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PSLE Foundation Mathematics

<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:

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

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

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.

Estimating Parameters from Summarised and Coded Data

Null and Alternative Hypotheses

Effects of Sample Size and Variance on Mean Tests

Two-Tailed Hypothesis Tests

Large-Sample Mean Tests with Estimated Variance

Interpreting Hypothesis Test Conclusions

Population and Random Samples

2025 HCI P2 Q8

Hypothesis Testing

Sampling

<p>A marine biologist claims that the mean mass of an adult sockeye salmon is at least $4.2$kg. The mass, $x$kg, of each adult sockeye salmon in a random sample of $100$ sockeye salmon is weighed. The results are summarised as follows.</p>
<p style="text-align: center">$\sum{\left( x-4.2 \right)}=-7.84$ $\sum{{{\left( x-4.2 \right)}^{2}}=\mathrm{32}\mathrm{.186}}$</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find unbiased estimates of the population mean and variance of the mass of adult sockeye salmons.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Test, at the $5\%$ level of significance, whether the marine biologist’s claim is valid.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Explain, in the context of the question, the meaning of ‘at the $5\%$ level of significance’.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Explain, without further calculation, whether the conclusion of the test in part <strong>(ii)</strong> will remain the same if the test were conducted with the same data but with only the following changes.</div><ol class="abc-list"><li><div class="list-item-paragraph">The marine biologist claims that the mean mass of an adult sockeye salmon is $4.2$ kg.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">The population variance is known and is smaller than the unbiased estimate of the population variance found in part <strong>(i)</strong>.</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

A marine biologist claims that the mean mass of an adult sockeye salmon is at least $4.2$kg. The mass, $x$kg, of each adult sockeye salmon in a random sample of $100$ sockeye salmon is weighed. The results are summarised as follows.

$\sum{\left( x-4.2 \right)}=-7.84$ $\sum{{{\left( x-4.2 \right)}^{2}}=\mathrm{32}\mathrm{.186}}$

Significance Levels and $p$-Values

One-Tailed Hypothesis Tests

Unbiased Estimates of Population Parameters

Hypothesis Tests Using Critical Regions and $p$-Values

Comparing One-Tailed and Two-Tailed Tests

2022 MI P2 Q10

<p>The amount customers spend is often correlated with the amount of time they spend in stores. According to a national survey of $500,000$ shopping trips, the average time customers spend in a clothing store is $30$ min. The first, second and third quartiles of the time spent, in min, are also given as follows</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760246455757-8tn4lt7dhi2.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAyNDY0NTU3NTctOHRuNGx0N2RoaTIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg2ODM3NiwiZXhwIjoxNzkyODY4Njc2fQ.Ud7FdeQUFtfYNEpD9Epql1gknvFnt5MRB_SjRrcm24k" width="400" class="aligncenter" /></figure>
<ol class="abc-list"><li><div class="list-item-paragraph">Explain why the data suggests that shopping times do not follow a normal distribution.</div><p style="text-align: right;">[1]</p></li></ol>
<p>Jack, a clothing store manager, suspects that customers spend less time in his store than the national average.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Determine whether Jack should carry out a $1$-tail or a $2$-tail test. State the hypotheses for the test and define any symbols you use in this context.</div><p style="text-align: right;">[3]</p></li></ol>
<p>Jack collected a random sample of $50$ shopping times. The results, $x$ (in min), are summarised as follows:</p>
<p style="text-align: center">$\sum{x=1475}$, $\sum{{{x}^{2}}=45029}$</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Test, at the $10\%$ significance level, whether Jack’s suspicion is valid.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">Explain what is meant by ‘at the $10\%$ significance level’ in the context of the question.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Explain why in performing the hypothesis test, there is no need for Jack to be concerned that the shopping times of customers in his store do not follow a normal distribution.</div><p style="text-align: right;">[1]</p></li></ol>
<p>Upon closer inspection of the data, Jack realises that he has made some miscalculations. While there is no mistake in the calculation of $\sum{x}$, the value of $\sum{{{x}^{2}}}$ should be larger than initially calculated.</p>
<ol class="abc-list" start="6"><li><div class="list-item-paragraph">Does this miscalculation affect Jack’s initial conclusion? Explain your answer.</div><p style="text-align: right;">[2]</p></li></ol>

The amount customers spend is often correlated with the amount of time they spend in stores. According to a national survey of $500,000$ shopping trips, the average time customers spend in a clothing store is $30$ min. The first, second and third quartiles of the time spent, in min, are also given as follows

Jack, a clothing store manager, suspects that customers spend less time in his store than the national average.

