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>A random sample of $90$ batteries, used in a particular model of mobile phone, is tested and the 'standby-time', $x$ hours, is measured. The results are summarised by $\sum x = 3040.8$ and $\sum x^2 = 115773.66$. Test, at the $1\%$ significance level, whether the mean standby-time is less than $36.0$ hours.</p>
<p>State, with a reason, whether in using the above test, it is necessary to assume that the 'standby-time' has a normal distribution.</p>
<p>In a test at the $5\%$ significance level, it is found that there is significant evidence that the population mean talk-time is less than $5$ hours. <strong>Using only this information, and giving a reason in each case</strong>, state whether each of the following statements is (i) necessarily true, (ii) necessarily false, or (iii) neither necessarily true nor false.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">There is significant evidence at the $10\%$ significance level that the population mean talk-time is less than $5$ hours.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">There is significant evidence at the $5\%$ significance level that the population mean talk-time is not $5$ hours.</div></li></ol>

A random sample of $90$ batteries, used in a particular model of mobile phone, is tested and the 'standby-time', $x$ hours, is measured. The results are summarised by $\sum x = 3040.8$ and $\sum x^2 = 115773.66$. Test, at the $1\%$ significance level, whether the mean standby-time is less than $36.0$ hours.

State, with a reason, whether in using the above test, it is necessary to assume that the 'standby-time' has a normal distribution.

In a test at the $5\%$ significance level, it is found that there is significant evidence that the population mean talk-time is less than $5$ hours. **Using only this information, and giving a reason in each case**, state whether each of the following statements is (i) necessarily true, (ii) necessarily false, or (iii) neither necessarily true nor false.

Assumptions and Validity of Mean Tests

Estimating Parameters from Summarised and Coded Data

Central Limit Theorem

Significance Levels and $p$-Values

One-Tailed Hypothesis Tests

Comparing One-Tailed and Two-Tailed Tests

Large-Sample Mean Tests with Estimated Variance

Sampling

Hypothesis Testing

<p>A company advertises that its rechargeable batteries can run non-stop for an average of $505$ minutes after a full charge. $100$ batteries are randomly selected, and their running time after a full charge, $x$ minutes, are measured. The data obtained is summarised by\[\sum (x - 500) = 320, \quad \sum (x - 500)^2 = 8416.\]</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Suggest a reason why, in this context, the given data is summarised in terms of $(x - 500)$ rather than $x$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance of the running time of the batteries.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph"><p>A hypothesis test is carried out at the $5\%$ level of significance. Determine whether the company is overestimating the average running time of the batteries.</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">State, giving a reason, whether it is necessary to assume a normal distribution for the test to be valid.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-e" data-list-item-path="item-e"><div class="list-item-paragraph"><p>After a change is made in the manufacturing process, another test at the $5\%$ significance level is carried out using a new random sample of size $50$. The sample standard deviation is found to be $8.5$ minutes. Find the range of values of the sample mean in order to conclude that the average running time of the batteries after a full charge is more than $505$ minutes.</p><span class="inline-marks">[4]</span></div></li></ol>

A company advertises that its rechargeable batteries can run non-stop for an average of $505$ minutes after a full charge. $100$ batteries are randomly selected, and their running time after a full charge, $x$ minutes, are measured. The data obtained is summarised by\[\sum (x - 500) = 320, \quad \sum (x - 500)^2 = 8416.\]

Suggest a reason why, in this context, the given data is summarised in terms of $(x - 500)$ rather than $x$.

Calculate unbiased estimates of the population mean and variance of the running time of the batteries.

State, giving a reason, whether it is necessary to assume a normal distribution for the test to be valid.

Parameter Bounds from Hypothesis Tests

Estimating Population Variance from Sample Variance

2025 YIJC P2 Q10

<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.

Sampling Techniques and Bias

Mean Tests for Normal Populations with Known Variance

Assumptions for Sampling

Critical Values and Critical Regions

2025 TJC P2 Q9

<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.

