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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 refrigerator manufacturer claims that the mean lifespan of refrigerators of a particular model is $12$ years. A consumers association representative suspects that the mean lifespan of the refrigerators is actually less than $12$ years.</p>
<p>The durations $x$, in years, of a random sample of $45$ refrigerators are summarised as follows.</p>
<p style="text-align: center">\[\sum(x-12) = -4.3 \qquad \sum(x-12)^2 = 17.08\]</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance of the lifespan of the refrigerators.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">State hypotheses that can be used to test if the mean lifespan of the refrigerators is less than $12$ years, defining any parameters you use. Test, at the $5\%$ level of significance, whether the mean lifespan of the refrigerators is less than $12$ years.<span class="inline-marks">[4]</span></div></li></ol>
<p>The manufacturer switches to a different coolant and decides to test whether the mean lifespan of the refrigerators has changed from $12$ years. He records the durations of a large random sample of $n$ refrigerators and finds that their mean lifespan is $12.4$ years and variance is $4.1$ square years.</p>
<p>A test at $5\%$ significance level, is carried out on the new random sample. The test shows that there is sufficient evidence that the mean lifespan of refrigerators has changed.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find the range of possible values of $n$.<span class="inline-marks">[4]</span></div></li></ol>

A refrigerator manufacturer claims that the mean lifespan of refrigerators of a particular model is $12$ years. A consumers association representative suspects that the mean lifespan of the refrigerators is actually less than $12$ years.

The durations $x$, in years, of a random sample of $45$ refrigerators are summarised as follows.

\[\sum(x-12) = -4.3 \qquad \sum(x-12)^2 = 17.08\]

The manufacturer switches to a different coolant and decides to test whether the mean lifespan of the refrigerators has changed from $12$ years. He records the durations of a large random sample of $n$ refrigerators and finds that their mean lifespan is $12.4$ years and variance is $4.1$ square years.

A test at $5\%$ significance level, is carried out on the new random sample. The test shows that there is sufficient evidence that the mean lifespan of refrigerators has changed.

Estimating Parameters from Summarised and Coded Data

One-Tailed Hypothesis Tests

Parameter Bounds from Hypothesis Tests

Estimating Population Variance from Sample Variance

Null and Alternative Hypotheses

Two-Tailed Hypothesis Tests

Large-Sample Mean Tests with Estimated Variance

2024 ACJC P2 Q8

Hypothesis Testing

Sampling

<p>A hand sanitiser company, Cleanser, claims that the mean volume of each bottle of their hand sanitiser is $15$ ml. The consumer association took a random sample of $80$ bottles and measured the volume of each bottle. It was found that the sample mean and the sample variance were $\overline{x}$ and $2$ ml$^2$ respectively. A test was conducted at the $5\%$ level of significance to test if Cleanser has overstated the mean volume of its hand sanitiser.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Explain, in the context of the question, the meaning of a random sample.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Explain, in the context of the question, the meaning of 'at the $5\%$ level of significance'.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Find the unbiased estimate of the population variance.<span class="inline-marks">[1]</span></div></li></ol>
<p>It was found that there was insufficient evidence to conclude that Cleanser had overstated the mean volume of their hand sanitiser.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">By defining any parameters used and stating the hypotheses clearly, find the range of values of $\overline{x}$, correct to $2$ decimal places.<span class="inline-marks">[5]</span></div></li></ol>

A hand sanitiser company, Cleanser, claims that the mean volume of each bottle of their hand sanitiser is $15$ ml. The consumer association took a random sample of $80$ bottles and measured the volume of each bottle. It was found that the sample mean and the sample variance were $\overline{x}$ and $2$ ml$^2$ respectively. A test was conducted at the $5\%$ level of significance to test if Cleanser has overstated the mean volume of its hand sanitiser.

It was found that there was insufficient evidence to conclude that Cleanser had overstated the mean volume of their hand sanitiser.

