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IB AA HL Math

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N(A) E Math

PSLE

PSLE Foundation Mathematics

<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.

Estimating Parameters from Summarised and Coded Data

One-Tailed Hypothesis Tests

Parameter Bounds from Hypothesis Tests

Mean Tests for Normal Populations with Known Variance

Null and Alternative Hypotheses

Two-Tailed Hypothesis Tests

Large-Sample Mean Tests with Estimated Variance

2025 TMJC P2 Q7

Sampling

Hypothesis Testing

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

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

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

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

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

State an assumption needed to carry out the hypothesis test.

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

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

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

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

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

Find an unbiased estimate of the population variance.

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

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

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

Assumptions and Validity of Mean Tests

Central Limit Theorem

Sampling Techniques and Bias

Assumptions for Sampling

Critical Values and Critical Regions

2025 TJC P2 Q9

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

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

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

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

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

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

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

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

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

2025 SAJC P2 Q11

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

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

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

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

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

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

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

Estimating Population Variance from Sample Variance

Interpreting Hypothesis Test Conclusions

2025 RI P2 Q9

<p>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.

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

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

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

Population and Random Samples

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 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.

<p>A new machine was used to pack coffee powder. A random sample of $5$ packets was taken and the mass of each packet was measured to the nearest gram and recorded:</p>
<p style="text-align: center">$199, 199, 200, 200, 202$</p>
<p>Assuming that the mass of a packet follows a normal distribution with mean $199\mathrm{g}$ and population variance $1.2\mathrm{g}^2$, perform a test at $6\%$ significance level to test whether there is an increase in the mean mass of the packets.</p>

A new machine was used to pack coffee powder. A random sample of $5$ packets was taken and the mass of each packet was measured to the nearest gram and recorded:

Assuming that the mass of a packet follows a normal distribution with mean $199\mathrm{g}$ and population variance $1.2\mathrm{g}^2$, perform a test at $6\%$ significance level to test whether there is an increase in the mean mass of the packets.

Classical Hypothesis and P-value Methods

<p>It is believed that the normal level of testosterone in blood is normally distributed with mean $25$ nmol/l and standard deviation $5.7$ nmol/l. Following a swimming competition, Joe Sueling gives five samples of blood with mean level of testosterone of $32$ nmol/l. Joe Sueling will be disqualified if his testosterone level is higher than the normal level.</p>
<p>Test at $7\%$ level of significance whether Joe Sueling will be disqualified.</p>

It is believed that the normal level of testosterone in blood is normally distributed with mean $25$ nmol/l and standard deviation $5.7$ nmol/l. Following a swimming competition, Joe Sueling gives five samples of blood with mean level of testosterone of $32$ nmol/l. Joe Sueling will be disqualified if his testosterone level is higher than the normal level.

Test at $7\%$ level of significance whether Joe Sueling will be disqualified.

<p>Raju is the star player of his school’s basketball team. He claims that on average he would score $21$ points per game. Raju’s career score is normally distributed with a standard deviation of $7$ points. A sample of $15$ sets of Raju’s games are collated and it is found that the mean sample score is $18.3$ points.</p>
<p>At $10\%$ level of significance, test whether Raju has overstated his claim.</p>

Raju is the star player of his school’s basketball team. He claims that on average he would score $21$ points per game. Raju’s career score is normally distributed with a standard deviation of $7$ points. A sample of $15$ sets of Raju’s games are collated and it is found that the mean sample score is $18.3$ points.

At $10\%$ level of significance, test whether Raju has overstated his claim.

<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.<br></br></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<br></br></p>
<p style="text-align: center">$\sum{x}=5314.5$, $\sum{{{x}^{2}}=1892040.25}$<br></br></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.<br></br>

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<br></br>

$\sum{x}=5314.5$, $\sum{{{x}^{2}}=1892040.25}$<br></br>

Worked Example 3

<p>Raju is the star player of his school’s basketball team. He claims that on average he would score $21$ points per game. Raju’s career score is normally distributed with a standard deviation of $7$ points. A sample of $15$ sets of Raju’s games are collated and it is found that the mean sample score is $18.3$ points.</p>
<p>At $10\%$ level of significance, test whether Raju has overstated his claim.</p>

Worked Example 2

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

Hypothesis Tests Using Critical Regions and $p$-Values

2024 RI P2 Q10

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

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

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

Significance Levels and $p$-Values

2024 NJC P2 Q9

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

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

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

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

Unbiased Estimates of Population Parameters

2023 DHS P2 Q10

<p>The owner of a fruit stall, Mr Lee, bought a large batch of durians from a durian farm.</p>
<p>The weights (in kg) of a sample of six durians from this batch are</p>
<p style="text-align: center">$1.0$,        $1.1$,       $1.3$,       $1.7$,       $a$,       $b$.</p>
<p>It is known that the unbiased estimates for the population mean and variance based on this sample are $1.25$kg and $0.095$kg$^{2}$ respectively.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the values of $a$ and $b$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Assuming that the weight of durians from Robert’s farm follows a normal distribution with mean $1.60$kg and standard deviation $0.3$kg, carry out an appropriate test at the $5%$ level to find out whether there is significant evidence that this batch of durians is not taken from Robert’s farm.</div><p style="text-align: right;">[4]</p></li></ol>
<p>The mean and standard deviation of the weight of papayas from Robert’s farm are $1.2$kg and $0.45$kg respectively. Mr Lee bought a batch of papayas from Robert’s farm. Find the least number of papayas in the batch such that the probability of the average weight exceeding $1.3$kg is less than $0.05$.</p><p style="text-align: right;">[3]</p>

The owner of a fruit stall, Mr Lee, bought a large batch of durians from a durian farm.

The weights (in kg) of a sample of six durians from this batch are

$1.0$,        $1.1$,       $1.3$,       $1.7$,       $a$,       $b$.

It is known that the unbiased estimates for the population mean and variance based on this sample are $1.25$kg and $0.095$kg$^{2}$ respectively.

The mean and standard deviation of the weight of papayas from Robert’s farm are $1.2$kg and $0.45$kg respectively. Mr Lee bought a batch of papayas from Robert’s farm. Find the least number of papayas in the batch such that the probability of the average weight exceeding $1.3$kg is less than $0.05$.

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