the 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><th style="text-align: left; --cell-justify: flex-start"></th><th style="text-align: center; --cell-justify: center">Mean</th><th style="text-align: center; --cell-justify: center">Standard Deviation</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$X$</td><td style="text-align: center; --cell-justify: center">$9.2$</td><td style="text-align: center; --cell-justify: center">$0.8$</td></tr><tr><td style="text-align: center; --cell-justify: center">$Y$</td><td style="text-align: center; --cell-justify: center">$8.5$</td><td style="text-align: center; --cell-justify: center">$0.5$</td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that the difference between thrice the weight of a male baby and the total weight of five female babies does not exceed $10$ kg.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that the mean weight of four randomly chosen male babies is at least $0.5$ kg more than the mean weight of three randomly chosen female babies.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">The probability that the total weight of two males and one female babies is less than $m$ kg is at least $0.9$. Find the least integral value of $m$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Four female babies are selected at random. Calculate the probability that exactly two of them weigh more than $9$ kg and exactly one of them weighs less than $8$ kg.</span><span class="inline-marks">[3]</span></div></li></ol>

The weights of a randomly chosen $10$-month old male baby, $X$ kg, and a randomly chosen female baby, $Y$ kg, are modelled as having independent normal distributions with means and standard deviations as shown in the table.

Find the probability that the difference between thrice the weight of a male baby and the total weight of five female babies does not exceed $10$ kg.

Find the probability that the mean weight of four randomly chosen male babies is at least $0.5$ kg more than the mean weight of three randomly chosen female babies.

The probability that the total weight of two males and one female babies is less than $m$ kg is at least $0.9$. Find the least integral value of $m$.

Four female babies are selected at random. Calculate the probability that exactly two of them weigh more than $9$ kg and exactly one of them weighs less than $8$ kg.

Comparing Independent Sample Means

RI Normal Distribution Tutorial Q4

<p>In this question you should state the parameters of any normal distributions you use.<br><br>A machine grades apples according to their mass. Apples with mass exceeding $120$ g are labelled as "large" and apples with mass less than $80$ g are labelled as "small". A large batch of apples is graded and it is found that&nbsp;$15\%$ are "large" and $10\%$ are "small". It is known that the masses of the apples are normally distributed.</p>
<ol class="abc-list"><li data-list-item-id="item-7a" data-list-item-path="item-7a"><div class="list-item-paragraph">Find the mean mass of a randomly chosen apple from the batch and show that the standard deviation is $17.3$ g, correct to $3$ significance figures.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-7b" data-list-item-path="item-7b"><div class="list-item-paragraph">Two apples are chosen in random. Find the probability that the difference in their masses is at least $20$ g.<span class="inline-marks">[3]</span></div></li></ol>
<p>The apples are labelled and packed into boxes. Each box contains $20$ apples.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-7c" data-list-item-path="item-7c"><div class="list-item-paragraph">Apples with mass between $110$ g and $140$ g are labelled as "premium". Find the expected number of "premium" apples in a randomly chosen box.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-7d" data-list-item-path="item-7d"><div class="list-item-paragraph">Find the probability that the mean mass of apples in a randomly chosen box is at most $105$ g.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-7e" data-list-item-path="item-7e"><div class="list-item-paragraph">The boxes of apples are sold by weight at a price of $\$2.90$ per kilogram. Find the probability that the total price of $3$ randomly chosen boxes of apples exceeds $\$18$.<span class="inline-marks">[3]</span></div></li></ol>

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

A machine grades apples according to their mass. Apples with mass exceeding $120$ g are labelled as "large" and apples with mass less than $80$ g are labelled as "small". A large batch of apples is graded and it is found that $15\%$ are "large" and $10\%$ are "small". It is known that the masses of the apples are normally distributed.

Find the mean mass of a randomly chosen apple from the batch and show that the standard deviation is $17.3$ g, correct to $3$ significance figures.

Two apples are chosen in random. Find the probability that the difference in their masses is at least $20$ g.

The apples are labelled and packed into boxes. Each box contains $20$ apples.

Apples with mass between $110$ g and $140$ g are labelled as "premium". Find the expected number of "premium" apples in a randomly chosen box.

Find the probability that the mean mass of apples in a randomly chosen box is at most $105$ g.

The boxes of apples are sold by weight at a price of $\$2.90$ per kilogram. Find the probability that the total price of $3$ randomly chosen boxes of apples exceeds $\$18$.

