A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>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 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

Estimating Parameters from Summarised and Coded Data

2024 ASRJC H1 Q10

Sampling

<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 $n$.

Assumptions and Validity of Mean Tests

Central Limit Theorem

Critical Values and Critical Regions

Population and Random Samples

2025 NJC P2 Q10

<p>A particular streaming service platform claims that the average time a user spends watching content on the platform per visit is more than $15$ minutes. A team of data analysts for the platform wants to verify this claim. They randomly select a sample of $80$ users and record the amount of time each user spends watching content in a single visit. The sample mean of time spent is found to be $16$ minutes. The population standard deviation of time spent is assumed to be $\sigma$ minutes.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Explain why the team should carry out a $1$-tail test.<span class="inline-marks">[1]</span></div></li></ol>
<p>A test at the $5\%$ significance level indicates that the platform's claim is valid.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Explain why the team is able to carry out a hypothesis test without knowing anything about the distribution of the times spent by the users.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the range of values of $\sigma$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Explain, in the context of the question, the meaning of 'at the $5\%$ significance level'.<span class="inline-marks">[1]</span></div></li></ol>

A particular streaming service platform claims that the average time a user spends watching content on the platform per visit is more than $15$ minutes. A team of data analysts for the platform wants to verify this claim. They randomly select a sample of $80$ users and record the amount of time each user spends watching content in a single visit. The sample mean of time spent is found to be $16$ minutes. The population standard deviation of time spent is assumed to be $\sigma$ minutes.

Explain why the team should carry out a $1$-tail test.

A test at the $5\%$ significance level indicates that the platform's claim is valid.

Explain why the team is able to carry out a hypothesis test without knowing anything about the distribution of the times spent by the users.

2025 JPJC P2 Q6

<p>A distributor for health products claimed that his slimming pills could help women reduce their masses. An experiment was carried out to test his claim. $80$ women whose original masses were $60$ kg were randomly chosen and they took the slimming pills for a period of time. Their masses were then taken and recorded. Their mean mass was found to be $59.5$ kg.<br><br>Denote the population standard deviation of the masses by $\sigma $ kg. A statistical test was carried out at $5\%$ level of significance and it was concluded that the slimming pills help to reduce the women’s masses. Find the range of values of $\sigma $.<br></p>

A distributor for health products claimed that his slimming pills could help women reduce their masses. An experiment was carried out to test his claim. $80$ women whose original masses were $60$ kg were randomly chosen and they took the slimming pills for a period of time. Their masses were then taken and recorded. Their mean mass was found to be $59.5$ kg.

Denote the population standard deviation of the masses by $\sigma $ kg. A statistical test was carried out at $5\%$ level of significance and it was concluded that the slimming pills help to reduce the women’s masses. Find the range of values of $\sigma $.


<p>A distributor for health products claimed that his slimming pills could help women achieve their desired mass of $49$ kg. An experiment was carried out to test his claim. $80$ women were randomly chosen and they took the slimming pills for a period of time. The mass, $x$, of each woman was then taken and recorded in kg. it was calculated that&nbsp;$\sum{(x-40)=837}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State, with a reason, whether it is necessary to assume that the masses of the women follows a normal distribution in order 	to test his claim at $2\%$ significance level. </div></li><li><div class="list-item-paragraph">Assuming that the population variance of the masses is $100$ kg$^{2}$, test at $4\%$ significance level whether the distributor was understating his claim. </div></li></ol>

A distributor for health products claimed that his slimming pills could help women achieve their desired mass of $49$ kg. An experiment was carried out to test his claim. $80$ women were randomly chosen and they took the slimming pills for a period of time. The mass, $x$, of each woman was then taken and recorded in kg. it was calculated that $\sum{(x-40)=837}$.

