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>At a school canteen, the average queuing time (in minutes) for a student to be served at a stall is known to be $4.5$ minutes. The school operation manager claims that with a recent change in students’ timetable, there is an improvement in the average queuing time.</p>
<p>A random sample of $10$ students’ queuing time is observed after the change in students’ timetable and the data is shown below.</p>
<p style="text-align: center">\[4.2, \quad 5.2, \quad 3.1, \quad 3.5, \quad 4.6, \quad 4.8, \quad 3.8, \quad 3.7, \quad 4.5, \quad 5.0\]</p>
<p>Given that the population variance of queuing time is known to be $0.48$ minutes$^2$, a one-tailed test is carried out to determine whether the manager’s claim is justified.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">State an assumption needed to carry out the hypothesis test.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Test whether the manager’s claim is justified at $5$% level of significance.</span><span class="inline-marks">[5]</span></div></li></ol>
<p>A student leader Jason suspected that with the change in students’ timetable, the average queuing time for a student to be served at a stall is in fact more than $\lambda$ minutes and also that the population variance of queuing time is no longer $0.48$ minutes$^2$. To investigate this, Jason recorded the queuing time of the first $50$ students who visited the stall from $12$ pm to $1$ pm.</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Give a reason why the sample collected is not a random sample.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>Jason decided to collect another set of data to do his investigation. He recorded the queuing time, $x$, in minutes, of a random sample of $50$ students. The summarized data is as shown below.</p>
<p style="text-align: center">\[\bar{x} = 4.6 \quad \text{and} \quad \sum x^2 = 1068\]</p>
<ol class="abc-list" start="4"><li><div class="list-item-content"><span class="list-item-text">Find an unbiased estimate of the population variance.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>Jason carried out a hypothesis test at $2$% level of significance with an alternative hypothesis of $\mu &gt; \lambda$ where&nbsp;$\mu$ is the population mean queuing time.</p>
<ol class="abc-list" start="5"><li><div class="list-item-content"><span class="list-item-text">Explain why the assumption in (a) is no longer necessary.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">By finding the critical region in terms of $z$-values, find the range of values of $\lambda$ such that the Jason’s suspicion is justified.</span><span class="inline-marks">[3]</span></div></li></ol>

At a school canteen, the average queuing time (in minutes) for a student to be served at a stall is known to be $4.5$ minutes. The school operation manager claims that with a recent change in students’ timetable, there is an improvement in the average queuing time.

A random sample of $10$ students’ queuing time is observed after the change in students’ timetable and the data is shown below.

\[4.2, \quad 5.2, \quad 3.1, \quad 3.5, \quad 4.6, \quad 4.8, \quad 3.8, \quad 3.7, \quad 4.5, \quad 5.0\]

Given that the population variance of queuing time is known to be $0.48$ minutes$^2$, a one-tailed test is carried out to determine whether the manager’s claim is justified.

State an assumption needed to carry out the hypothesis test.

Test whether the manager’s claim is justified at $5$% level of significance.

A student leader Jason suspected that with the change in students’ timetable, the average queuing time for a student to be served at a stall is in fact more than $\lambda$ minutes and also that the population variance of queuing time is no longer $0.48$ minutes$^2$. To investigate this, Jason recorded the queuing time of the first $50$ students who visited the stall from $12$ pm to $1$ pm.

Give a reason why the sample collected is not a random sample.

Jason decided to collect another set of data to do his investigation. He recorded the queuing time, $x$, in minutes, of a random sample of $50$ students. The summarized data is as shown below.

\[\bar{x} = 4.6 \quad \text{and} \quad \sum x^2 = 1068\]

Find an unbiased estimate of the population variance.

Jason carried out a hypothesis test at $2$% level of significance with an alternative hypothesis of $\mu > \lambda$ where $\mu$ is the population mean queuing time.

Explain why the assumption in (a) is no longer necessary.

By finding the critical region in terms of $z$-values, find the range of values of $\lambda$ such that the Jason’s suspicion is justified.