2025 NYJC P2 Q11

<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

<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>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 class="paragraph-header">IB AA SL and HL: core and optional content</p>
<p>Core: the single-boy weight percentile and one girl/one boy on opposite sides of 47 kg. Optional: the difference of two girls’ weights and the pooled mean weight.</p>
<p style="margin-bottom: 1rem">The weights of girls and boys in a college have independent normal distributions with their means and standard deviations given 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: 33.3333%;"><col style="width: 33.3333%;"><col style="width: 33.3333%;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Mean</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Standard Deviation</p></th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Girls</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$44$ kg</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$5$ kg</p></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Boys</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$50$ kg</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$4$ kg</p></td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-1766738012901-jtlmfsu4l" data-list-item-path="item-1766738012901-jtlmfsu4l"><div class="list-item-paragraph">Find the greatest weight (correct to nearest $0.1$ kg) exceeded by at least $10\%$ of the boys.	<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1766738013636-kdtjc6hwh" data-list-item-path="item-1766738013636-kdtjc6hwh"><div class="list-item-paragraph">Find the probability that the difference of the weights of $2$ randomly chosen girls is less than $1.2$ kg.	<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1766738014910-e5zwkl8df" data-list-item-path="item-1766738014910-e5zwkl8df"><div class="list-item-paragraph">One girl and one boy are chosen at random. Find the probability that one of them is at least $47$ kg and the other is 	less than $47$ kg.	<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1766738057847-f2agt0w5u" data-list-item-path="item-1766738057847-f2agt0w5u"><div class="list-item-paragraph">Find the probability that the average of $2$ randomly chosen girls and $8$ randomly chosen boys is less than $50$ kg.<span class="inline-marks">[3]</span></div></li></ol>
<p>Modelling assumption for this exercise: weights of distinct selected students are mutually independent and follow the stated distribution for their group.</p>

IB AA SL and HL: core and optional content

Core: the single-boy weight percentile and one girl/one boy on opposite sides of 47 kg. Optional: the difference of two girls’ weights and the pooled mean weight.

The weights of girls and boys in a college have independent normal distributions with their means and standard deviations given in the following table.

Modelling assumption for this exercise: weights of distinct selected students are mutually independent and follow the stated distribution for their group.

Linear Combinations, Statistical Modeling, and Probabilistic Inference

<p class="paragraph-header">IB AA SL: core and optional content</p>
<p>Core: single-bulb interval and binomial box rejection. Optional: difference involving the average fluorescent-bulb lifespan.</p>
<p>The lifespan of a halogen bulb is normally distributed with mean $160$ hours with standard deviation $10$ hours, while the lifespan of a fluorescent bulb is normally distributed with mean $240$ hours and standard deviation $12$ hours. The lifespan of any bulb is independent of one another.</p>
<ol class="i-list"><li data-list-item-id="item-1766737467487-3wxb55s61" data-list-item-path="item-1766737467487-3wxb55s61"><div class="list-item-paragraph">A halogen bulb is randomly chosen. Find the greatest value of $a$, correct to three significant figures, if the probability that its lifespan lies in the range of $(160-a,160+a)$ is at most $0.4$.	<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1766737483514-wqwhbhklk" data-list-item-path="item-1766737483514-wqwhbhklk"><div class="list-item-paragraph">Find the probability that the difference between the average lifespan of two fluorescent bulbs and twice the lifespan of a halogen bulb does not exceed $70$ hours. 	<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1767017879941-iz2l41fyi" data-list-item-path="item-1767017879941-iz2l41fyi"><div class="list-item-paragraph">The halogen bulbs are packed in boxes of $n$ bulbs. If there are more than $10$ bulbs that have lifespans of less than $150$ hours, the box will be rejected. Find the greatest value of $n$ so that the probability that a box will be rejected is less than $0.2$. <span class="inline-marks">[5]</span></div></li></ol>

Core: single-bulb interval and binomial box rejection. Optional: difference involving the average fluorescent-bulb lifespan.

The lifespan of a halogen bulb is normally distributed with mean $160$ hours with standard deviation $10$ hours, while the lifespan of a fluorescent bulb is normally distributed with mean $240$ hours and standard deviation $12$ hours. The lifespan of any bulb is independent of one another.