<p>A distributor for musicals claimed that the ticket sales for a particular musical hit an average of $1220$ tickets a day, with variance $3136$. A random sample of $61$ days was taken and the sales, $x$ tickets, for each day were recorded. It was found that</p>
<p style="text-align: center">$\sum(x-1100)=6608$</p>
<ol class="i-list"><li><div class="list-item-paragraph">State, with a reason, whether it is necessary to assume that the ticket sales follow a normal distribution in order to test his claim at $10\%$ significance level.</div></li><li><div class="list-item-paragraph">Test at $10\%$ significance level whether the distributor was overstating his claim.</div></li></ol>

A distributor for musicals claimed that the ticket sales for a particular musical hit an average of $1220$ tickets a day, with variance $3136$. A random sample of $61$ days was taken and the sales, $x$ tickets, for each day were recorded. It was found that

Population with Known Variance And Not Normally Distributed

<p>A bakery claimed that the average number of loaves of bread sold per day is more than $150$, with variance $2500$. A random sample of $50$ days was taken and the sales, $x$ loaves, for each day were recorded. It was found that</p>
<p style="text-align: center">$\sum(x-140)=-120$</p>
<ol class="i-list"><li><div class="list-item-paragraph">State, with a reason, whether it is necessary to assume that the bread sales follow a normal distribution in order to test this claim at the $10\%$ significance level.</div></li><li><div class="list-item-paragraph">Test at the $10\%$ significance level whether the bakery’s claim was valid.</div></li></ol>

A bakery claimed that the average number of loaves of bread sold per day is more than $150$, with variance $2500$. A random sample of $50$ days was taken and the sales, $x$ loaves, for each day were recorded. It was found that

Population with Known Variance And Not Normally Distributed Q1

<p>A bakery claimed that the average number of loaves of bread sold per day is more than $150$, with variance $2500$. A random sample of $50$days was taken and the sales, $x$ loaves, for each day were recorded. It was found that<br></br></p>
<p style="text-align: center">$\sum{\left( x-140 \right)}=-120$<br></br></p>
<ol class="i-list"><li><div class="list-item-paragraph">State, with a reason, whether it is necessary to assume that the bread sales follow a normal distribution in order to test this claim at the $10\%$ significance level.</div></li><li><div class="list-item-paragraph">Test at the $10\%$ significance level whether the bakery’s claim was valid.</div></li></ol>

A bakery claimed that the average number of loaves of bread sold per day is more than $150$, with variance $2500$. A random sample of $50$days was taken and the sales, $x$ loaves, for each day were recorded. It was found that<br></br>

$\sum{\left( x-140 \right)}=-120$<br></br>

Worked Example 4

<p>The number of employees of a company, classified by department and gender, is shown below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 500px;"><colgroup><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"></colgroup><thead><tr style="height: 25px;"><th style="height: 25px; min-height: 25px; text-align: left; --cell-justify: flex-start"></th><th style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">Production</th><th style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">Development</th><th style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">Administration</th><th style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">Finance</th></tr></thead><tbody><tr style="height: 25px;"><td style="height: 25px; min-height: 25px; text-align: left; --cell-justify: flex-start">Male</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$2345$</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$1013$</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$237$</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$344$</td></tr><tr style="height: 25px;"><td style="height: 25px; min-height: 25px; text-align: left; --cell-justify: flex-start">Female</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$867$</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$679$</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$591$</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$523$</td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-paragraph">The directors wish to survey a sample of $100$ of the employees. This sample is to be a stratified sample,based on department and gender.</div><ol class="abc-list"><li><div class="list-item-paragraph">How many males should be in the sample?<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">How many females from the Development department should be in the sample?<span class="inline-marks">[1]</span></div></li></ol></li></ol>
<p>The Managing Director knows that, some years ago, the mean age of employees was $37$ years. He believes that the mean age of employees now is less than $37$ years.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">State why the stratified sample from part <strong>(i)</strong> should not be used for a hypothesis test of the Managing Director's belief.<span class="inline-marks">[1]</span></div></li></ol>
<p>The Company Secretary obtains a suitable sample of $80$ employees in order to carry out a hypothesis test of the Managing Director’s belief that the mean age of the employees now is less than $37$ years. You are given that the population variance of the ages is $140$ years$^{2}$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Write down appropriate hypotheses to test the Managing Director’s belief. You are given that the result of the test, using a $5%$ significance level, is that the Managing Director’s belief should be accepted. Determine the set of possible values of the mean age of the sample of employees.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">You are given instead that the mean age of the sample of employees is $35.2$ years, and that the result of a test at the $\alpha %$ significance level is that the Managing Director’s belief should not be accepted. Find the set of possible values of $\alpha $.<span class="inline-marks">[3]</span></div></li></ol>