Assumptions and Validity of Mean Tests

Estimating Parameters from Summarised and Coded Data

Central Limit Theorem

One-Tailed Hypothesis Tests

Parameter Bounds from Hypothesis Tests

Sampling Techniques and Bias

Mean Tests for Normal Populations with Known Variance

Assumptions for Sampling

Large-Sample Mean Tests with Estimated Variance

Critical Values and Critical Regions

2025 TJC P2 Q9

Hypothesis Testing

Sampling

<p>A beverage company produces bottled orange juice, claiming that the mean vitamin $C$ content in each $250$ ml bottle is $50$ mg. It is known that the vitamin $C$ content is normally distributed with a standard deviation of $2$ mg</p>
<p>Due to feedback from consumers, the quality control manager wants to test, at the $4$% level of significance, whether the mean vitamin $C$ content is, in fact, $50$ mg. He randomly samples $12$ bottles and measures the vitamin $C$ content.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State the hypotheses and find the critical region for this test.<span class="inline-marks">[3]</span></div></li></ol>
<p>Recently, the company launched a new "Vitamin Boost" orange juice line, advertised to contain an average of $60$ mg of vitamin $C$ content per $250$ ml bottle. After some complaints that the juice is too sour, the manager suspects that the vitamin $C$ content might be higher than advertised. He decides to conduct a hypothesis test to investigate. This time, he randomly samples $30$ bottles from the new juice line.</p>
<p><strong>(ii)</strong> Explain why the manager takes a sample of $30$ bottles for this test when he only took a sample of $12$ bottles in his earlier test.<span class="inline-marks">[2]</span></p>
<p>The vitamin $C$ content, $y$ mg, in the sample of $30$ bottles is summarized as follows:</p>
<p>\[\sum y = 1865, \quad \sum y^2 = 116275\]</p>
<p><strong>(iii)</strong> Calculate unbiased estimates of the population mean and variance for the vitamin $C$ content in the bottles of the new juice line.<span class="inline-marks">[2]</span></p>
<p><strong>(iv)</strong> At the $5$% level of significance, test whether the mean vitamin $C$ content in the bottles of the new juice line is greater than $60$ mg. Clearly state your hypotheses and define any symbols you use.<span class="inline-marks">[4]</span></p>

A beverage company produces bottled orange juice, claiming that the mean vitamin $C$ content in each $250$ ml bottle is $50$ mg. It is known that the vitamin $C$ content is normally distributed with a standard deviation of $2$ mg

Due to feedback from consumers, the quality control manager wants to test, at the $4$% level of significance, whether the mean vitamin $C$ content is, in fact, $50$ mg. He randomly samples $12$ bottles and measures the vitamin $C$ content.

Recently, the company launched a new "Vitamin Boost" orange juice line, advertised to contain an average of $60$ mg of vitamin $C$ content per $250$ ml bottle. After some complaints that the juice is too sour, the manager suspects that the vitamin $C$ content might be higher than advertised. He decides to conduct a hypothesis test to investigate. This time, he randomly samples $30$ bottles from the new juice line.

**(ii)** Explain why the manager takes a sample of $30$ bottles for this test when he only took a sample of $12$ bottles in his earlier test.

The vitamin $C$ content, $y$ mg, in the sample of $30$ bottles is summarized as follows:

\[\sum y = 1865, \quad \sum y^2 = 116275\]

**(iii)** Calculate unbiased estimates of the population mean and variance for the vitamin $C$ content in the bottles of the new juice line.

**(iv)** At the $5$% level of significance, test whether the mean vitamin $C$ content in the bottles of the new juice line is greater than $60$ mg. Clearly state your hypotheses and define any symbols you use.

Null and Alternative Hypotheses

Two-Tailed Hypothesis Tests

2025 SAJC P2 Q11

<p>Based on observations over a long period, the mean time taken by male students in Griffles Junior College (GJC) to complete a $2.4$ km run is known to be $11.3$ minutes with a standard deviation of $2.2$ minutes. As part of a review of the current physical training programme, the Physical Education (PE) department in GJC decided to test, at the $2.5\%$ level of significance, whether the mean time taken by male students in GJC to complete a $2.4$ km run has changed. The time taken, $x$ minutes, by a random sample of $8$ Year $5$ male students from GJC to complete a $2.4$ km run, are as follows:</p>
<p style="text-align: center">${11 \quad 11.5 \quad 10.8 \quad 11.2 \quad 11.4 \quad 11 \quad 11.8 \quad 12.5}$</p>
<ol class="abc-list"><li data-list-item-id="item-a-4" data-list-item-path="item-a-4"><div class="list-item-paragraph">Write down the null and alternative hypotheses for this test, defining any symbols you use.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-4" data-list-item-path="item-b-4"><div class="list-item-paragraph">Stating a necessary assumption, find the critical region for this test. Hence state the conclusion of the test in the context of the question.<span class="inline-marks">[6]</span></div></li></ol>
<p>The PE department in GJC also conducted a second test, at the $3\%$ level of significance, to determine whether the mean time taken by female students in GJC to complete a $2.4$ km run is less than $14.5$ minutes. A random sample of $n$ female students from GJC is taken, where&nbsp;$n$ is large. The mean and standard deviation of the time taken by this sample to complete a $2.4$ km run are found to be $14.2$ minutes and $1.5$ minutes respectively.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c-4" data-list-item-path="item-c-4"><div class="list-item-paragraph">Given that the PE department concludes that the mean time taken by female students to complete a $2.4$ km run is less than $14.5$ minutes, find the range of values that&nbsp;$n$ can take.<span class="inline-marks">[5]</span></div></li></ol>