Interpreting Hypothesis Test Conclusions

2025 RI P2 Q9

<p>Helen is a conservationist who monitors the health of fish in a river. She is investigating whether a new water-treatment plant has affected the mass of adult trout in the river. Previously, the mean mass of adult trout in the river has been $w$ kg. Helen carries out a test, at the $10$% level of significance to find out whether there has been a change in the mean mass of adult trout in the river.</p>
<p>Helen catches $45$ adult trout and measures the mass, $X$, kilograms, of each fish. Her summarised data are as follows.</p>
<p style="text-align: center">\[n=45 \quad \sum x=80.1 \quad \sum (x-\bar{x})^2=24.29\]</p>
<ol class="abc-list"><li data-list-item-id="item-4a" data-list-item-path="item-4a"><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance of the mass of adult trout.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-4b" data-list-item-path="item-4b"><div class="list-item-paragraph">Use an algebraic method to calculate the set of values of $w$ for which there is insufficient evidence to conclude that there has been a change in the mean mass of adult trout in the river. You should state your hypotheses and define any symbols you use.<span class="inline-marks">[6]</span></div></li><li data-list-item-id="item-4c" data-list-item-path="item-4c"><div class="list-item-paragraph">Explain why there is no need for Helen to know anything about the population distribution of the mass of adult trout.<span class="inline-marks">[2]</span></div></li></ol>

Helen is a conservationist who monitors the health of fish in a river. She is investigating whether a new water-treatment plant has affected the mass of adult trout in the river. Previously, the mean mass of adult trout in the river has been $w$ kg. Helen carries out a test, at the $10$% level of significance to find out whether there has been a change in the mean mass of adult trout in the river.

Helen catches $45$ adult trout and measures the mass, $X$, kilograms, of each fish. Her summarised data are as follows.

\[n=45 \quad \sum x=80.1 \quad \sum (x-\bar{x})^2=24.29\]

Calculate unbiased estimates of the population mean and variance of the mass of adult trout.

Use an algebraic method to calculate the set of values of $w$ for which there is insufficient evidence to conclude that there has been a change in the mean mass of adult trout in the river. You should state your hypotheses and define any symbols you use.

Explain why there is no need for Helen to know anything about the population distribution of the mass of adult trout.

2025 NYJC P2 Q8

<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

<p>A particular streaming service platform claims that the average time a user spends watching content on the platform per visit is more than $15$ minutes. A team of data analysts for the platform wants to verify this claim. They randomly select a sample of $80$ users and record the amount of time each user spends watching content in a single visit. The sample mean of time spent is found to be $16$ minutes. The population standard deviation of time spent is assumed to be $\sigma$ minutes.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Explain why the team should carry out a $1$-tail test.<span class="inline-marks">[1]</span></div></li></ol>
<p>A test at the $5\%$ significance level indicates that the platform's claim is valid.</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">Explain why the team is able to carry out a hypothesis test without knowing anything about the distribution of the times spent by the users.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the range of values of $\sigma$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Explain, in the context of the question, the meaning of 'at the $5\%$ significance level'.<span class="inline-marks">[1]</span></div></li></ol>

A particular streaming service platform claims that the average time a user spends watching content on the platform per visit is more than $15$ minutes. A team of data analysts for the platform wants to verify this claim. They randomly select a sample of $80$ users and record the amount of time each user spends watching content in a single visit. The sample mean of time spent is found to be $16$ minutes. The population standard deviation of time spent is assumed to be $\sigma$ minutes.

Explain why the team should carry out a $1$-tail test.

A test at the $5\%$ significance level indicates that the platform's claim is valid.

Explain why the team is able to carry out a hypothesis test without knowing anything about the distribution of the times spent by the users.