<p class="paragraph-header">IB AA SL: core and optional content</p>
<p>Core: individual-height probability in(b)(i) and qualitative model discussion in(a)(i)–(ii). Optional: distribution of a sample mean in(a)(iii), total-height probability in(b)(ii) and dependent comparison.</p>
<p style="margin-bottom: 1.25rem">A school has a large number of girls. For each year-group it may be assumed that the heights of girls are normally distributed, with average heights and standard deviations as given 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: 33.3333%;"><col style="width: 33.3333%;"><col style="width: 33.3333%;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Year-group</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Average height</p><p style="text-align: center;"></p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Standard deviation</p></th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$11$-year old girls</p><p style="text-align: left;"></p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p style="text-align: center;">$1.1$ m</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$0.2$ m</p></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$16$-year old girls</p><p style="text-align: left;"></p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p style="text-align: center;">$1.7$ m</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$0.3$ m</p></td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-1766728490353-i5w228g92" data-list-item-path="item-1766728490353-i5w228g92"><div class="list-item-paragraph">For each of the following cases, state, with a reason, whether or not a normal model is likely to be appropriate.</div><ol class="i-list"><li data-list-item-id="subitem-1766728494748-kbnojr0i6" data-list-item-path="item-1766728490353-i5w228g92/subitem-1766728494748-kbnojr0i6"><div class="list-item-paragraph">The number of days since her last birthday of a randomly chosen $11$-year old girl from the school.</div></li><li data-list-item-id="subitem-1775716884584-mxk34q59k" data-list-item-path="item-1766728490353-i5w228g92/subitem-1775716884584-mxk34q59k"><div class="list-item-paragraph">The height of a pupil chosen at random from the combined group of $11$-year old and $16$-year old girls in the school.<br></div></li><li data-list-item-id="subitem-1775716885544-inq04xl2l" data-list-item-path="item-1766728490353-i5w228g92/subitem-1775716885544-inq04xl2l"><div class="list-item-paragraph">The average height of a randomly chosen sample of eight $11$-year old girls from the school</div></li></ol></li><li data-list-item-id="item-1766728576542-z28lfk77q" data-list-item-path="item-1766728576542-z28lfk77q"><div class="list-item-paragraph">Find the probability such that</div><ol class="i-list"><li data-list-item-id="subitem-1766728588677-nznerw1f9" data-list-item-path="item-1766728576542-z28lfk77q/subitem-1766728588677-nznerw1f9"><div class="list-item-paragraph">each of the two randomly chosen $11$-year old girls is taller than $1.2$ m.</div></li><li data-list-item-id="subitem-1766728605102-ss7we938s" data-list-item-path="item-1766728576542-z28lfk77q/subitem-1766728605102-ss7we938s"><div class="list-item-paragraph">the total height of two randomly chosen $11$-year old girls is more than $2.4$ m.</div></li></ol></li></ol>
<p>Explain why the answer to <strong>(b)(ii)</strong> is higher than that of <strong>(b)(i)</strong>.</p>

Core: individual-height probability in(b)(i) and qualitative model discussion in(a)(i)–(ii). Optional: distribution of a sample mean in(a)(iii), total-height probability in(b)(ii) and dependent comparison.

A school has a large number of girls. For each year-group it may be assumed that the heights of girls are normally distributed, with average heights and standard deviations as given in the following table.

<p>Average height</p><p style="text-align: center;"></p>

Explain why the answer to **(b)(ii)** is higher than that of **(b)(i)**.

Assessing the Suitability of a Normal Model

Linear Combinations, Parameter Estimation, and Real-World Modeling

<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

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

Assumptions For Sampling

<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>The height of a species of monkey has a normal distribution with mean $56.2$ cm and standard deviation of $\sqrt{15}$ cm. A random sample of $20$ monkeys is taken and the mean height is calculated. Find the probability that</p>
<ol class="i-list"><li><div class="list-item-paragraph">The sample mean lies between $55$ cm and $57$ cm,</div></li><li><div class="list-item-paragraph">$P\left( X_{1}+X_{2}+\dots+X_{20}&gt;11\mathrm{ m} \right)$</div></li></ol>

The height of a species of monkey has a normal distribution with mean $56.2$ cm and standard deviation of $\sqrt{15}$ cm. A random sample of $20$ monkeys is taken and the mean height is calculated. Find the probability that

Sample Mean

<p>The height of a species of plant has a normal distribution with mean $34$ cm and standard deviation of $\sqrt{78}$ cm. A random sample of $13$ plants is taken and the mean height is calculated. Find the probability that</p>
<ol class="i-list"><li><div class="list-item-paragraph">The sample mean lies between $29$ cm and $35$ cm,</div></li><li><div class="list-item-paragraph">$P\left( X_{1}+X_{2}+...+X_{13}&gt;450\mathrm{ cm} \right)$</div></li></ol>

The height of a species of plant has a normal distribution with mean $34$ cm and standard deviation of $\sqrt{78}$ cm. A random sample of $13$ plants is taken and the mean height is calculated. Find the probability that

<p>Given that the random variables $X$ and $Y$ have distributions $N(60, 25)$ and $N(54, 64)$ respectively, and that $\bar{X}$ and $\bar{Y}$ denote the means of $7$ independent observations of $X$ and $11$ independent observations of $Y$ respectively.</p>
<p>State the sampling distributions of</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\bar{X}-\bar{Y}$</div></li><li><div class="list-item-paragraph">$2\bar{X}+5\bar{Y}$</div></li><li><div class="list-item-paragraph">$4\bar{X}-3\bar{Y}$ </div></li><li><div class="list-item-paragraph">$\frac{{{X}_{1}}+{{X}_{2}}+{{X}_{3}}-{{Y}_{1}}-{{Y}_{2}}-...-{{Y}_{5}}}{8}$</div></li><li><div class="list-item-paragraph">$\frac{{{X}_{1}}+{{X}_{2}}+\ldots +{{X}_{10}}+{{Y}_{1}}+{{Y}_{2}}+...+{{Y}_{5}}}{15}$ </div></li></ol>

Given that the random variables $X$ and $Y$ have distributions $N(60, 25)$ and $N(54, 64)$ respectively, and that $\bar{X}$ and $\bar{Y}$ denote the means of $7$ independent observations of $X$ and $11$ independent observations of $Y$ respectively.

Math Question Bank