Based on observations over a long period, the mean time taken by male students in Griffles Junior College (GJC) to complete a $2.4$ km run is known to be $11.3$ minutes with a standard deviation of $2.2$ minutes. As part of a review of the current physical training programme, the Physical Education (PE) department in GJC decided to test, at the $2.5\%$ level of significance, whether the mean time taken by male students in GJC to complete a $2.4$ km run has changed. The time taken, $x$ minutes, by a random sample of $8$ Year $5$ male students from GJC to complete a $2.4$ km run, are as follows:

${11 \quad 11.5 \quad 10.8 \quad 11.2 \quad 11.4 \quad 11 \quad 11.8 \quad 12.5}$

Write down the null and alternative hypotheses for this test, defining any symbols you use.

Stating a necessary assumption, find the critical region for this test. Hence state the conclusion of the test in the context of the question.

The PE department in GJC also conducted a second test, at the $3\%$ level of significance, to determine whether the mean time taken by female students in GJC to complete a $2.4$ km run is less than $14.5$ minutes. A random sample of $n$ female students from GJC is taken, where $n$ is large. The mean and standard deviation of the time taken by this sample to complete a $2.4$ km run are found to be $14.2$ minutes and $1.5$ minutes respectively.

Given that the PE department concludes that the mean time taken by female students to complete a $2.4$ km run is less than $14.5$ minutes, find the range of values that $n$ can take.

Estimating Population Variance from Sample Variance

Interpreting Hypothesis Test Conclusions

2025 RI P2 Q9

<p style="text-align: center">---</p>
<p>The production manager of a food container manufacturing company wishes to take a random sample of a certain type of lunch box from the thousands produced one day at his factory, for quality control purposes. He wishes to check that the mean height of the lunch boxes is $12.8$ cm.</p>
<ol class="i-list"><li data-list-item-id="item-q3-i" data-list-item-path="item-q3-i"><div class="list-item-paragraph">State what it means for a sample to be random in this context.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-q3-ii" data-list-item-path="item-q3-ii"><div class="list-item-paragraph">Explain why the manager should take a sample of at least $30$ lunch boxes to carry out a hypothesis test.<span class="inline-marks">[1]</span></div></li></ol>
<p>The heights, $x$ cm, of a random sample of $48$ lunch boxes are summarised as follows:\[\sum(x-13) = -3.6 \quad \text{and} \quad \sum(x-13)^2 = 12.02\]</p>
<ol class="i-list" start="3"><li data-list-item-id="item-q3-iii" data-list-item-path="item-q3-iii"><div class="list-item-paragraph">Calculate the unbiased estimates of the population mean and variance of the height of the lunch boxes.<span class="inline-marks">[2]</span></div></li></ol>
<p>The production manager claims that the mean height of the lunch boxes is $12.8$ cm.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-q3-iv" data-list-item-path="item-q3-iv"><div class="list-item-paragraph">Test his claim at the $10\%$ level of significance, stating the hypotheses clearly.<span class="inline-marks">[4]</span></div></li></ol>
<p>In the hope of keeping the mean height of the lunch boxes not to exceed $12.8$ cm, the production manager decides to replace the production line such that the population variance is reduced to $0.04$ cm$^2$. He then carries out a hypothesis test at $2\%$ level of significance using another random sample of $n$ lunch boxes.</p>
<ol class="i-list" start="5"><li data-list-item-id="item-q3-v" data-list-item-path="item-q3-v"><div class="list-item-paragraph">Assuming that&nbsp;$n$ is sufficiently large, find the critical region for this test in terms of $n$.<span class="inline-marks">[4]</span></div></li></ol>

The production manager of a food container manufacturing company wishes to take a random sample of a certain type of lunch box from the thousands produced one day at his factory, for quality control purposes. He wishes to check that the mean height of the lunch boxes is $12.8$ cm.

State what it means for a sample to be random in this context.

Explain why the manager should take a sample of at least $30$ lunch boxes to carry out a hypothesis test.

The heights, $x$ cm, of a random sample of $48$ lunch boxes are summarised as follows:\[\sum(x-13) = -3.6 \quad \text{and} \quad \sum(x-13)^2 = 12.02\]

Calculate the unbiased estimates of the population mean and variance of the height of the lunch boxes.

The production manager claims that the mean height of the lunch boxes is $12.8$ cm.