2025 JPJC P2 Q6

<p>EastHam is a football club in a competitive league. It is found that the average time taken by EastHam to score their first goal in a match is $50$ minutes. The coach trialled a new play tactic and the time taken, $x$ minutes, to score the first goal is recorded for a random sample of $35$ matches in the season. The total time taken is found to be $1650$ minutes and the variance of the time is $250$ minutes$^2$.</p>
<p>The coach wants to test if the average time taken to score the first goal has decreased with the implementation of the new play tactic.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Explain why the coach is able to carry out a hypothesis test without any assumption about the distribution of the time taken to score the first goal.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find unbiased estimates of the population mean and variance. Give your answers correct to $3$ decimal places.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Carry out the test at $10\%$ level of significance. You should state your hypotheses and define any symbols you use.</span><span class="inline-marks">[4]</span></div></li></ol>
<p>A fan of another football club in the same league, NewPalace, claims that the average time taken by NewPalace to score their first goal in a match is $40$ minutes. The time taken by NewPalace to score the first goal in a match can be assumed to be normally distributed with variance $280$ minutes$^2$.</p>
<ol class="abc-list" start="4"><li><div class="list-item-content"><span class="list-item-text">A random sample of $50$ matches is taken and a hypothesis test is carried out on whether the fan's claim is valid. Find the range of values of the mean time of this sample for which the fan's claim would be rejected at $10\%$ level of significance.</span><span class="inline-marks">[4]</span></div></li></ol>

EastHam is a football club in a competitive league. It is found that the average time taken by EastHam to score their first goal in a match is $50$ minutes. The coach trialled a new play tactic and the time taken, $x$ minutes, to score the first goal is recorded for a random sample of $35$ matches in the season. The total time taken is found to be $1650$ minutes and the variance of the time is $250$ minutes$^2$.

The coach wants to test if the average time taken to score the first goal has decreased with the implementation of the new play tactic.

Explain why the coach is able to carry out a hypothesis test without any assumption about the distribution of the time taken to score the first goal.

Find unbiased estimates of the population mean and variance. Give your answers correct to $3$ decimal places.

Carry out the test at $10\%$ level of significance. You should state your hypotheses and define any symbols you use.

A fan of another football club in the same league, NewPalace, claims that the average time taken by NewPalace to score their first goal in a match is $40$ minutes. The time taken by NewPalace to score the first goal in a match can be assumed to be normally distributed with variance $280$ minutes$^2$.

A random sample of $50$ matches is taken and a hypothesis test is carried out on whether the fan's claim is valid. Find the range of values of the mean time of this sample for which the fan's claim would be rejected at $10\%$ level of significance.

2025 CJC P2 Q11

<p>A distributor for health products claimed that his slimming pills could help women achieve their desired mass of $49$ kg. An experiment was carried out to test his claim. $80$ women were randomly chosen and they took the slimming pills for a period of time. The mass, $x$, of each woman was then taken and recorded in kg. it was calculated that&nbsp;$\sum{(x-40)=837}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State, with a reason, whether it is necessary to assume that the masses of the women follows a normal distribution in order 	to test his claim at $2\%$ significance level. </div></li><li><div class="list-item-paragraph">Assuming that the population variance of the masses is $100$ kg$^{2}$, test at $4\%$ significance level whether the distributor was understating his claim. </div></li></ol>

A distributor for health products claimed that his slimming pills could help women achieve their desired mass of $49$ kg. An experiment was carried out to test his claim. $80$ women were randomly chosen and they took the slimming pills for a period of time. The mass, $x$, of each woman was then taken and recorded in kg. it was calculated that $\sum{(x-40)=837}$.

<p>A distributor for musicals claimed that the ticket sales for a particular musical hit an average of $1220$ tickets a day, with variance $3136$. A random sample of $61$ days was taken and the sales, $x$ tickets, for each day were recorded. It was found that</p>
<p style="text-align: center">$\sum(x-1100)=6608$</p>
<ol class="i-list"><li><div class="list-item-paragraph">State, with a reason, whether it is necessary to assume that the ticket sales follow a normal distribution in order to test his claim at $10\%$ significance level.</div></li><li><div class="list-item-paragraph">Test at $10\%$ significance level whether the distributor was overstating his claim.</div></li></ol>

A distributor for musicals claimed that the ticket sales for a particular musical hit an average of $1220$ tickets a day, with variance $3136$. A random sample of $61$ days was taken and the sales, $x$ tickets, for each day were recorded. It was found that