The number of employees of a company, classified by department and gender, is shown below.

The Managing Director knows that, some years ago, the mean age of employees was $37$ years. He believes that the mean age of employees now is less than $37$ years.

The Company Secretary obtains a suitable sample of $80$ employees in order to carry out a hypothesis test of the Managing Director’s belief that the mean age of the employees now is less than $37$ years. You are given that the population variance of the ages is $140$ years$^{2}$.

Sampling Techniques and Bias

Assumptions for Sampling

Stratified Sampling

N2016 P2 Q6

<p>The manufacturer of Mola snack biscuits claimed that the mean biscuit content in a packet is $15$g. A sample of $80$ packets is taken and the weight, $x$g, of the content in each packet is measured. The results are summarised by</p>
<p style="text-align: center">$\sum{\left( x-15 \right)=53.6}$ and $\sum{{{\left( x-15 \right)}^{2}}=795.7}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance.</div></li><li><div class="list-item-paragraph">Stating a necessary assumption, test whether the mean weight differs from $15$g at $4%$ level of significance.</div></li><li><div class="list-item-paragraph">If the test is conducted at $\alpha %$ level of significance and the null hypothesis is rejected in favour of the alternative hypothesis based on the values from this sample, determine the set of values of $\alpha $.</div></li><li><div class="list-item-paragraph">Another large random sample of $n$ packets is taken. Assume now that the population standard deviation is $3$g. Find the range of values of the sample mean, in terms of $n$, for which the claim that the mean weight differs from $15$g is supported at $4%$ level of significance.</div></li></ol>

The manufacturer of Mola snack biscuits claimed that the mean biscuit content in a packet is $15$g. A sample of $80$ packets is taken and the weight, $x$g, of the content in each packet is measured. The results are summarised by

$\sum{\left( x-15 \right)=53.6}$ and $\sum{{{\left( x-15 \right)}^{2}}=795.7}$.

Unbiased Estimates of Population Parameters

Hypothesis Tests Using Critical Regions and $p$-Values

2015 TJC P2 Q7 [Modified]

<p>The Electronic Road Pricing (ERP) system is the primary method of regulating traffic in Singapore. ERP rates are determined based on traffic conditions. If the average traffic speed rises above $45$km/h on expressways, ERP charges at that gantry will be reduced. Conversely, ERP rates will be increased if traffic moves slower than the average speed of $45$km/h on expressways.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">The authority reviewed the traffic conditions on a particular expressway by measuring the speeds of $150$ randomly selected cars as they pass a speed camera. The speed, $x$ km/h was recorded. The results are summarised by</div><div class="list-item-paragraph" style="text-align: center">$\sum{\left( x-40 \right)=\mathrm{585}}$,         $\sum{{{\left( x-40 \right)}^{2}}=\mathrm{10998}}$.</div><ol class="i-list"><li><div class="list-item-paragraph">Find the unbiased estimates of the population mean and variance of the speed of a car on the expressway.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Test, at the $5\%$ level of significance, whether the ERP rate needs to be increased for this expressway.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Explain the meaning of ‘$p$-value’ in the context of the question in <strong>(ii)</strong>.</div><p style="text-align: right;">[1]</p></li></ol></li><li><div class="list-item-paragraph">A new speed camera is installed along the expressway and it is now known that the population standard deviation of the speed of a car along this expressway is $7.79$ km/h. A large random sample of n cars is observed and the average speed of $45.9$km/h is recorded. A new test is carried out at $5\%$ level of significance. Find the least value of $n$ that results in the reduction of the ERP rate along this expressway.</div><p style="text-align: right;">[4]</p></li></ol>