Significance Levels and $p$-Values

Population and Random Samples

<p>In a factory, salt is packed into bags of $25$ grams by a filling machine.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">The quality control manager selects a random sample of $50$ bags of salt. The masses, $x$ grams, are summarized by\[\sum x = 995.7 \text{ and}\hspace{0.5em} \sum x^2 = 26884.37\]Test, at the $5\%$ level of significance, whether the filling machine produces underweight bags of salt.<span class="inline-marks">[7]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Following a slight adjustment to the filling machine and assuming that the mass of salt in a bag follows a normal distribution, the quality control manager takes a new random sample of $50$ bags to check if the mean mass of salt per bag is $25$ grams. The mean and standard deviation of the sample is found to be $22$ grams and $12.0$ grams respectively. A test at the $\alpha\%$ significance level gives insufficient evidence to conclude that the mean mass of salt in each bag differs from $25$ grams. Find the range of possible values of $\alpha$.</span><span class="inline-marks">[4]</span></div></li></ol>

In a factory, salt is packed into bags of $25$ grams by a filling machine.

Following a slight adjustment to the filling machine and assuming that the mass of salt in a bag follows a normal distribution, the quality control manager takes a new random sample of $50$ bags to check if the mean mass of salt per bag is $25$ grams. The mean and standard deviation of the sample is found to be $22$ grams and $12.0$ grams respectively. A test at the $\alpha\%$ significance level gives insufficient evidence to conclude that the mean mass of salt in each bag differs from $25$ grams. Find the range of possible values of $\alpha$.

2024 TJC H1 Q11

<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.

Assumptions and Validity of Mean Tests

Central Limit Theorem

2025 YIJC P2 Q10

<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.

Mean Tests for Normal Populations with Known Variance

Assumptions for Sampling

Interpreting Hypothesis Test Conclusions

Critical Values and Critical Regions

2025 RI P2 Q9

<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>The wingspan of a certain endangered species of giant butterfly found in the rainforest in New Guinea is a normally distributed random variable with mean $18.0$ cm. A lepidopterist who just arrived in New Guinea is keen to find out if climate changes have caused the size of the butterflies to decrease. He measures the wingspan of a random sample of $45$ such butterflies and uses the data to perform a hypothesis test.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State appropriate hypotheses for the test, defining any symbols that you use. <span class="inline-marks">[2]</span></div></li></ol>
<p>The sample mean wingspan is denoted by $\bar{x}$ cm and the sample variance is $1.2$ cm$^{2}$.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the set of values of $\bar{x}$ for which the null hypothesis would be rejected at $5\%$ level of significance<span class="inline-marks">[4]</span></div></li></ol>
<p>The lepidopterist later discovered some old records and noted that the wingspan of the butterfly has a population variance of $1.3$ cm$^{2}$. Using this information, he carried out an appropriate test at $5\%$ significance level to determine if the size of the butterflies has changed.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find the set of values of $\bar{x}$ for which the null hypothesis would not be rejected.	<span class="inline-marks">[4]</span></div></li></ol>

The wingspan of a certain endangered species of giant butterfly found in the rainforest in New Guinea is a normally distributed random variable with mean $18.0$ cm. A lepidopterist who just arrived in New Guinea is keen to find out if climate changes have caused the size of the butterflies to decrease. He measures the wingspan of a random sample of $45$ such butterflies and uses the data to perform a hypothesis test.

The sample mean wingspan is denoted by $\bar{x}$ cm and the sample variance is $1.2$ cm$^{2}$.

The lepidopterist later discovered some old records and noted that the wingspan of the butterfly has a population variance of $1.3$ cm$^{2}$. Using this information, he carried out an appropriate test at $5\%$ significance level to determine if the size of the butterflies has changed.

2015 RI P2 Q9 [Modified]

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

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

Finding The Population Variance Given Sample Variance

<p>A sample, of size $80$, has a variance of $6.77$. Estimate the population variance. 