Population with Known Variance And Not Normally Distributed

<p>A bakery claimed that the average number of loaves of bread sold per day is more than $150$, with variance $2500$. A random sample of $50$ days was taken and the sales, $x$ loaves, for each day were recorded. It was found that</p>
<p style="text-align: center">$\sum(x-140)=-120$</p>
<ol class="i-list"><li><div class="list-item-paragraph">State, with a reason, whether it is necessary to assume that the bread sales follow a normal distribution in order to test this claim at the $10\%$ significance level.</div></li><li><div class="list-item-paragraph">Test at the $10\%$ significance level whether the bakery’s claim was valid.</div></li></ol>

A bakery claimed that the average number of loaves of bread sold per day is more than $150$, with variance $2500$. A random sample of $50$ days was taken and the sales, $x$ loaves, for each day were recorded. It was found that

Population with Known Variance And Not Normally Distributed Q1

<p>A bakery claimed that the average number of loaves of bread sold per day is more than $150$, with variance $2500$. A random sample of $50$days was taken and the sales, $x$ loaves, for each day were recorded. It was found that<br></br></p>
<p style="text-align: center">$\sum{\left( x-140 \right)}=-120$<br></br></p>
<ol class="i-list"><li><div class="list-item-paragraph">State, with a reason, whether it is necessary to assume that the bread sales follow a normal distribution in order to test this claim at the $10\%$ significance level.</div></li><li><div class="list-item-paragraph">Test at the $10\%$ significance level whether the bakery’s claim was valid.</div></li></ol>

A bakery claimed that the average number of loaves of bread sold per day is more than $150$, with variance $2500$. A random sample of $50$days was taken and the sales, $x$ loaves, for each day were recorded. It was found that<br></br>

$\sum{\left( x-140 \right)}=-120$<br></br>

Worked Example 4

<p>The number of employees of a company, classified by department and gender, is shown below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 500px;"><colgroup><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"></colgroup><thead><tr style="height: 25px;"><th style="height: 25px; min-height: 25px; text-align: left; --cell-justify: flex-start"></th><th style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">Production</th><th style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">Development</th><th style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">Administration</th><th style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">Finance</th></tr></thead><tbody><tr style="height: 25px;"><td style="height: 25px; min-height: 25px; text-align: left; --cell-justify: flex-start">Male</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$2345$</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$1013$</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$237$</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$344$</td></tr><tr style="height: 25px;"><td style="height: 25px; min-height: 25px; text-align: left; --cell-justify: flex-start">Female</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$867$</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$679$</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$591$</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$523$</td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-paragraph">The directors wish to survey a sample of $100$ of the employees. This sample is to be a stratified sample,based on department and gender.</div><ol class="abc-list"><li><div class="list-item-paragraph">How many males should be in the sample?<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">How many females from the Development department should be in the sample?<span class="inline-marks">[1]</span></div></li></ol></li></ol>
<p>The Managing Director knows that, some years ago, the mean age of employees was $37$ years. He believes that the mean age of employees now is less than $37$ years.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">State why the stratified sample from part <strong>(i)</strong> should not be used for a hypothesis test of the Managing Director's belief.<span class="inline-marks">[1]</span></div></li></ol>
<p>The Company Secretary obtains a suitable sample of $80$ employees in order to carry out a hypothesis test of the Managing Director’s belief that the mean age of the employees now is less than $37$ years. You are given that the population variance of the ages is $140$ years$^{2}$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Write down appropriate hypotheses to test the Managing Director’s belief. You are given that the result of the test, using a $5%$ significance level, is that the Managing Director’s belief should be accepted. Determine the set of possible values of the mean age of the sample of employees.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">You are given instead that the mean age of the sample of employees is $35.2$ years, and that the result of a test at the $\alpha %$ significance level is that the Managing Director’s belief should not be accepted. Find the set of possible values of $\alpha $.<span class="inline-marks">[3]</span></div></li></ol>

The number of employees of a company, classified by department and gender, is shown below.