In the hope of keeping the mean height of the lunch boxes not to exceed $12.8$ cm, the production manager decides to replace the production line such that the population variance is reduced to $0.04$ cm$^2$. He then carries out a hypothesis test at $2\%$ level of significance using another random sample of $n$ lunch boxes.

Assuming that $n$ is sufficiently large, find the critical region for this test in terms of $n$.

Large-Sample Mean Tests with Known Variance

Population and Random Samples

2025 NJC P2 Q10

<ol class="abc-list"><li data-list-item-id="item-1760850952814-32z7ow6ou" data-list-item-path="item-1760850952814-32z7ow6ou"><div class="list-item-paragraph">A company has a machine designed to fill bags with, on average, ${{\mu }_{0}}$ kg of salt. The mass of salt in a randomly chosen bag has a normal distribution with population standard deviation denoted by $\sigma $ kg. The production manager wishes to investigate if the machine is adjusted correctly. He takes a random sample of $n$ bags and carries out a hypothesis test at the $1\%$ level of significance.</div><ol class="i-list"><li data-list-item-id="subitem-1760850961980-lolsv1a5c" data-list-item-path="item-1760850952814-32z7ow6ou/subitem-1760850961980-lolsv1a5c"><div class="list-item-paragraph">State null and alternative hypotheses for the manager’s test, defining any parameters you use.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760850962602-trl0qp3k4" data-list-item-path="item-1760850952814-32z7ow6ou/subitem-1760850962602-trl0qp3k4"><div class="list-item-paragraph">Find, in terms of ${{\mu }_{0}}$, $\sigma $ and $n$, the critical region(s) for this test.<span class="inline-marks">[4]</span></div></li></ol></li><li data-list-item-id="item-1760850953602-o2afx6akc" data-list-item-path="item-1760850953602-o2afx6akc"><div class="list-item-paragraph">The company has a different machine which fills bags with, on average, $25$kg of low sodium salt. One of the company’s production supervisor has reported that some of the workers suspect the machine is no longer set correctly, and the average mass of low sodium salt in the bags may in fact be more than $25$kg.The production supervisor decides to carry out a hypothesis test at the $0.5\%$ level of significance with a random sample of $80$ bags of low sodium salt. Summary data for the mass, $y$ kg, of low sodium salt in these bags is as follows.</div><div class="list-item-paragraph" style="text-align: center"><p>${n = 80} \qquad \sum(y - 25) = 27.2 \qquad \sum(y - 25)^2 = 85.1$</p></div><ol class="i-list"><li data-list-item-id="subitem-1760850984412-x30rh5ujw" data-list-item-path="item-1760850953602-o2afx6akc/subitem-1760850984412-x30rh5ujw"><div class="list-item-paragraph">Carry out the test and state the conclusion of the test in the context of the question.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="subitem-1760850985537-82yqb97f4" data-list-item-path="item-1760850953602-o2afx6akc/subitem-1760850985537-82yqb97f4"><div class="list-item-paragraph">State, giving a reason, whether any assumptions about the population are needed in order for the test to be valid.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

Hypothesis Tests Using Critical Regions and $p$-Values

2024 RI P2 Q10

<p>A factory manufactures cans and bottles of iced tea. Machine A is used to fill the cans with iced tea and Machine B is used to fill the bottles with iced tea. Machine A is set to fill each can with $300$ millilitres (ml) of iced tea. A random sample of $60$ filled cans of iced tea was taken and the volume, $x$ ml, of iced tea in each can was measured. The following summarised data was obtained.</p>
<p style="text-align: center">${\sum(x - 300) = -112.8} \qquad \sum(x - 300)^2 = 4532.87$</p>
<ol class="abc-list"><li data-list-item-id="item-1760849433683-34w1pnzuk" data-list-item-path="item-1760849433683-34w1pnzuk"><div class="list-item-paragraph">Defining clearly any symbols you use, test at the $8\%$ level of significance, whether the mean volume of iced tea per can is $300$ ml.<span class="inline-marks">[6]</span></div></li><li data-list-item-id="item-1760849434763-jpuebb6k9" data-list-item-path="item-1760849434763-jpuebb6k9"><div class="list-item-paragraph">Explain in the context of the question, the meaning of ‘at the $8\%$ level of significance’.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760849435377-hizobfurn" data-list-item-path="item-1760849435377-hizobfurn"><div class="list-item-paragraph"><p>The manager of the factory claims that the mean volume of iced tea that Machine B fills per bottle with is at least $500$ ml. It is found that the volumes of iced tea in the bottles filled by Machine B follow a normal distribution with standard deviation $5$ ml. A random sample of $35$ filled bottles was taken and a test for the validity of the manager’s claim was carried out at the $4\%$ level of significance. Find the critical region for this test, correct to 1 decimal place.</p><span class="inline-marks">[3]</span></div></li></ol>

A factory manufactures cans and bottles of iced tea. Machine A is used to fill the cans with iced tea and Machine B is used to fill the bottles with iced tea. Machine A is set to fill each can with $300$ millilitres (ml) of iced tea. A random sample of $60$ filled cans of iced tea was taken and the volume, $x$ ml, of iced tea in each can was measured. The following summarised data was obtained.