Jack collected a random sample of $50$ shopping times. The results, $x$ (in min), are summarised as follows:

Upon closer inspection of the data, Jack realises that he has made some miscalculations. While there is no mistake in the calculation of $\sum{x}$, the value of $\sum{{{x}^{2}}}$ should be larger than initially calculated.

Assumptions and Validity of Mean Tests

Central Limit Theorem

Assessing the Suitability of a Normal Model

2023 EJC P2 Q11

Continuous Random Variables

<p>A company sells a certain type of fitness tracker batteries. The company claims that the mean lifetime of the batteries is at least $2000$ hours. The production manager of the company wishes to take a random sample of batteries from the thousands produced in a day at his factory, for quality control purposes.</p>
<ol class="i-list"><li data-list-item-id="item-1759920878341-7zg9yw9xk" data-list-item-path="item-1759920878341-7zg9yw9xk"><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>A random sample of $200$ batteries is taken and it is found that the mean lifetime was $1995$ hours and the standard deviation was $25.5$ hours.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1759920899414-8cd6c7odw" data-list-item-path="item-1759920899414-8cd6c7odw"><div class="list-item-paragraph">Test, at the $1\%$ significance level, whether there is any evidence to doubt the company’s claim.<span class="inline-marks">[6]</span></div></li><li data-list-item-id="item-1759920909596-imv44khzz" data-list-item-path="item-1759920909596-imv44khzz"><div class="list-item-paragraph">State two assumptions or approximations that are involved when carrying out the significance test using the above sample data.<span class="inline-marks">[2]</span></div></li></ol>
<p>The company wishes to test whether the average lifetime of a second type of battery is longer than $2000$ hours. It can be assumed that the lifetime of the second type of battery has a normal distribution with variance ${{\sigma }^{2}}$. $8$ randomly chosen batteries of the second type $2$ are tested and it is found that their lifetimes, in hours, are as follows.</p>
<p style="text-align: center">${2001.5, \qquad 1997.3, \qquad 1989.0, \qquad 2018.7, \qquad 2030.7, \qquad 2023.0, \qquad 2027.5, \qquad 2013.3}$</p>
<p>The company carries out a hypothesis test at the $5\%$ level of significance.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-1759920947141-uqs9wcv8w" data-list-item-path="item-1759920947141-uqs9wcv8w"><div class="list-item-paragraph">Explain, with justification, how the population variance of the lifetimes, ${{\sigma }^{2}}$, will affect the conclusion.<span class="inline-marks">[4]</span></div></li></ol>

A company sells a certain type of fitness tracker batteries. The company claims that the mean lifetime of the batteries is at least $2000$ hours. The production manager of the company wishes to take a random sample of batteries from the thousands produced in a day at his factory, for quality control purposes.

A random sample of $200$ batteries is taken and it is found that the mean lifetime was $1995$ hours and the standard deviation was $25.5$ hours.

The company wishes to test whether the average lifetime of a second type of battery is longer than $2000$ hours. It can be assumed that the lifetime of the second type of battery has a normal distribution with variance ${{\sigma }^{2}}$. $8$ randomly chosen batteries of the second type $2$ are tested and it is found that their lifetimes, in hours, are as follows.

${2001.5, \qquad 1997.3, \qquad 1989.0, \qquad 2018.7, \qquad 2030.7, \qquad 2023.0, \qquad 2027.5, \qquad 2013.3}$

The company carries out a hypothesis test at the $5\%$ level of significance.