The Managing Director knows that, some years ago, the mean age of employees was $37$ years. He believes that the mean age of employees now is less than $37$ years.

The Company Secretary obtains a suitable sample of $80$ employees in order to carry out a hypothesis test of the Managing Director’s belief that the mean age of the employees now is less than $37$ years. You are given that the population variance of the ages is $140$ years$^{2}$.

Stratified Sampling

N2016 P2 Q6

<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>For quality control, the production manager of a company wishes to take a random sample of a certain type of chocolate bar from his factory. He wants to check that the mean mass of the bars is $52$ grams, as stated on the packets.</p>
<ol class="abc-list"><li data-list-item-id="item-1760850761017-7e5flxaik" data-list-item-path="item-1760850761017-7e5flxaik"><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 masses, $x$ grams, of a random sample of $80$ chocolate bars are summarised as follows.</p>
<p style="text-align: center">${n = 80} \qquad \sum(x - 52) = -37 \qquad \sum(x - 52)^2 = 310.7$</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760850778522-476owwb43" data-list-item-path="item-1760850778522-476owwb43"><div class="list-item-paragraph">Calculate the unbiased estimates of the population mean and variance.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760850780480-bwbw010tn" data-list-item-path="item-1760850780480-bwbw010tn"><div class="list-item-paragraph">What do you understand by the term ‘unbiased estimate’?<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760850781162-b2709b40d" data-list-item-path="item-1760850781162-b2709b40d"><div class="list-item-paragraph">Carry out a suitable hypothesis test, explaining the choice between a $1$-tail test and a $2$-tail test.</div><div class="list-item-paragraph">You should state your hypotheses and define any symbols you use.</div><div class="list-item-paragraph">Referring to the $p$-value for your test, explain what it indicates.<span class="inline-marks">[6]</span></div></li><li data-list-item-id="item-1760850795277-w2prh92ix" data-list-item-path="item-1760850795277-w2prh92ix"><div class="list-item-paragraph">Explain whether there is a need for the manager to know anything about the population distribution of the masses of the chocolate bars.<span class="inline-marks">[2]</span></div></li></ol>

For quality control, the production manager of a company wishes to take a random sample of a certain type of chocolate bar from his factory. He wants to check that the mean mass of the bars is $52$ grams, as stated on the packets.

The masses, $x$ grams, of a random sample of $80$ chocolate bars are summarised as follows.