${\sum(x - 300) = -112.8} \qquad \sum(x - 300)^2 = 4532.87$

Significance Levels and $p$-Values

2024 NJC P2 Q9

<p>From a school-wide survey done in 2022, it was found that the students in a certain school spend an average of $45.5$ hours per week on social media platforms. In 2023, the school’s Student Well-being team decides to carry out a survey on a group of $50$ randomly chosen students.</p>
<ol class="abc-list"><li data-list-item-id="item-1760684452937-xysm00fag" data-list-item-path="item-1760684452937-xysm00fag"><div class="list-item-paragraph">State what it means for a sample to be random in this context.<span class="inline-marks">[1]</span></div></li></ol>
<p>The time, $x$ hours, spent per week on social media platforms by the $50$ students are summarised below.</p>
<p style="text-align: center">$\sum{\left( x-45.4 \right)}=55$                             ${{\sum{\left( x-45.4 \right)}}^{2}}=\mathrm{1494}\mathrm{.74}$</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760684470202-rrxs6w0ni" data-list-item-path="item-1760684470202-rrxs6w0ni"><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance of the time spent by students per week on social media platforms.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760684470848-563dfjb4y" data-list-item-path="item-1760684470848-563dfjb4y"><div class="list-item-paragraph">State hypotheses that can be used to test if the mean time spent by students per week differs from $45.5$ hours. Work out the critical region in this case, and use it to carry out the test at the $5\%$ level of significance, giving your conclusion in the context of the question.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760684492282-0bdzmhofl" data-list-item-path="item-1760684492282-0bdzmhofl"><div class="list-item-paragraph">Explain why it is not necessary to assume that the time spent by students per week on social media platforms follows a normal distribution when performing the test in part <strong>(c)</strong>.<span class="inline-marks">[1]</span></div></li></ol>
<p>It is now given that the time spent by students per week on social media platforms follows a normal distribution with a variance of $25$ hours$^{2}$. Based on observations, the Head of Student Well-being, Ms Tan, believes that the mean time spent by students per week on social media platforms has in fact increased.</p>
<ol class="abc-list" start="5"><li data-list-item-id="item-1760684510702-pgqmc86s3" data-list-item-path="item-1760684510702-pgqmc86s3"><div class="list-item-paragraph">By carrying out the test at $\alpha %$ level of significance using data collected from $50$ randomly chosen students with a mean of $47$ hours, determine the set of values of $\alpha $ for which Ms Tan’s belief should be accepted.<span class="inline-marks">[4]</span></div></li></ol>

From a school-wide survey done in 2022, it was found that the students in a certain school spend an average of $45.5$ hours per week on social media platforms. In 2023, the school’s Student Well-being team decides to carry out a survey on a group of $50$ randomly chosen students.

The time, $x$ hours, spent per week on social media platforms by the $50$ students are summarised below.

$\sum{\left( x-45.4 \right)}=55$                             ${{\sum{\left( x-45.4 \right)}}^{2}}=\mathrm{1494}\mathrm{.74}$

It is now given that the time spent by students per week on social media platforms follows a normal distribution with a variance of $25$ hours$^{2}$. Based on observations, the Head of Student Well-being, Ms Tan, believes that the mean time spent by students per week on social media platforms has in fact increased.

Unbiased Estimates of Population Parameters

2023 RI P2 Q8

<p>Osoi is frequently late for school. His mean travelling time to school is $24.5$ minutes. The Year Head asks Osoi to take an alternative route to school and to record his travelling time, $t$ minutes, from home to school every morning on the alternative route. The results for&nbsp;$72$ randomly chosen mornings when Osoi took the alternative route are summarised as follows:</p>
<p style="text-align: center">${\sum(t - 20) = 215\text{ ;}} \qquad \sum(t - 20)^2 = 3234$</p>
<ol class="i-list"><li data-list-item-id="item-1760684338150-761yco7qa" data-list-item-path="item-1760684338150-761yco7qa"><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance of the travelling time for the alternative route.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760684338713-9l1b7mzt1" data-list-item-path="item-1760684338713-9l1b7mzt1"><div class="list-item-paragraph">Using a $5\%$ level of significance, test whether Osoi’s mean travelling time from home to school has shortened after taking the alternative route to school. You should state your hypotheses and define any symbols that you use.<span class="inline-marks">[5]</span></div></li></ol>
<p>Suppose the Year Head wants to investigate if the alternative route to school has made a difference to Osoi’s mean travelling time in the morning instead of whether it has shortened his mean travelling time in the morning, at the same level of significance.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760684364617-qmlgysq8p" data-list-item-path="item-1760684364617-qmlgysq8p"><div class="list-item-paragraph">Find the critical region for this test.<span class="inline-marks">[3]</span></div></li></ol>