The Electronic Road Pricing (ERP) system is the primary method of regulating traffic in Singapore. ERP rates are determined based on traffic conditions. If the average traffic speed rises above $45$km/h on expressways, ERP charges at that gantry will be reduced. Conversely, ERP rates will be increased if traffic moves slower than the average speed of $45$km/h on expressways.

2022 JPJC P2 Q10

<p>A virologist claims that due to the introduction of vaccine, the recovery period from a certain type of flu-like virus is on average $7$ days. However, an analyst who works with such patients on a day-to-day basis believes that the average recovery period quoted by the virologist is understated. He carries out a survey to investigate this by recording the recovery period of $120$ randomly chosen patients. The recovery period of each of these $120$ patients are denoted by $x$ days. The summarised results are given by</p>
<p style="text-align: center">$\sum{\left( x-7 \right)=24}$ and $\sum{{{\left( x-7 \right)}^{2}}=\mathrm{14908}}$</p>
<ol class="i-list"><li><div class="list-item-paragraph">Explain in context, the meaning of a random sample of $120$ patients.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Calculate the unbiased estimates of the population mean and variance of the recovery period of the patients.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Carry out a test at $5\%$ significance level, on the claim that the average recovery period of $7$ days is understated.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Explain if the test in part <strong>(iii)</strong> would still be valid if the analyst had asked for recovery periods from a random sample of $15$ patients.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Suppose instead, the analyst wants to test if the average recovery period differs from $7$ days. The analyst collects data from another random sample of $120$ patients. Given that the population standard deviation is now known to be $1$ day, find the range of values of the sample mean, $\overline{x}$, such that the null hypothesis is rejected at $1\%$ level of significance.</div><p style="text-align: right;">[4]</p></li></ol>

A virologist claims that due to the introduction of vaccine, the recovery period from a certain type of flu-like virus is on average $7$ days. However, an analyst who works with such patients on a day-to-day basis believes that the average recovery period quoted by the virologist is understated. He carries out a survey to investigate this by recording the recovery period of $120$ randomly chosen patients. The recovery period of each of these $120$ patients are denoted by $x$ days. The summarised results are given by

$\sum{\left( x-7 \right)=24}$ and $\sum{{{\left( x-7 \right)}^{2}}=\mathrm{14908}}$

2022 CJC P2 Q12

<p>Researchers at University $S$ claimed that Fungus A, a strain of fungi that is typically found in soil and plants, was able to break down polypropylene, a plastic that is hard to recycle, that had been pre-treated with either UV light or heat by at least $27$ grams per $100$ grams of polypropylene over $90$ days.</p>
<p>Researchers at University $M$ believe that the researchers at University $S$ overstated their claim and decided to perform a hypothesis test on a random sample of $120$ batches of polypropylene, each of mass $100$ grams. The researchers recorded the amount $x$ grams of polypropylene for each batch that Fungus A was able to break down over a $90$-day period. The results are summarised as follows:</p>
<p style="text-align: center">$\sum{\left( x-27 \right)=-81}$, $\sum{{{\left( x-27 \right)}^{2}}=2070.8}$</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Calculate the unbiased estimates of the population mean and variance.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Test, at $5\%$ level of significance, the claim that the population mean mass of polypropylene per $100$ grams that were broken down is at least $27$ grams. You should state your hypotheses and define the symbols you use.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">State, in the context of the question, what you understand by the expression ‘at $5\%$ level of significance’ found in part <strong>(b)</strong>.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Explain if it is necessary for University $M$ to assume a normal distribution for the mass of polypropylene per $100$ grams that Fungus A was able to break down for this test to be valid.</div><p style="text-align: right;">[1]</p></li></ol>
<p>University $S$ responded by experimenting on $50$ batches of polypropylene, each of mass $100$ grams, and recorded the amount $y$ grams of polypropylene for each batch that Fungus A was able to break down over a $90$-day period. The null and alternative hypotheses remain the same as the ones stated by University $M$ in part <strong>(b)</strong>.</p>
<p>Based on the research findings from University $S$, the population standard deviation of the mass of polypropylene per $100$ grams that Fungus A was able to break down is known to be $4$ grams.</p>
<ol class="abc-list" start="5"><li><div class="list-item-paragraph">Find the set of values of sample mean mass, $\bar{y}$ grams, of polypropylene per $100$ grams from University $S$ such that the null hypothesis is not rejected in favour of the alternative hypothesis at $5\%$ level of significance.</div><p style="text-align: right;">[3]</p></li></ol>