${n = 80} \qquad \sum(x - 52) = -37 \qquad \sum(x - 52)^2 = 310.7$

Calculating and Interpreting $p$-Values

Interpreting Unbiased Estimators

Unbiased Estimates of Population Parameters

2024 DHS P2 Q10

<p>In a drinks factory, a machine is programmed to dispense $500$ ml of green tea into empty bottles.</p>
<ol class="abc-list"><li data-list-item-id="item-1760849155823-cwfmudd7t" data-list-item-path="item-1760849155823-cwfmudd7t"><div class="list-item-paragraph">After a routine check of the machine, the production manager suspects that the machine is dispensing more green tea than expected. A random sample of $50$ bottles is taken for a hypothesis test at $\alpha \%$ level of significance and the data is as follows:</div><div class="list-item-paragraph">Taking $\alpha =2$, test the production manager’s suspicion and state the meaning of the $p$-value in the context of the question.<span class="inline-marks">[5]</span></div><div class="table-container"><table style="table-layout: fixed; width: 530px;"><colgroup><col style="width: 250px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"></colgroup><thead><tr><th style="text-align: left; --cell-justify: flex-start"><p>Volume of green tea (correct to nearest ml)</p></th><th style="text-align: center; --cell-justify: center"><p>$498$</p></th><th style="text-align: center; --cell-justify: center"><p>$499$</p></th><th style="text-align: center; --cell-justify: center"><p>$500$</p></th><th style="text-align: center; --cell-justify: center"><p>$501$</p></th><th style="text-align: center; --cell-justify: center"><p>$502$</p></th><th style="text-align: center; --cell-justify: center"><p>$503$</p></th><th style="text-align: center; --cell-justify: center"><p>$504$</p></th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start"><p>Number of bottles</p></td><td style="text-align: center; --cell-justify: center"><p>$5$</p></td><td style="text-align: center; --cell-justify: center"><p>$11$</p></td><td style="text-align: center; --cell-justify: center"><p>$10$</p></td><td style="text-align: center; --cell-justify: center"><p>$11$</p></td><td style="text-align: center; --cell-justify: center"><p>$10$</p></td><td style="text-align: center; --cell-justify: center"><p>$1$</p></td><td style="text-align: center; --cell-justify: center"><p>$2$</p></td></tr></tbody></table></div><ol class="i-list"><li data-list-item-id="subitem-1760849179335-gzzo4fifa" data-list-item-path="item-1760849155823-cwfmudd7t/subitem-1760849179335-gzzo4fifa"><div class="list-item-paragraph">Determine the set of values of $\alpha $ for which the production manager’s suspicion is valid.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1760849181660-e5nlmtzad" data-list-item-path="item-1760849155823-cwfmudd7t/subitem-1760849181660-e5nlmtzad"><div class="list-item-paragraph">Explain why the production manager is able to carry out the hypothesis test without knowing anything about the distribution of the volume of green tea dispensed by the machine.<span class="inline-marks">[1]</span></div></li></ol></li><li data-list-item-id="item-1760849182808-rbkv3l3rx" data-list-item-path="item-1760849182808-rbkv3l3rx"><div class="list-item-paragraph">The machine is being recalibrated to dispense $500$ ml of green tea. Another random sample of $50$ bottles of green tea is taken and the mean and standard deviation of this sample are $502$ ml and $k$ ml respectively. A hypothesis test, at the $5\%$ significance level, concluded that the recalibration is done accurately. Find the set of values that&nbsp;$k$ can take.<span class="inline-marks">[5]</span></div></li></ol>

In a drinks factory, a machine is programmed to dispense $500$ ml of green tea into empty bottles.

2024 JPJC P2 Q10

<p>The average waiting time (in minutes) for a customer to be served at a fast-food restaurant is $k$ minutes. The manager claims that with subsequent increased staff strength, the restaurant can now better serve the customers and the average waiting time has improved. The population variance of waiting time is known to be $0.22$ minutes$^{2}$.</p>
<p>The observed waiting times (in minutes) of a random sample of $9$ customers after the increased staff strength are:</p>
<p style="text-align: center">${3.6, \qquad 2.5, \qquad 2.4, \qquad 3.5, \qquad 2.8, \qquad 2.9, \qquad 3.1, \qquad 3.7, \qquad 3.0}$</p>
<ol class="abc-list"><li data-list-item-id="item-1760684824142-jq0g84ts5" data-list-item-path="item-1760684824142-jq0g84ts5"><div class="list-item-paragraph">Find the unbiased estimates of the population mean and variance of the waiting time.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760684824907-8w247vqp9" data-list-item-path="item-1760684824907-8w247vqp9"><div class="list-item-paragraph">Using the value $k=3.3$.</div><ol class="i-list"><li data-list-item-id="subitem-1760684841991-37dkm5kbj" data-list-item-path="item-1760684824907-8w247vqp9/subitem-1760684841991-37dkm5kbj"><div class="list-item-paragraph">Carry out a one-tailed test at $5\%$ level of significance to determine whether the manager’s claim is valid, stating clearly the $p$-value and any assumption(s) you have made.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="subitem-1760684843476-u3qbyt7l3" data-list-item-path="item-1760684824907-8w247vqp9/subitem-1760684843476-u3qbyt7l3"><div class="list-item-paragraph">Explain the meaning of $p$-value in the context of the question.<span class="inline-marks">[1]</span></div></li></ol></li><li data-list-item-id="item-1760684827232-z625w5ft8" data-list-item-path="item-1760684827232-z625w5ft8"><div class="list-item-paragraph">Using the same assumption in part <strong>(b)(i)</strong>, a one-tailed test was carried out at $10\%$ and it was found that the manager’s claim was justified. Find the range of possible values of $k$.<span class="inline-marks">[4]</span></div></li></ol>

The average waiting time (in minutes) for a customer to be served at a fast-food restaurant is $k$ minutes. The manager claims that with subsequent increased staff strength, the restaurant can now better serve the customers and the average waiting time has improved. The population variance of waiting time is known to be $0.22$ minutes$^{2}$.

