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PSLE

PSLE Foundation Mathematics

<p>The mean of a random variable $X$ is denoted by $\mu$. A sample of 50 random observations of $X$ is taken and the results are summarised by $\sum x = 527.1$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">It is given that the population variance is 15. Carry out a 2-tail test of the null hypothesis $\mu = 9.5$, at the 5% significance level.</div></li><li><div class="list-item-paragraph">It is given instead that&nbsp;$\sum x^2 = 6172.31$. In a 1-tail test of the null hypothesis $\mu = 11.5$, the alternative hypothesis is accepted. State the alternative hypothesis, and find an inequality satisfied by the significance level of the test.</div></li></ol>

The mean of a random variable $X$ is denoted by $\mu$. A sample of 50 random observations of $X$ is taken and the results are summarised by $\sum x = 527.1$.

Significance Levels and $p$-Values

One-Tailed Hypothesis Tests

Parameter Bounds from Hypothesis Tests

Null and Alternative Hypotheses

Two-Tailed Hypothesis Tests

Large-Sample Mean Tests with Estimated Variance

Large-Sample Mean Tests with Known Variance

Hypothesis Testing

<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

Estimating Population Variance from Sample Variance

2024 ACJC P2 Q8

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.

Population and Random Samples

<p>A coffee shop owner claims that its signature coffee has a mean caffeine content of $120$ mg per cup. To investigate this claim, a random sample of $60$ cups of coffee is selected and the caffeine content, $x$ mg per cup, is summarised as follows.\[\sum x = 6960, \quad \sum x^2 = 822465\]</p>
<ol class="abc-list"><li data-list-item-id="item-a3" data-list-item-path="item-a3"><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance for the caffeine content per cup.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b3" data-list-item-path="item-b3"><div class="list-item-paragraph"><p>Test, at the $5\%$ level of significance, whether the coffee shop owner’s claim is supported by the data. You should state your hypotheses and define any symbols that you use.</p><span class="inline-marks">[4]</span></div></li></ol>
<p>The coffee shop also sells premium coffee, where its caffeine content is normally distributed with population variance $200$ mg${}^2$. The coffee shop owner now claims that its premium coffee has a mean caffeine content less than $120$ mg per cup. Another large random sample of $n$ cups of premium coffee is selected and the sample mean caffeine content per cup is found to be $116.6$ mg. A test is carried out at the $2\frac{1}{2}\%$ level of significance and the result supports the owner’s claim.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c3" data-list-item-path="item-c3"><div class="list-item-paragraph">Find the set of values that&nbsp;$n$ can take.<span class="inline-marks">[4]</span></div></li></ol>

A coffee shop owner claims that its signature coffee has a mean caffeine content of $120$ mg per cup. To investigate this claim, a random sample of $60$ cups of coffee is selected and the caffeine content, $x$ mg per cup, is summarised as follows.\[\sum x = 6960, \quad \sum x^2 = 822465\]

Calculate unbiased estimates of the population mean and variance for the caffeine content per cup.

The coffee shop also sells premium coffee, where its caffeine content is normally distributed with population variance $200$ mg${}^2$. The coffee shop owner now claims that its premium coffee has a mean caffeine content less than $120$ mg per cup. Another large random sample of $n$ cups of premium coffee is selected and the sample mean caffeine content per cup is found to be $116.6$ mg. A test is carried out at the $2\frac{1}{2}\%$ level of significance and the result supports the owner’s claim.

Find the set of values that $n$ can take.

Mean Tests for Normal Populations with Known Variance

2025 TMJC P2 Q7

<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.

Assumptions and Validity of Mean Tests

Central Limit Theorem

Critical Values and Critical Regions

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.

Assumptions for Sampling

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 $n$.

2025 NJC P2 Q10

<p>A food producer claims that the mean mass of a can of beans it produces is $425$ g. Following customer feedback, the production manager wishes to test if the mean mass of a can of beans is indeed $425$ g.</p>
<p>The production manager took a random sample of size $50$ and the mass of each can, in $x$ g, is recorded and the results are shown below:</p>
<p style="text-align: center">\[\sum x = 21209, \sum(x - 424.18)^2 = 522\]</p>
<ol class="abc-list"><li data-list-item-id="item-6a" data-list-item-path="item-6a"><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-6b" data-list-item-path="item-6b"><div class="list-item-paragraph">Find the unbiased estimates for the population mean and variance.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-6c" data-list-item-path="item-6c"><div class="list-item-paragraph">State the hypotheses for the manager's test, defining any parameters you use. Carry out the test at the $5\%$ level of significance, giving your conclusion in the context of the question.<span class="inline-marks">[5]</span></div></li></ol>
<p>The production manager wishes to test whether the mean mass of a can of beans has increased using the alternative manufacturing process. He finds that the mean mass of $55$ randomly chosen cans is $426.5$ g. He carries out a hypothesis test at $10\%$ level of significance.</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-6d" data-list-item-path="item-6d"><div class="list-item-paragraph">Explain, with justification, how the population standard deviation of the mass of a can produced under the alternative process will affect the conclusion made by the production manager.<span class="inline-marks">[3]</span></div></li></ol>

A food producer claims that the mean mass of a can of beans it produces is $425$ g. Following customer feedback, the production manager wishes to test if the mean mass of a can of beans is indeed $425$ g.