Osoi is frequently late for school. His mean travelling time to school is $24.5$ minutes. The Year Head asks Osoi to take an alternative route to school and to record his travelling time, $t$ minutes, from home to school every morning on the alternative route. The results for $72$ randomly chosen mornings when Osoi took the alternative route are summarised as follows:

${\sum(t - 20) = 215\text{ ;}} \qquad \sum(t - 20)^2 = 3234$

Suppose the Year Head wants to investigate if the alternative route to school has made a difference to Osoi’s mean travelling time in the morning instead of whether it has shortened his mean travelling time in the morning, at the same level of significance.

2022 NJC P2 Q9

<p>Carbon steel is made by adding carbon to iron; this makes the iron stronger, though less flexible. A steel manufacturing company makes carbon steel in the form of round bars. The process is designed to manufacture bars in which the amount of carbon in each bar, by weight, is $1.5\%$. It is known that the percentage of carbon in the steel bars is distributed normally, and that the standard deviation is $0.09\%$.</p>
<p>After comments from customers, the production manager wishes to test, at the $5\%$ level of significance, if the percentage of carbon in the steel bars is, in fact, $1.5\%$. He examines a random sample of $15$ bars to determine the percentage of carbon in each bar.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the critical region for this test.</div><p style="text-align: right;">[4]</p></li></ol>
<p>The company recently launched a new line of flat bars made from mild steel. In mild steel the amount of carbon in each bar, by weight, is $0.25\%$.</p>
<p>Comments from customers suggests that these bars are not sufficiently flexible, which makes the production manager suspect that too much carbon has been added. He decides to perform a hypothesis test on a random sample of $40$ of the new flat bars to find out if this is the case.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Explain why the production manager takes a sample of $40$ flat bars for this test when he only took a sample of $15$ round bars in his earlier test.</div><p style="text-align: right;">[2]</p></li></ol>
<p>The amount of carbon, $x\%$, in a random sample of $40$ flat bars are summarised as follows.</p>
<p style="text-align: center">$n=40$       $\sum{x=10.16}$       $\sum{{{x}^{2}}=2.586342}$</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance for the percentage amount of carbon in the flat bars.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Test, at the $2\frac{1}{2}\%$ level of significance, whether the mean amount of carbon in the flat bars is more than $0.25\%$. You should state your hypotheses and define any symbols that you use.</div><p style="text-align: right;">[5]</p></li></ol>

Carbon steel is made by adding carbon to iron; this makes the iron stronger, though less flexible. A steel manufacturing company makes carbon steel in the form of round bars. The process is designed to manufacture bars in which the amount of carbon in each bar, by weight, is $1.5\%$. It is known that the percentage of carbon in the steel bars is distributed normally, and that the standard deviation is $0.09\%$.

After comments from customers, the production manager wishes to test, at the $5\%$ level of significance, if the percentage of carbon in the steel bars is, in fact, $1.5\%$. He examines a random sample of $15$ bars to determine the percentage of carbon in each bar.

The company recently launched a new line of flat bars made from mild steel. In mild steel the amount of carbon in each bar, by weight, is $0.25\%$.

Comments from customers suggests that these bars are not sufficiently flexible, which makes the production manager suspect that too much carbon has been added. He decides to perform a hypothesis test on a random sample of $40$ of the new flat bars to find out if this is the case.

The amount of carbon, $x\%$, in a random sample of $40$ flat bars are summarised as follows.