The observed waiting times (in minutes) of a random sample of $9$ customers after the increased staff strength are:

${3.6, \qquad 2.5, \qquad 2.4, \qquad 3.5, \qquad 2.8, \qquad 2.9, \qquad 3.1, \qquad 3.7, \qquad 3.0}$

2023 DHS P2 Q10

<p>The manufacturer of Mola snack biscuits claimed that the mean biscuit content in a packet is $15$g. A sample of $80$ packets is taken and the weight, $x$g, of the content in each packet is measured. The results are summarised by</p>
<p style="text-align: center">$\sum{\left( x-15 \right)=53.6}$ and $\sum{{{\left( x-15 \right)}^{2}}=795.7}$.</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">Stating a necessary assumption, test whether the mean weight differs from $15$g at $4%$ level of significance.</div></li><li><div class="list-item-paragraph">If the test is conducted at $\alpha %$ level of significance and the null hypothesis is rejected in favour of the alternative hypothesis based on the values from this sample, determine the set of values of $\alpha $.</div></li><li><div class="list-item-paragraph">Another large random sample of $n$ packets is taken. Assume now that the population standard deviation is $3$g. Find the range of values of the sample mean, in terms of $n$, for which the claim that the mean weight differs from $15$g is supported at $4%$ level of significance.</div></li></ol>

The manufacturer of Mola snack biscuits claimed that the mean biscuit content in a packet is $15$g. A sample of $80$ packets is taken and the weight, $x$g, of the content in each packet is measured. The results are summarised by

$\sum{\left( x-15 \right)=53.6}$ and $\sum{{{\left( x-15 \right)}^{2}}=795.7}$.

2015 TJC P2 Q7 [Modified]

<p>A virologist claims that due to the introduction of vaccine, the recovery period from a certain type of flu-like virus is on average $7$ days. However, an analyst who works with such patients on a day-to-day basis believes that the average recovery period quoted by the virologist is understated. He carries out a survey to investigate this by recording the recovery period of $120$ randomly chosen patients. The recovery period of each of these $120$ patients are denoted by $x$ days. The summarised results are given by</p>
<p style="text-align: center">$\sum{\left( x-7 \right)=24}$ and $\sum{{{\left( x-7 \right)}^{2}}=\mathrm{14908}}$</p>
<ol class="i-list"><li><div class="list-item-paragraph">Explain in context, the meaning of a random sample of $120$ patients.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Calculate the unbiased estimates of the population mean and variance of the recovery period of the patients.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Carry out a test at $5\%$ significance level, on the claim that the average recovery period of $7$ days is understated.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Explain if the test in part <strong>(iii)</strong> would still be valid if the analyst had asked for recovery periods from a random sample of $15$ patients.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Suppose instead, the analyst wants to test if the average recovery period differs from $7$ days. The analyst collects data from another random sample of $120$ patients. Given that the population standard deviation is now known to be $1$ day, find the range of values of the sample mean, $\overline{x}$, such that the null hypothesis is rejected at $1\%$ level of significance.</div><p style="text-align: right;">[4]</p></li></ol>

A virologist claims that due to the introduction of vaccine, the recovery period from a certain type of flu-like virus is on average $7$ days. However, an analyst who works with such patients on a day-to-day basis believes that the average recovery period quoted by the virologist is understated. He carries out a survey to investigate this by recording the recovery period of $120$ randomly chosen patients. The recovery period of each of these $120$ patients are denoted by $x$ days. The summarised results are given by

$\sum{\left( x-7 \right)=24}$ and $\sum{{{\left( x-7 \right)}^{2}}=\mathrm{14908}}$

Math Question Bank