</p>

A sample, of size $80$, has a variance of $6.77$. Estimate the population variance. 









<p>A sample, of size $65$, has a standard deviation of $3.612$. Estimate the population variance.</p>

A sample, of size $65$, has a standard deviation of $3.612$. Estimate the population variance.

<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.

Calculating and Interpreting $p$-Values

Hypothesis Tests Using Critical Regions and $p$-Values

2024 JPJC P2 Q10

<p>In a large busy restaurant, the mean time taken for a server to clear a table and set it for the next guest is $4.5$ minutes. In order to improve the quality of service and maintain a clean environment, the restaurant manager introduced a new routine to clear tables. A random sample of $30$ servers is taken and the time taken by each server to clear a table, $x$ minutes, is recorded. The data are summarised as follows:</p>
<p style="text-align: center">$\sum{x=132}$ and $\sum{{{x}^{2}}=\mathrm{583}\mathrm{.96}}$.</p>
<ol class="i-list"><li data-list-item-id="item-1760358318404-6mot9auhx" data-list-item-path="item-1760358318404-6mot9auhx"><div class="list-item-paragraph">Find unbiased estimates of the population mean and variance.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760358320144-siphcm6rd" data-list-item-path="item-1760358320144-siphcm6rd"><div class="list-item-paragraph">Test, at the $10\%$ significance level, whether the mean time taken for a server to clear a table has changed.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-1760358345021-v9ibabikb" data-list-item-path="item-1760358345021-v9ibabikb"><div class="list-item-paragraph">Using the results in part <strong>(ii)</strong>, state the range of values of $\alpha $, where&nbsp;$\alpha \%$ is the level of significance, at which a test would lead the manager to conclude that the mean time has not changed.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760358346011-1kxquuqf3" data-list-item-path="item-1760358346011-1kxquuqf3"><div class="list-item-paragraph">Without carrying out another test, state with reasons, the conclusion if the test at $5\%$ significance level was to determine whether the mean time taken to clear a table is less than $4.5$ minutes.<span class="inline-marks">[2]</span></div></li></ol>
<p>After a few weeks, based on feedback from servers and guests, the manager modifies the routine. Another random sample of $30$ servers is taken and their time taken to clear a table has mean $\bar{x}$ minutes and variance $1.01$ minutes$^{2}$.</p>
<ol class="i-list" start="5"><li data-list-item-id="item-1760358379988-udt6gpuft" data-list-item-path="item-1760358379988-udt6gpuft"><div class="list-item-paragraph">Find the set of values of $\bar{x}$ for which a test at $1\%$ significance level concludes that the mean time taken to clear a table is greater than $4.5$ minutes, giving your answer to $3$ decimal places.<span class="inline-marks">[4]</span></div></li></ol>

In a large busy restaurant, the mean time taken for a server to clear a table and set it for the next guest is $4.5$ minutes. In order to improve the quality of service and maintain a clean environment, the restaurant manager introduced a new routine to clear tables. A random sample of $30$ servers is taken and the time taken by each server to clear a table, $x$ minutes, is recorded. The data are summarised as follows:

$\sum{x=132}$ and $\sum{{{x}^{2}}=\mathrm{583}\mathrm{.96}}$.

After a few weeks, based on feedback from servers and guests, the manager modifies the routine. Another random sample of $30$ servers is taken and their time taken to clear a table has mean $\bar{x}$ minutes and variance $1.01$ minutes$^{2}$.