The production manager took a random sample of size $50$ and the mass of each can, in $x$ g, is recorded and the results are shown below:

\[\sum x = 21209, \sum(x - 424.18)^2 = 522\]

Find the unbiased estimates for the population mean and variance.

The production manager wishes to test whether the mean mass of a can of beans has increased using the alternative manufacturing process. He finds that the mean mass of $55$ randomly chosen cans is $426.5$ g. He carries out a hypothesis test at $10\%$ level of significance.

Explain, with justification, how the population standard deviation of the mass of a can produced under the alternative process will affect the conclusion made by the production manager.

Effects of Sample Size and Variance on Mean Tests

2025 HCI P2 Q8

<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>Flour is packed in bags of $1.5$ kg, as stated on the packaging, by two separate machines $A$ and $B$ in a factory. During a quality control inspection, the production manager selects a random sample of $40$ bags filled by machine $A$ for checking. The masses, $x$ kg, of these $40$ bags are recorded as follows.</p>
<p style="text-align: center">\[\sum x = 60.32 \quad \sum x^2 = 90.993\]</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">State what it means for a sample to be random in this context.</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 of the mass of the filled bags.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Test, at the $5$% level of significance, whether machine $A$ produces overweight bags. You should state your hypotheses and define any symbols you use.</span><span class="inline-marks">[5]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain in the context of the question, the meaning of '$p$-value'.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">The manager also wants to test whether the mean mass of the bags filled by machine $B$ differs from $1.5$ kg. It is known that the mass of the bags filled by machine $B$ is normally distributed with variance $0.00198$ kg${}^2$. A sample of size $n$ is randomly selected and its mean is found to be $1.489$ kg. Find the range of values that&nbsp;$n$ can take if there is sufficient evidence to conclude that the mean mass of the filled bags differs from $1.5$ kg, at the $5$% level of significance.<span class="inline-marks">[3]</span></div></li></ol>

Flour is packed in bags of $1.5$ kg, as stated on the packaging, by two separate machines $A$ and $B$ in a factory. During a quality control inspection, the production manager selects a random sample of $40$ bags filled by machine $A$ for checking. The masses, $x$ kg, of these $40$ bags are recorded as follows.

\[\sum x = 60.32 \quad \sum x^2 = 90.993\]

Find unbiased estimates of the population mean and variance of the mass of the filled bags.

Test, at the $5$% level of significance, whether machine $A$ produces overweight bags. You should state your hypotheses and define any symbols you use.

Explain in the context of the question, the meaning of '$p$-value'.

Calculating and Interpreting $p$-Values

2025 ASRJC P2 Q12

<p>A machine is designed to make paper with mean thickness $56.8$ micrometres. The thickness, $T$ micrometres, of a piece of paper made by the machine is normally distributed with standard deviation $2.85$ micrometres. A random sample of $300$ pieces of the paper is taken and a test is carried out, at the $5\%$ significance level, to determine whether the machine is producing paper of the desired thickness.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State appropriate hypotheses for the test.<span class="inline-marks">[1]</span></div></li></ol>
<p>The sample mean thickness is denoted by $\bar{t}$ micrometres.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Use an algebraic method to calculate the range of values of $\bar{t}$ for which the null hypothesis would be rejected.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">State the conclusion of the test if&nbsp;$\bar{t}=56.9$.	<span class="inline-marks">[2]</span></div></li></ol>

A machine is designed to make paper with mean thickness $56.8$ micrometres. The thickness, $T$ micrometres, of a piece of paper made by the machine is normally distributed with standard deviation $2.85$ micrometres. A random sample of $300$ pieces of the paper is taken and a test is carried out, at the $5\%$ significance level, to determine whether the machine is producing paper of the desired thickness.

The sample mean thickness is denoted by $\bar{t}$ micrometres.

2013 JJC P2 Q6

<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>Write down the null and alternative hypotheses for each of the following situations:</p>
<p>The average IQ in a school ($\mu $) over a long period of time has been $102$. It is thought that changing the mathematics teacher might influence the average IQ.</p>

Write down the null and alternative hypotheses for each of the following situations:

The average IQ in a school ($\mu $) over a long period of time has been $102$. It is thought that changing the mathematics teacher might influence the average IQ.