Researchers at University $S$ claimed that Fungus A, a strain of fungi that is typically found in soil and plants, was able to break down polypropylene, a plastic that is hard to recycle, that had been pre-treated with either UV light or heat by at least $27$ grams per $100$ grams of polypropylene over $90$ days.

Researchers at University $M$ believe that the researchers at University $S$ overstated their claim and decided to perform a hypothesis test on a random sample of $120$ batches of polypropylene, each of mass $100$ grams. The researchers recorded the amount $x$ grams of polypropylene for each batch that Fungus A was able to break down over a $90$-day period. The results are summarised as follows:

$\sum{\left( x-27 \right)=-81}$, $\sum{{{\left( x-27 \right)}^{2}}=2070.8}$

University $S$ responded by experimenting on $50$ batches of polypropylene, each of mass $100$ grams, and recorded the amount $y$ grams of polypropylene for each batch that Fungus A was able to break down over a $90$-day period. The null and alternative hypotheses remain the same as the ones stated by University $M$ in part **(b)**.

Based on the research findings from University $S$, the population standard deviation of the mass of polypropylene per $100$ grams that Fungus A was able to break down is known to be $4$ grams.

2023 HCI P2 Q11

<p>Yummy Berries Farm produces blueberries and raspberries packed in boxes.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Yummy Berries Farm claims that the mean mass, $x$ grams, of each box of blueberries is no less than $125$ grams. After receiving complaints from customers, the Consumers Association of Singapore (CASE) took a random sample of $50$ boxes of blueberries from Yummy Berries Farm and the mass of each box was recorded. The data obtained are summarised in the table.</div><div class="list-item-image"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760251739564-5ixww3pypg5.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAyNTE3Mzk1NjQtNWl4d3czcHlwZzUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg2ODQ3OCwiZXhwIjoxNzkyODY4Nzc4fQ.XniQ1RdGF999qTRJooyfp0RSOtMxm53lTC_bUEBkLxc" alt="List item image" /></div><ol class="i-list"><li><div class="list-item-paragraph">Find the unbiased estimates of the population mean and variance.</div></li><li><div class="list-item-paragraph">Test, at the $10\%$ level of significance, whether Yummy Berries Farm has overstated its claim.</div><div class="list-item-paragraph">State, giving a reason, whether any assumptions about the masses of boxes of blueberries are needed in order for the test to be valid.</div></li></ol></li><li><div class="list-item-paragraph">The masses of boxes of raspberries, each of $y$ grams, are assumed to have a mean of $170$ grams with standard deviation $15$ grams. CASE took a large random sample of $n$ boxes of raspberries and the mean mass of boxes of raspberries from the sample is found to be $165$ grams. A test is to be carried out at the $5\%$ level of significance to determine if the mean mass of the boxes of raspberries is not $170$ grams.</div><div class="list-item-paragraph">Find the minimum number of boxes of raspberries to be taken for which the result of the test would be to reject the null hypothesis.</div></li></ol>

Yummy Berries Farm produces blueberries and raspberries packed in boxes.