Unbiased Estimates of Population Parameters

Comparing One-Tailed and Two-Tailed Tests

2018 CJC P2 Q10

<p>In a public swimming centre, the time spent by a randomly chosen user in using its facilities is $T$ minutes, is known to be normally distributed. The centre manager claims that its users spend an average of $50$ minutes to use its facilities. To check this claim, time spent by a random sample of $60$ users were recorded. The data recorded has an average of $47$ minutes and a standard deviation of $16.4$ minutes.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find an unbiased estimate of the population variance, giving your answer correct to $2$ decimal places.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Test, at the $5\%$ significance level, whether the centre manager overstated the average time spent.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Another sample of size $n\left( n>30 \right)$ that was collected independently is now used to test, at the $5\%$ significance level, whether the centre manager’s claim is valid. For this sample, the mean time taken is $46$ minutes. If the result of the test using this information and the unbiased estimate of the population variance in part <strong>(i)</strong> is that the null hypothesis is rejected, find the least possible value that $n$ can take.</div><p style="text-align: right;">[4]</p></li></ol>

In a public swimming centre, the time spent by a randomly chosen user in using its facilities is $T$ minutes, is known to be normally distributed. The centre manager claims that its users spend an average of $50$ minutes to use its facilities. To check this claim, time spent by a random sample of $60$ users were recorded. The data recorded has an average of $47$ minutes and a standard deviation of $16.4$ minutes.

2022 ASRJC P2 Q8

<p>In a shopping mall, the number of hours that customers spend shopping in the mall follows a normal distribution. The manager of the mall claims that the average number of hours customers spend in the mall is at least $2$ hours. To test this claim, a random sample of $60$ customers is taken and the time, $x$ hours, spent is measured. The results are summarized by $\sum{x}=110.5$ and $\sum{{{x}^{2}}}=220.3$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find unbiased estimates of the population mean and variance.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Test at the $2\%$ significance level whether the manager's claim is valid.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Explain, in the context of the question, the meaning of “at the $2\%$ significance level”.<span class="inline-marks">[1]</span></div></li></ol>
<p>After an upgrading project to improve the facilities in the mall, a new random sample of $55$ customers is taken. The data gives a sample mean of $\bar{x}$ with sample standard deviation $0.51$ hours. A test at the $2\%$ significance level indicates that the manager’s claim is valid with the improved facilities.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Using an algebraic method, calculate the range of values of $\bar{x}$.<span class="inline-marks">[5]</span></div></li></ol>

In a shopping mall, the number of hours that customers spend shopping in the mall follows a normal distribution. The manager of the mall claims that the average number of hours customers spend in the mall is at least $2$ hours. To test this claim, a random sample of $60$ customers is taken and the time, $x$ hours, spent is measured. The results are summarized by $\sum{x}=110.5$ and $\sum{{{x}^{2}}}=220.3$.

After an upgrading project to improve the facilities in the mall, a new random sample of $55$ customers is taken. The data gives a sample mean of $\bar{x}$ with sample standard deviation $0.51$ hours. A test at the $2\%$ significance level indicates that the manager’s claim is valid with the improved facilities.

2013 IJC P2 Q11 [Modified]

<p>It is found that certain kinds of meat lose weight as a result of being cooked. A restaurant chef is prepared to accept up to $10\%$ loss but suspects that the recent consignments have a higher percentage weight loss. She decides to carry out a hypothesis test on a random sample of steaks.</p>
<ol class="i-list"><li data-list-item-id="item-1760242787976-hvdi0d192" data-list-item-path="item-1760242787976-hvdi0d192"><div class="list-item-paragraph">Explain the meaning of ‘a random sample’ in the context of the question and why the chef should sample a large number of steaks.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760242788554-bwp5oqbne" data-list-item-path="item-1760242788554-bwp5oqbne"><div class="list-item-paragraph">State suitable hypotheses for the test, defining any symbols that you use.<span class="inline-marks">[2]</span></div></li></ol>
<p>The chef takes a random sample of $40$ steaks, and calculate the percentage weight loss of each steak, $x$ (in percent). The mean and variance of the percentage weight loss of the $40$ steaks are $10.48\%$ with variance $3.37\%$ respectively.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760242816244-54660qzba" data-list-item-path="item-1760242816244-54660qzba"><div class="list-item-paragraph">Test, at the $5\%$ significance level, to determine whether the sample supports the chef’s suspicion.<span class="inline-marks">[3]</span></div></li></ol>
<p>The chef carries out another test using $100$ readings obtained from chefs of other restaurants. The percentage weight loss of each steak, $y$ (in percent), is summarised by:</p>
<p style="text-align: center">$\sum{y=\mathrm{1055}\mathrm{.2}}$, $\sum{{{y}^{2}}=k}$.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-1760242854472-ui9ke0ika" data-list-item-path="item-1760242854472-ui9ke0ika"><div class="list-item-paragraph">Find the range of values of $k$ such that the chef’s suspicion is not valid at the $5\%$ level of significance, giving your answer correct to $2$ decimal places.<span class="inline-marks">[5]</span></div></li></ol>