$n=40$       $\sum{x=10.16}$       $\sum{{{x}^{2}}=2.586342}$

N2020 P2 Q10

<p>The manufacturer of a particular toy drone claims that its average flying time (AFT) is $12$ minutes. The quality control manager suspects that the AFT is different from what the manufacturer has claimed and decides to carry out a hypothesis test on a sample of these drones.</p>
<p>The flight durations, $x$ minutes, of a random sample of $100$ drones are summarised as follows:</p>
<p style="text-align: center">$\sum{x}=1171$ $\sum{{{x}^{2}}}=14023$</p>
<ol class="abc-list"><li><div class="list-item-paragraph">State what it means for a sample to be random in this context.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance of the AFT of the toy drones.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Test, at the $5\%$ level of significance, the manufacturer’s claim that the AFT of toy drones is $12$ minutes. You should state your hypotheses and define any symbols you use.</div><p style="text-align: right;">[5]</p></li></ol>
<p>Over a long period of time, it is established that the AFT of the toy drone is normally distributed with mean and standard deviation of $12$ minutes and $2$ minutes respectively. However, changes in the manufacturing process due to an upgrade of the drone’s rechargeable battery, is expected to improve the AFT of the drone. To verify this, a random sample of $n$ drones is taken with sample mean AFT of $m$. If $m$ is greater than some critical value, it will be accepted that the AFT of the toy drone has in fact increased.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find in terms of $n$, the critical value for a $5\%$ significance level.</div><p style="text-align: right;">[3]</p></li></ol>

The manufacturer of a particular toy drone claims that its average flying time (AFT) is $12$ minutes. The quality control manager suspects that the AFT is different from what the manufacturer has claimed and decides to carry out a hypothesis test on a sample of these drones.

The flight durations, $x$ minutes, of a random sample of $100$ drones are summarised as follows:

Over a long period of time, it is established that the AFT of the toy drone is normally distributed with mean and standard deviation of $12$ minutes and $2$ minutes respectively. However, changes in the manufacturing process due to an upgrade of the drone’s rechargeable battery, is expected to improve the AFT of the drone. To verify this, a random sample of $n$ drones is taken with sample mean AFT of $m$. If $m$ is greater than some critical value, it will be accepted that the AFT of the toy drone has in fact increased.

2023 NYJC P2 Q9

<ol class="abc-list"><li><div class="list-item-paragraph">Experimental data concerning a variable $X$, which measures the reliability of a certain electronic component, is as follows: $\sum{x}=1164.2$, $\sum{{{x}^{2}}}=13911.6$, $n=100$. Calculate the unbiased estimates of the population mean and variance from these figures. Explain whether, on the evidence of this sample, you would reject the hypothesis that the mean value of $X$ is $12$ at $5\%$ level of significance.</div></li><li><div class="list-item-paragraph">A second random sample of $50$ electronic components of this type is to be tested and the measurement, $y$, are summarized by</div><div class="list-item-paragraph" style="text-align: center">$\sum{y}=582.1$, $\sum{{{y}^{2}}}=7055.9$.</div><div class="list-item-paragraph">Combining the two samples into a single sample, carry out a test, at the $5\%$ significance level, of the same null and alternative hypotheses.</div></li><li><div class="list-item-paragraph">Figures collected over a long period have established that the mean and standard deviation of $X$ are $12$ and $2$ respectively. After a change in the manufacturing process, it is expected that the mean will have increased, but it may be assumed that the standard deviation remains equal to $2$. A sample of $n$ values of $X$ is taken, with sample mean $m$; if $m$ is greater than some critical value, it will be accepted that the mean has in fact increased, but if $m$ is less than the critical value, the increase is not established. State carefully appropriate null and alternative hypotheses for this situation, and find, in terms of $n$, the critical value for a $1\%$ significance level.</div></li></ol>

Combining Samples for Parameter Estimation

NYJC Hypothesis Testing Tutorial Q8

<p>Exposure to Volatile Organic Compounds (VOCs), which have been identified in indoor air, is suspected as a cause of headaches and respiratory systems.Indoor plants have not only a positive psychological effect on humans, but may also improve the air quality.Certain species of indoor plants were found to be effective removers of VOCs.</p>
<p>A commonly known VOC is Benzene. The following data gives the benzene levels, $x$(in ppm) in $40$ test chambers containing the indoor plant Epipremnum aureum.</p>
<p style="text-align: center">$n=40$, $\sum{\left( x-26.0 \right)=-30.1}$, ${{\sum{\left( x-26.0 \right)}}^{2}}=214.61$.</p>
<p>The initial mean Benzene level (in ppm) without Epipremnum aureum was found to be $26.0$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Test, at the $5$% level of significance, the claim that the mean Benzene level, $\mu $(in ppm), has decreased as a result of the indoor plant Epipremnum aureum. You should state your hypotheses clearly.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">State, giving a reason, whether there is a need to make any assumptions about the population distribution of the Benzene level in order for the test to be valid.</div><p style="text-align: right;">[2]</p></li></ol>
<p>The Benzene levels of another $50$ test chambers containing the indoor plant Epipremnum aureum were recorded. The sample mean is $\bar{x}$ppm and the sample variance is $8.33$ ppm$^{2}$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">The acceptance region of a test of the null hypothesis $\mu =26.0$ is $\bar{x}>25.1$. State the alternative hypothesis and find the level of significance of the test.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">If the null hypothesis is $\mu ={{\mu }_{0}}$, where ${{\mu }_{0}}>26.0$, would the significance level of a test with the same acceptance region in part <strong>(iii)</strong> be larger or smaller than that found in <strong>(iii)</strong>? Give a reason for your answer.</div><p style="text-align: right;">[2]</p></li></ol>

Exposure to Volatile Organic Compounds (VOCs), which have been identified in indoor air, is suspected as a cause of headaches and respiratory systems.Indoor plants have not only a positive psychological effect on humans, but may also improve the air quality.Certain species of indoor plants were found to be effective removers of VOCs.

