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IB AA HL Math

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N(A) E Math

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 company claims that the mean weight of a certain type of packaged product is 500 grams. A quality control manager suspects that the company underestimates the mean weight of the packages. To test his hypothesis, the manager takes a random sample of $40$ packages from the production line. The weights of the packages, $x$ grams, are summarized as follows:\[\sum(x-500) = 211 \text{ and}\hspace{0.5em} \sum(x-500)^2 = 9479\]</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Calculate unbiased estimates of the population mean and variance for the weights of the packages.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">At the $1$% significance level, test whether the manager's suspicion is supported by the data.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain $p$-value in the context of the question.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>A new random sample of $80$ packages is taken from a different batch. The mean weight in this new sample is $k$ grams, and the population variance is known to be $225$ grams$^2$. A hypothesis test at the $2.5$% significance level gives sufficient evidence to conclude that the mean weight of the packages differs from $500$ grams.</p>
<ol class="abc-list" start="4"><li><div class="list-item-content"><span class="list-item-text">Find the range of possible values of $k$.</span><span class="inline-marks">[3]</span></div></li></ol>

A company claims that the mean weight of a certain type of packaged product is 500 grams. A quality control manager suspects that the company underestimates the mean weight of the packages. To test his hypothesis, the manager takes a random sample of $40$ packages from the production line. The weights of the packages, $x$ grams, are summarized as follows:\[\sum(x-500) = 211 \text{ and } \sum(x-500)^2 = 9479\]

Calculate unbiased estimates of the population mean and variance for the weights of the packages.

At the $1$% significance level, test whether the manager's suspicion is supported by the data.

Explain $p$-value in the context of the question.

A new random sample of $80$ packages is taken from a different batch. The mean weight in this new sample is $k$ grams, and the population variance is known to be $225$ grams$^2$. A hypothesis test at the $2.5$% significance level gives sufficient evidence to conclude that the mean weight of the packages differs from $500$ grams.

Find the range of possible values of $k$.

Calculating and Interpreting $p$-Values

2024 ASRJC H1 Q10

<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>The branch manager of a bank claims that the mean waiting time for customers to be served by a bank consultant is 12.4 minutes. To investigate this claim, the branch manager plans to conduct a survey on a sample of customers an hour before the bank's closing time.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Explain why this method of sampling will not give a random sample for the purpose of the investigation.<span class="inline-marks">[1]</span></div></li></ol>
<p>After the survey was conducted, the branch manager realised that the waiting time for each customer before they were served at the counter was of a major concern. He then suspected that the waiting time per customer could be longer than 12.4 minutes. He decided to record the waiting time, $t$ minutes, for 50 customers at the branch.</p>
<p>The results are summarised by</p>
<p style="text-align: center">\[\sum (t - 12.4) = 5.5\ ,\ \sum (t - 12.4)^2 = 10.5\]</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Find the unbiased estimates of the population mean and variance.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Test, at the 5% significance level, whether this set of data supports the branch manager's suspicion<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Explain what is meant by the $p$-value obtained in part <strong>(c)</strong> in the context of the question.<span class="inline-marks">[1]</span></div></li></ol>
<p>The branch manager reviewed some processes for the counter service at his branch. A different random sample of 50 customers was chosen and it was found that the mean waiting time for each customer before they were served at the counter was $k$ minutes.</p>
<p>A significance test of the null hypothesis $\mu = 12.4$ against the alternative hypothesis $\mu \neq 12.4$ is carried out at 4% level of significance. It is assumed that the unbiased estimate of the population variance remains the same as in part <strong>(b)</strong>,</p>
<ol class="abc-list" start="5"><li><div class="list-item-paragraph">Find the set of possible values of the sample mean that would lead to the null hypothesis being rejected.<span class="inline-marks">[4]</span></div></li></ol>

The branch manager of a bank claims that the mean waiting time for customers to be served by a bank consultant is 12.4 minutes. To investigate this claim, the branch manager plans to conduct a survey on a sample of customers an hour before the bank's closing time.

After the survey was conducted, the branch manager realised that the waiting time for each customer before they were served at the counter was of a major concern. He then suspected that the waiting time per customer could be longer than 12.4 minutes. He decided to record the waiting time, $t$ minutes, for 50 customers at the branch.

\[\sum (t - 12.4) = 5.5\ ,\ \sum (t - 12.4)^2 = 10.5\]

The branch manager reviewed some processes for the counter service at his branch. A different random sample of 50 customers was chosen and it was found that the mean waiting time for each customer before they were served at the counter was $k$ minutes.

A significance test of the null hypothesis $\mu = 12.4$ against the alternative hypothesis $\mu \neq 12.4$ is carried out at 4% level of significance. It is assumed that the unbiased estimate of the population variance remains the same as in part **(b)**,

Sampling Techniques and Bias

2024 SAJC H1 Q12

<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 $n$.

Population and Random Samples

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'.