It is found that certain kinds of meat lose weight as a result of being cooked. A restaurant chef is prepared to accept up to $10\%$ loss but suspects that the recent consignments have a higher percentage weight loss. She decides to carry out a hypothesis test on a random sample of steaks.

The chef takes a random sample of $40$ steaks, and calculate the percentage weight loss of each steak, $x$ (in percent). The mean and variance of the percentage weight loss of the $40$ steaks are $10.48\%$ with variance $3.37\%$ respectively.

The chef carries out another test using $100$ readings obtained from chefs of other restaurants. The percentage weight loss of each steak, $y$ (in percent), is summarised by:

$\sum{y=\mathrm{1055}\mathrm{.2}}$, $\sum{{{y}^{2}}=k}$.

2020 TJC P2 Q11

<ol class="abc-list" start="2"><li><div class="list-item-paragraph">In general, the marks obtained by students in their Mathematics examination may be assumed to be a random variable with mean $65$ and variance $32$. After the introduction of a new program of revision classes, it is found that there are improvements to the marks of the some of the students.</div><div class="list-item-paragraph">A random sample of the marks of $50$ students is taken and the mean is found to be $66.4$. A test is conducted and it shows that there is insufficient evidence that the new program of revision classes is effective. Find an inequality satisfied by the significance level of the test.A random sample of the marks of $50$ students is taken and the mean is found to be $66.4$. A test is conducted and it shows that there is insufficient evidence that the new program of revision classes is effective. Find an inequality satisfied by the significance level of the test.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">A sample of $n$ observations is taken form a population with mean $5$. The sample mean is found to be $5.1$ and sample standard deviation $0.7$. A 2-tailed normal test ($z$-test) is conducted and the null hypothesis is rejected at the $5\%$ significance level. Find the least value of $n$.</div></li></ol>