Estimating Population Variance from Sample Variance

Mean Tests for Normal Populations with Known Variance

Assumptions for Sampling

2020 VJC P2 Q9

<p>A company uses a machine to produce chocolate bars. The machine is designed to produce chocolate bars with average fat content of $30$g. After using the machine for many years, the manager wishes to test, at $5\%$ level of significance, if the machine still maintains the average fat content in the chocolate bars at $30$g. He selects a sample of $40$ chocolate bars for testing.</p>
<ol class="i-list"><li data-list-item-id="item-1759728699944-cxvfzdaa3" data-list-item-path="item-1759728699944-cxvfzdaa3"><div class="list-item-paragraph">State, giving a reason, whether it is necessary to assume that the fat content of the chocolate bars is normally distributed for the test to be valid.<span class="inline-marks">[1]</span></div></li></ol>
<p>After some years of usage, the machine broke down. While waiting for the new machine to arrive, the company reverted to their traditional mode of producing handmade chocolate bars. The manager suspects that handmade chocolate bars have higher average fat content than those made by the machines. To verify his suspicion, the manager asked his staff to perform a hypothesis test at $5\%$ significance level, on a random sample of $40$ handmade chocolate bars.</p>
<p>The fat content, $x$ grams, of the $40$ handmade chocolate bars are summarised as follows:</p>
<p style="text-align: center">${\sum x = 1220,} \qquad \sum(x - 30.5)^2 = 50$</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1759728747103-uzzhqr9g6" data-list-item-path="item-1759728747103-uzzhqr9g6"><div class="list-item-paragraph">Calculate the unbiased estimates of the population mean and variance of the fat content in handmade chocolate bars.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759728757040-di2ol4sp0" data-list-item-path="item-1759728757040-di2ol4sp0"><div class="list-item-paragraph">The sample mean fat content of the handmade chocolate bars is denoted by $\bar{x}$ grams. State the null and alternate hypotheses and calculate the range of values of $\bar{x}$ for which the null hypothesis would be rejected at $5\%$ level of significance. Hence conclude if the manager’s suspicion is valid at $5\%$ significance level.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1759728757858-jd6yy6v1k" data-list-item-path="item-1759728757858-jd6yy6v1k"><div class="list-item-paragraph"><p>The manager now knows the population variance of the fat content in handmade chocolate bars. It is given that the population variance is smaller than the unbiased estimate of the population variance calculated in part <strong>(ii)</strong>. Without carrying out another hypothesis test, explain with justification, if there will be a change in the manager’s conclusion about his suspicion at $5\%$ significance level.</p><span class="inline-marks">[1]</span></div></li></ol>

A company uses a machine to produce chocolate bars. The machine is designed to produce chocolate bars with average fat content of $30$g. After using the machine for many years, the manager wishes to test, at $5\%$ level of significance, if the machine still maintains the average fat content in the chocolate bars at $30$g. He selects a sample of $40$ chocolate bars for testing.

After some years of usage, the machine broke down. While waiting for the new machine to arrive, the company reverted to their traditional mode of producing handmade chocolate bars. The manager suspects that handmade chocolate bars have higher average fat content than those made by the machines. To verify his suspicion, the manager asked his staff to perform a hypothesis test at $5\%$ significance level, on a random sample of $40$ handmade chocolate bars.

The fat content, $x$ grams, of the $40$ handmade chocolate bars are summarised as follows:

${\sum x = 1220,} \qquad \sum(x - 30.5)^2 = 50$

Critical Values and Critical Regions

2022 HCI P2 Q7

<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 $\bar{x}$ 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.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Explain, in the context of the question, the meaning of ‘at the $5$% level of significance’.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the unbiased estimate of the population variance.</div><p style="text-align: right;">[1]</p></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 $\bar{x}$, correct to $2$ decimal places.</div><p style="text-align: right;">[5]</p></li></ol>
<p>Another random sample of $80$ bottles of hand sanitiser was taken from another hand sanitiser company, Germsfree. The distribution of the volume of the $80$ bottles of Germsfree sanitisers were given in the table below.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759644668304-njrxl5461vf.jpg?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTk2NDQ2NjgzMDQtbmpyeGw1NDYxdmYuanBnIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg2ODM3NiwiZXhwIjoxNzkyODY4Njc2fQ.mNnnMMEWrjJjKCbjaI3nq_KCif4nVVd1sofynFiOqrE" width="400" class="aligncenter" /></figure>
<ol class="i-list" start="5"><li><div class="list-item-paragraph">Find an unbiased estimate for the population mean and variance of volume of a bottle of Germsfree sanitiser.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">The same test was performed on the sample from Germsfree to see if Germsfree had also overstated the mean volume of its hand sanitiser.</div><p style="text-align: right;">[2]</p><div class="list-item-paragraph">It was found that both samples from Cleanser and Germsfree had the same mean. Let ${{p}_{1}}$ and ${{p}_{2}}$ be the $p$-value from the test on the sample from Cleanser and Germsfree respectively. Without doing any further calculation, state with justification an inequality relating ${{p}_{1}}$ and ${{p}_{2}}$.</div><p style="text-align: right;">[1]</p></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 $\bar{x}$ 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.

Another random sample of $80$ bottles of hand sanitiser was taken from another hand sanitiser company, Germsfree. The distribution of the volume of the $80$ bottles of Germsfree sanitisers were given in the table below.

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