2017 HCI P2 Q11 [Modified]

<p>The production manager of a badminton racket company wishes to check that the mean mass of the badminton rackets produced is $800$g as stated in the design specifications. The population standard deviation of the badminton rackets is $20$g. He carries out a hypothesis test on a large sample of $n$ badminton rackets. The sample mean mass is $807$g and the null hypothesis is rejected at the $2\%$ level of significance.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Use an algebraic method to find the least value of $n$ that will support this conclusion.</div><p style="text-align: right;">[6]</p></li><li><div class="list-item-paragraph">Explain whether there is a need for the production manager to know anything about the population distribution of the masses of the badminton rackets.</div><p style="text-align: right;">[1]</p></li></ol>

The production manager of a badminton racket company wishes to check that the mean mass of the badminton rackets produced is $800$g as stated in the design specifications. The population standard deviation of the badminton rackets is $20$g. He carries out a hypothesis test on a large sample of $n$ badminton rackets. The sample mean mass is $807$g and the null hypothesis is rejected at the $2\%$ level of significance.

2023 JPJC P2 Q6

<p>The manufacturer of a particular type of battery claims that the lifetime is distributed with mean $150$ hours and standard deviation $16.877$ hours. A random sample of $n$ batteries, where&nbsp;$n$ is large, is taken and the lifetime, $t$ hours, of each battery is measured. It is found that&nbsp;$\sum{t}=10\mathrm{ }317$.A test is carried out to determine whether the manufacturer has overstated the mean lifetime of the batteries, and the $p$-value is found to be $0.0975$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Write down appropriate hypotheses for the test, defining any symbols you use. State, in context, the conclusion of the test if the level of significance is $8\%$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Show that&nbsp;$n$ satisfies the equation $\frac{10317}{n}+\frac{21.871}{\sqrt{n}}-150=0$, and solve it numerically.<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">State, with a reason, whether it is necessary to assume that the lifetime of the batteries is normally distributed in order to carry out the calculation in <strong>(ii)</strong>.<span class="inline-marks">[1]</span></div></li></ol>

The manufacturer of a particular type of battery claims that the lifetime is distributed with mean $150$ hours and standard deviation $16.877$ hours. A random sample of $n$ batteries, where $n$ is large, is taken and the lifetime, $t$ hours, of each battery is measured. It is found that $\sum{t}=10\mathrm{ }317$.A test is carried out to determine whether the manufacturer has overstated the mean lifetime of the batteries, and the $p$-value is found to be $0.0975$.

Interpreting Hypothesis Test Conclusions

2013 RI P2 Q8

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

Estimating Population Variance from Sample Variance

2007 IJC P2 Q11

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

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

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

2022 ACJC P2 Q10

<p>The length of string in the balls of string made by a particular manufacturer has mean $\mu $m and variance $27.4$ m$^{2}$.</p>
<p>The manufacturer claims that $\mu =300$. A random sample of $100$ balls of string is taken and the sample sum (total length) is found to be $S$m.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">If $S=29920$, test whether this provides significant evidence, at the $3%$ level, that the manufacturer’s claim overstates the value of $\mu $.</div></li><li><div class="list-item-paragraph">Find the set of values of $S$ (to one decimal place) for which the test concludes that the manufacturer’s claim overstates the value of $\mu $ at the $5%$ significance level.</div></li></ol>
<p>State, giving a reason, whether it is necessary for the length of string in the balls of string to have a normal distribution for the test to be valid.</p>

The length of string in the balls of string made by a particular manufacturer has mean $\mu $m and variance $27.4$ m$^{2}$.

The manufacturer claims that $\mu =300$. A random sample of $100$ balls of string is taken and the sample sum (total length) is found to be $S$m.

State, giving a reason, whether it is necessary for the length of string in the balls of string to have a normal distribution for the test to be valid.

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