Large-Sample Mean Tests with Known Variance

2007 IJC P2 Q11

<p>A nasi lemak stall holder uses fresh chicken wings as an ingredient for fried chicken wings. Based on his past years records, his mean daily profit was $\$535$. With the recent lack of fresh chicken supply, the stall holder substituted fresh chicken wings with frozen chicken wings as the ingredient. His wife was hesitant to the change and claimed that the mean daily profit will decrease. To test his wife’s claim, the stall holder takes a random sample of $45$ days and recorded the daily profits, $\$x$.</p>
<ol class="i-list"><li data-list-item-id="item-1760170818976-ettfcivxm" data-list-item-path="item-1760170818976-ettfcivxm"><div class="list-item-paragraph">State appropriate hypotheses to test the wife’s claim and define any symbols that you use.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760170819591-bhsf0knrt" data-list-item-path="item-1760170819591-bhsf0knrt"><div class="list-item-paragraph">State, with a reason, whether it is necessary to assume that his past years records of daily profits are normally distributed for the test to be valid.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760170820108-s6z87lhln" data-list-item-path="item-1760170820108-s6z87lhln"><div class="list-item-paragraph">Based on the past years records, it is assumed that the population variance of the daily profit is $2591$. If the test shows that there is sufficient evidence that the wife’s claim is accepted at $5\%$ level of significance, determine the set of possible values of $\bar{x}$, the mean daily profit in the $45$ days.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760170821211-p70xnq96n" data-list-item-path="item-1760170821211-p70xnq96n"><div class="list-item-paragraph">The stall holder found that&nbsp;$\bar{x}=\$520$ and suspects that the population variance of $\mathrm{2591}$ may be incorrect. Hence, he decided to use the sample variance value of $2008$ to test his wife’s claim. State the conclusion of the test, showing your workings clearly.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760170843329-916j68q67" data-list-item-path="item-1760170843329-916j68q67"><div class="list-item-paragraph">State the largest significance level that the stall holder should use so that the conclusion in <strong>(iv)</strong> will be different. Leave your answer in $2$ decimal places.<span class="inline-marks">[1]</span></div></li></ol>
<p>The stall holder now suspects that the mean daily profit does not differ from $\$535$, even if frozen chicken wings are used instead of fresh chicken wings. To test his claim, the stall holder decides to increase the number of randomly selected days, $n$, to record his daily profit.</p>
<ol class="i-list" start="6"><li data-list-item-id="item-1760170874626-yhqc2unfn" data-list-item-path="item-1760170874626-yhqc2unfn"><div class="list-item-paragraph">It is given that&nbsp;$\bar{x}=526$ and the population variance is assumed to be $\mathrm{2591}$. Determine the greatest value of $n$, so that the conclusion of the test shows that there is no reason to reject the null hypothesis at $8\%$ level of significance.<span class="inline-marks">[4]</span></div></li></ol>

A nasi lemak stall holder uses fresh chicken wings as an ingredient for fried chicken wings. Based on his past years records, his mean daily profit was $\$535$. With the recent lack of fresh chicken supply, the stall holder substituted fresh chicken wings with frozen chicken wings as the ingredient. His wife was hesitant to the change and claimed that the mean daily profit will decrease. To test his wife’s claim, the stall holder takes a random sample of $45$ days and recorded the daily profits, $\$x$.

The stall holder now suspects that the mean daily profit does not differ from $\$535$, even if frozen chicken wings are used instead of fresh chicken wings. To test his claim, the stall holder decides to increase the number of randomly selected days, $n$, to record his daily profit.

2022 ACJC P2 Q10

<p>A popular investment manager claims that the mean dividend yield of his clients is $12\%$. A research firm investigates further by collecting a random sample of $40$ of the manager’s clients. The sample was found to have a mean dividend yield of $10.9\%$ and a standard deviation of $2.5\%$</p>
<ol class="i-list"><li data-list-item-id="item-1760146831141-merkf1sc2" data-list-item-path="item-1760146831141-merkf1sc2"><div class="list-item-paragraph">State, giving a reason, whether there is a need to make any assumption about the distribution of the dividend yield earned by the manager’s clients in order for a hypothesis test to be valid.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760146831808-06hgjwt33" data-list-item-path="item-1760146831808-06hgjwt33"><div class="list-item-paragraph">Test, at the $5%$ level of significance, whether the manager overstated his claim.<span class="inline-marks">[5]</span></div></li></ol>
<p>It was later known that the standard deviation of the dividend yield of the popular investment manager’s clients is $2.3\%$. A potential investor collected the dividend yield data of a small sample size, $n$, of the manager’s clients. The sample was found to have a mean dividend yield of $10.2\%$.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760146857450-6s5b0h49a" data-list-item-path="item-1760146857450-6s5b0h49a"><div class="list-item-paragraph">State, in context, two assumptions that are needed for you to carry out a test to examine the manager’s claim that the mean dividend yield of his clients is $12\%$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760146858095-htzb9i4vk" data-list-item-path="item-1760146858095-htzb9i4vk"><div class="list-item-paragraph">Hence find the least value of $n$ such that the manager’s claim at the $5\%$ level of significance, will be rejected. You should state your hypothesis clearly.<span class="inline-marks">[4]</span></div></li></ol>

A popular investment manager claims that the mean dividend yield of his clients is $12\%$. A research firm investigates further by collecting a random sample of $40$ of the manager’s clients. The sample was found to have a mean dividend yield of $10.9\%$ and a standard deviation of $2.5\%$

It was later known that the standard deviation of the dividend yield of the popular investment manager’s clients is $2.3\%$. A potential investor collected the dividend yield data of a small sample size, $n$, of the manager’s clients. The sample was found to have a mean dividend yield of $10.2\%$.