A commonly known VOC is Benzene. The following data gives the benzene levels, $x$(in ppm) in $40$ test chambers containing the indoor plant Epipremnum aureum.

$n=40$, $\sum{\left( x-26.0 \right)=-30.1}$, ${{\sum{\left( x-26.0 \right)}}^{2}}=214.61$.

The initial mean Benzene level (in ppm) without Epipremnum aureum was found to be $26.0$.

The Benzene levels of another $50$ test chambers containing the indoor plant Epipremnum aureum were recorded. The sample mean is $\bar{x}$ppm and the sample variance is $8.33$ ppm$^{2}$.

2019 VJC P2 Q9

<p>A company uses a machine to produce chocolate bars. The machine is designed to produce chocolate bars with average fat content of $30$g. After using the machine for many years, the manager wishes to test, at $5\%$ level of significance, if the machine still maintains the average fat content in the chocolate bars at $30$g. He selects a sample of $40$ chocolate bars for testing.</p>
<ol class="i-list"><li data-list-item-id="item-1759728699944-cxvfzdaa3" data-list-item-path="item-1759728699944-cxvfzdaa3"><div class="list-item-paragraph">State, giving a reason, whether it is necessary to assume that the fat content of the chocolate bars is normally distributed for the test to be valid.<span class="inline-marks">[1]</span></div></li></ol>
<p>After some years of usage, the machine broke down. While waiting for the new machine to arrive, the company reverted to their traditional mode of producing handmade chocolate bars. The manager suspects that handmade chocolate bars have higher average fat content than those made by the machines. To verify his suspicion, the manager asked his staff to perform a hypothesis test at $5\%$ significance level, on a random sample of $40$ handmade chocolate bars.</p>
<p>The fat content, $x$ grams, of the $40$ handmade chocolate bars are summarised as follows:</p>
<p style="text-align: center">${\sum x = 1220,} \qquad \sum(x - 30.5)^2 = 50$</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1759728747103-uzzhqr9g6" data-list-item-path="item-1759728747103-uzzhqr9g6"><div class="list-item-paragraph">Calculate the unbiased estimates of the population mean and variance of the fat content in handmade chocolate bars.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759728757040-di2ol4sp0" data-list-item-path="item-1759728757040-di2ol4sp0"><div class="list-item-paragraph">The sample mean fat content of the handmade chocolate bars is denoted by $\bar{x}$ grams. State the null and alternate hypotheses and calculate the range of values of $\bar{x}$ for which the null hypothesis would be rejected at $5\%$ level of significance. Hence conclude if the manager’s suspicion is valid at $5\%$ significance level.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1759728757858-jd6yy6v1k" data-list-item-path="item-1759728757858-jd6yy6v1k"><div class="list-item-paragraph"><p>The manager now knows the population variance of the fat content in handmade chocolate bars. It is given that the population variance is smaller than the unbiased estimate of the population variance calculated in part <strong>(ii)</strong>. Without carrying out another hypothesis test, explain with justification, if there will be a change in the manager’s conclusion about his suspicion at $5\%$ significance level.</p><span class="inline-marks">[1]</span></div></li></ol>

A company uses a machine to produce chocolate bars. The machine is designed to produce chocolate bars with average fat content of $30$g. After using the machine for many years, the manager wishes to test, at $5\%$ level of significance, if the machine still maintains the average fat content in the chocolate bars at $30$g. He selects a sample of $40$ chocolate bars for testing.

After some years of usage, the machine broke down. While waiting for the new machine to arrive, the company reverted to their traditional mode of producing handmade chocolate bars. The manager suspects that handmade chocolate bars have higher average fat content than those made by the machines. To verify his suspicion, the manager asked his staff to perform a hypothesis test at $5\%$ significance level, on a random sample of $40$ handmade chocolate bars.

The fat content, $x$ grams, of the $40$ handmade chocolate bars are summarised as follows:

${\sum x = 1220,} \qquad \sum(x - 30.5)^2 = 50$

Math Question Bank