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>A manufacturer invented a new device to be attached to the engine of cars. A test is to be carried out to investigate the effect of the device on the rate of fuel consumption of cars. The device was attached to the engines of $100$ cars of a particular model whose rate of fuel consumption is known to have a mean of $\mu_{0}$ km per litre. The results are summarised as follows:</p>
<p style="text-align: center">$\sum (x-16)=173.4$ and $\sum (x-16)^{2}=323.12$</p>
<p>A statistical test is carried out at $8\%$ level of significance and it is found that the device has no impact on the rate of fuel consumption.</p>
<p>Find the range of values of $\mu_{0}$.</p>

A manufacturer invented a new device to be attached to the engine of cars. A test is to be carried out to investigate the effect of the device on the rate of fuel consumption of cars. The device was attached to the engines of $100$ cars of a particular model whose rate of fuel consumption is known to have a mean of $\mu_{0}$ km per litre. The results are summarised as follows:

$\sum (x-16)=173.4$ and $\sum (x-16)^{2}=323.12$

A statistical test is carried out at $8\%$ level of significance and it is found that the device has no impact on the rate of fuel consumption.

Deducing of Inequalities Using Classical Hypothesis Testing

<p>A sample of $80$ light bulbs is taken from a population with a mean lifetime of $1500$ hours. The lifetime, $x$ hours, of each lightbulb was recorded and the following data are obtained:</p>
<p style="text-align: center">$\sum x = 110200$ and $\sum x^{2} = 180512000$</p>
<p>A statistical test is carried out at an $8\%$ level of significance, and it is found that the population mean lifetime is not $1500$ hours.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State what you understand by the expression at the $8\%$ significance level in the context of the question.</div></li><li><div class="list-item-paragraph">Find the range of the level of significance at which the test will indicate that the population mean lifetime is not $1500$ hours.</div></li></ol>

A sample of $80$ light bulbs is taken from a population with a mean lifetime of $1500$ hours. The lifetime, $x$ hours, of each lightbulb was recorded and the following data are obtained:

$\sum x = 110200$ and $\sum x^{2} = 180512000$

A statistical test is carried out at an $8\%$ level of significance, and it is found that the population mean lifetime is not $1500$ hours.

Meaning of level of significance and its relation to p-value

<p>A farmer knows from experience that the height of apple trees is normally distributed with mean of $2.7$ m and standard deviation $0.7$ m. He buys a new orchard and wants to test whether the average height of apple trees is different. He assumes that the standard deviation of heights is still $0.7$ m. He measures the heights of $45$ trees and finds their average to be $2.3$ m. Test at $10\%$ level of significance, the outcome of the test.</p>

A farmer knows from experience that the height of apple trees is normally distributed with mean of $2.7$ m and standard deviation $0.7$ m. He buys a new orchard and wants to test whether the average height of apple trees is different. He assumes that the standard deviation of heights is still $0.7$ m. He measures the heights of $45$ trees and finds their average to be $2.3$ m. Test at $10\%$ level of significance, the outcome of the test.

Two-tailed Test

<p>A food manufacturer produces thousands of packets of flour each day. The mass of flour in a packet follows a normal distribution with mean $505$ g and standard deviation $0.8$ g.</p>
<p>Following a slight adjustment to the filling machine, the mass, $x$ g, of the flour in $12$ randomly chosen packets is summarised by</p>
<p style="text-align: center">$\sum x = 6063.5$, $\sum x^{2} = 3063841.81$</p>
<p>Assuming that the variance of the distribution is unaltered by the adjustment, test at the $5\%$ significance level whether there is any change in the mean mass of the flour.</p>

A food manufacturer produces thousands of packets of flour each day. The mass of flour in a packet follows a normal distribution with mean $505$ g and standard deviation $0.8$ g.

Following a slight adjustment to the filling machine, the mass, $x$ g, of the flour in $12$ randomly chosen packets is summarised by

$\sum x = 6063.5$, $\sum x^{2} = 3063841.81$

Assuming that the variance of the distribution is unaltered by the adjustment, test at the $5\%$ significance level whether there is any change in the mean mass of the flour.

<p>A beverage company produces thousands of cans of soda each day. The volume of a soda in a can follows a normal distribution with mean $355$ ml and standard deviation $1.2$ ml.</p>
<p>Following the maintenance of the filling machine, there may have been a slight adjustment. To determine if the machine still produces $355$ ml of soda per can, $15$ randomly chosen cans are chosen. The volume, $x$ ml, of the $15$ cans of soda is summarised by</p>
<p style="text-align: center">$\sum x = 5314.5$, $\sum x^{2} = 1892040.25$</p>
<p>Assuming that the variance of the distribution is unaltered by the adjustment, test at the $6\%$ significance level whether there is any change in the mean volume of the soda.</p>

A beverage company produces thousands of cans of soda each day. The volume of a soda in a can follows a normal distribution with mean $355$ ml and standard deviation $1.2$ ml.

Following the maintenance of the filling machine, there may have been a slight adjustment. To determine if the machine still produces $355$ ml of soda per can, $15$ randomly chosen cans are chosen. The volume, $x$ ml, of the $15$ cans of soda is summarised by

$\sum x = 5314.5$, $\sum x^{2} = 1892040.25$

Assuming that the variance of the distribution is unaltered by the adjustment, test at the $6\%$ significance level whether there is any change in the mean volume of the soda.

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