Worked Example 1

<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$^{2}$.

Sampling Techniques and Bias

Stratified Sampling

N2016 P2 Q6

<p>A giant panda’s daily diet consists almost entirely of leaves, stems and shoots of various bamboo species. A panda foundation claims that giant pandas consume on average at most $25$ kg of bamboo every day to meet their energy needs. A random sample of $40$ pandas is taken and the mass, $x$ kilograms, of bamboo consumed per day by this group of pandas is summarised below.</p>
<p style="text-align: center">$\sum{\left( x-25 \right)=80.4}$, $\sum{{{\left( x-25 \right)}^{2}}=1893.5}$</p>
<ol class="abc-list"><li data-list-item-id="item-1760866778439-rlxe6anr4" data-list-item-path="item-1760866778439-rlxe6anr4"><div class="list-item-paragraph">Calculate the unbiased estimates of the population mean and variance for the mass of bamboo consumed per day by pandas.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760866779249-vsjw6yz4e" data-list-item-path="item-1760866779249-vsjw6yz4e"><div class="list-item-paragraph">State the hypotheses that can be used to test whether the panda foundation’s claim is valid. By finding the $p$-value, carry out the test at $2\%$ level of significance, giving your conclusion in the context of the question.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760866780007-j5mfs0lwb" data-list-item-path="item-1760866780007-j5mfs0lwb"><div class="list-item-paragraph">State, in context, the meaning of the $p$-value found in part <strong>(b)</strong>.<span class="inline-marks">[1]</span></div></li></ol>
<p>Another random sample of $n$ pandas is selected by zookeepers in a certain country and the sample mean was found to be $26.9$ kg per day. A test was carried out at $2\%$ level of significance.</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-1760866826326-avndpkr6a" data-list-item-path="item-1760866826326-avndpkr6a"><div class="list-item-paragraph">By using the unbiased estimate of the population variance found in <strong>(a)</strong>, find the least possible value of $n$ such that this new sample achieves a different conclusion from that in <strong>(b)</strong>.</div></li></ol>

A giant panda’s daily diet consists almost entirely of leaves, stems and shoots of various bamboo species. A panda foundation claims that giant pandas consume on average at most $25$ kg of bamboo every day to meet their energy needs. A random sample of $40$ pandas is taken and the mass, $x$ kilograms, of bamboo consumed per day by this group of pandas is summarised below.

$\sum{\left( x-25 \right)=80.4}$, $\sum{{{\left( x-25 \right)}^{2}}=1893.5}$

Another random sample of $n$ pandas is selected by zookeepers in a certain country and the sample mean was found to be $26.9$ kg per day. A test was carried out at $2\%$ level of significance.

Unbiased Estimates of Population Parameters

Hypothesis Tests Using Critical Regions and $p$-Values

2024 CJC P2 Q8

<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>

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.

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

Interpreting Unbiased Estimators

2024 DHS P2 Q10

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

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

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

2024 NJC P2 Q9

<p>A mobile application states that the mean time taken to travel from Town A to Park B is $14.5$ minutes. A data analyst would like to test if the mean travel time is more than $14.5$ minutes. The data analyst gathers a random sample of $35$ travel times of various people. The travel times, $x$ minutes, are summarized below.</p>
<p style="text-align: center">$\sum{x}=539$        $\sum{{{x}^{2}}}=8647.4$</p>
<ol class="i-list"><li data-list-item-id="item-1760684654012-ke9mvneea" data-list-item-path="item-1760684654012-ke9mvneea"><div class="list-item-paragraph">Calculate the unbiased estimates of the population mean and variance of the travel time from Town A to Park B.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760684655204-nm2dc8g5g" data-list-item-path="item-1760684655204-nm2dc8g5g"><div class="list-item-paragraph">Test, at the $5\%$ level of significance, whether the mean travel time from Town A to Park B is more than $14.5$ minutes. You should state your hypotheses clearly and define any symbols used.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760684657117-whru1yw9o" data-list-item-path="item-1760684657117-whru1yw9o"><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 data-list-item-id="item-1760684658197-2pho65349" data-list-item-path="item-1760684658197-2pho65349"><div class="list-item-paragraph">Explain, without performing another test, whether the conclusion in part <strong>(ii)</strong> would have changed if instead the data analyst was testing if the mean travel time from Town A to Park B was not $14.5$ minutes.<span class="inline-marks">[1]</span></div></li></ol>

A mobile application states that the mean time taken to travel from Town A to Park B is $14.5$ minutes. A data analyst would like to test if the mean travel time is more than $14.5$ minutes. The data analyst gathers a random sample of $35$ travel times of various people. The travel times, $x$ minutes, are summarized below.

$\sum{x}=539$        $\sum{{{x}^{2}}}=8647.4$

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