2020 ACJC P2 Q10

<ol class="abc-list"><li data-list-item-id="item-1760090781612-2w45xgq4i" data-list-item-path="item-1760090781612-2w45xgq4i"><div class="list-item-paragraph"><p>A confectionery is selling mooncakes during the Mid-Autumn Festival. The owner claims that the mean mass of the mooncakes is at least $150$ g. The distribution of the mass of a mooncake has standard deviation $6.73$ g. A customer bought a random selection of $9$ mooncakes from the confectionery with masses, in g, as follows:</p></div><div class="list-item-paragraph" style="text-align: center">$145$    $148$    $153$    $156$    $141$    $151$    $143$    $157$    $138$</div><div class="list-item-paragraph">Test at the $10\%$ significance level whether the owner’s claim is valid. State an assumption about the population distribution of the mass of the mooncakes.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-1760090782663-5oy69erny" data-list-item-path="item-1760090782663-5oy69erny"><div class="list-item-paragraph">In a particular country, the mean working hours of teachers in schools is $60$ hours per week. After implementation of the Home-Based Learning in all schools, the Ministry claims that the mean working hours of teachers remained unchanged. A random sample of $50$ teachers was taken and the number of working hours per week was recorded. The mean working hours and standard deviation for the sample were found to be $62$ hours and $k$ hours respectively. A hypothesis test was conducted and it was found that there is sufficient evidence to reject the Ministry’s claim at $5\%$ significance level. Find the set of values of $k$.<span class="inline-marks">[4]</span></div></li></ol>

2022 DHS P2 Q9

<p>Based on past records, shoppers spend an average of $1.5$ hours at MJ mall per visit. After a revamp and expansion to include more stores, the mall management wishes to conduct a hypothesis test to check if the duration that shoppers spend in the mall per visit has increased. A random sample of $50$ shoppers were surveyed and the duration of time, $x$, in hours per visit is recorded. The sample sum is $82.5$ hours and the sample variance is $0.425$ hours$^{2}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State appropriate hypotheses for the test and find the unbiased estimates of the population mean and variance.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Explain why the mall management is able to carry out a hypothesis test without making any assumptions about the distribution of the duration shoppers spend at MJ mall per visit.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Calculate the $p$-value and state its meaning in context of the question. Hence state the set of values of the level of significance $\alpha \%$ for which the management’s hypothesis is valid.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">The mall management believes that the mean spending per shopper per visit is $\$150$. A test at $5\%$ level of significance found that there is significant evidence that the population mean spending is less than $\$150$. Using only this information, explain if the following statements are necessarily true, necessarily false, or neither necessarily true nor necessarily false.</div><ol class="abc-list"><li><div class="list-item-paragraph">There is significant evidence at the $10\%$ level of significance that the population mean spending is less than $\$150$.</div></li><li><div class="list-item-paragraph">There is significant evidence at the $5\%$ level of significance that the population mean spending differs from $\$150$.</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

Based on past records, shoppers spend an average of $1.5$ hours at MJ mall per visit. After a revamp and expansion to include more stores, the mall management wishes to conduct a hypothesis test to check if the duration that shoppers spend in the mall per visit has increased. A random sample of $50$ shoppers were surveyed and the duration of time, $x$, in hours per visit is recorded. The sample sum is $82.5$ hours and the sample variance is $0.425$ hours$^{2}$.

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