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E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

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IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>A parcel company claims that the population mean time taken by its automated system to sort a parcel is less than $42$ seconds. A one-tailed test is carried out. At the $6\%$ significance level, it is found that there is insufficient evidence that the population mean sorting time is less than $42$ seconds. Using only this information, and giving a reason in each case, state whether each of the following statements is <strong>(i)</strong> necessarily true, <strong>(ii)</strong> necessarily false, or <strong>(iii)</strong> neither necessarily true nor false.</p>
<ol class="abc-list"><li data-list-item-id="q2-statements-item-1" data-list-item-path="q2-statements-item-1"><div class="list-item-paragraph"><p>There is significant evidence at the $2\%$ significance level that the population mean sorting time is less than $42$ seconds.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="q2-statements-item-2" data-list-item-path="q2-statements-item-2"><div class="list-item-paragraph"><p>There is significant evidence at the $10\%$ significance level that the population mean sorting time is less than $42$ seconds.</p><span class="inline-marks">[2]</span></div></li></ol>

A parcel company claims that the population mean time taken by its automated system to sort a parcel is less than $42$ seconds. A one-tailed test is carried out. At the $6\%$ significance level, it is found that there is insufficient evidence that the population mean sorting time is less than $42$ seconds. Using only this information, and giving a reason in each case, state whether each of the following statements is **(i)** necessarily true, **(ii)** necessarily false, or **(iii)** neither necessarily true nor false.

Significance Levels and $p$-Values

Sorting-time conclusions from an inconclusive test

Hypothesis Testing

<p>A cereal producer states that the mean mass of cereal in its packets is $500$ g. A one-tailed test is carried out. At the $4\%$ significance level, it is found that there is significant evidence that the population mean mass exceeds $500$ g. Using only this information, and giving a reason in each case, state whether each of the following statements is (i) necessarily true, (ii) necessarily false, or (iii) neither necessarily true nor false.</p>
<ol class="abc-list"><li data-list-item-id="q1-statements-item-1" data-list-item-path="q1-statements-item-1"><div class="list-item-paragraph"><p>There is significant evidence at the $7\%$ significance level that the population mean mass exceeds $500$ g.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="q1-statements-item-2" data-list-item-path="q1-statements-item-2"><div class="list-item-paragraph"><p>There is significant evidence at the $2\%$ significance level that the population mean mass exceeds $500$ g.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1788345830691-s4mr2sp8p" data-list-item-path="item-1788345830691-s4mr2sp8p"><div class="list-item-paragraph"><p>Suppose instead a <strong>two-tailed test</strong> is carried out at the $4 \%$ significance level. There is significant evidence that the population mean mass is <strong>not equal to 500 g</strong>.</p><span class="inline-marks">[2]</span></div></li></ol>

A cereal producer states that the mean mass of cereal in its packets is $500$ g. A one-tailed test is carried out. At the $4\%$ significance level, it is found that there is significant evidence that the population mean mass exceeds $500$ g. Using only this information, and giving a reason in each case, state whether each of the following statements is (i) necessarily true, (ii) necessarily false, or (iii) neither necessarily true nor false.

Comparing One-Tailed and Two-Tailed Tests

Packet-mass conclusions at different significance levels

<p>A random sample of $90$ batteries, used in a particular model of mobile phone, is tested and the 'standby-time', $x$ hours, is measured. The results are summarised by $\sum x = 3040.8$ and $\sum x^2 = 115773.66$. Test, at the $1\%$ significance level, whether the mean standby-time is less than $36.0$ hours.</p>
<p>State, with a reason, whether in using the above test, it is necessary to assume that the 'standby-time' has a normal distribution.</p>
<p>In a test at the $5\%$ significance level, it is found that there is significant evidence that the population mean talk-time is less than $5$ hours. <strong>Using only this information, and giving a reason in each case</strong>, state whether each of the following statements is (i) necessarily true, (ii) necessarily false, or (iii) neither necessarily true nor false.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">There is significant evidence at the $10\%$ significance level that the population mean talk-time is less than $5$ hours.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">There is significant evidence at the $5\%$ significance level that the population mean talk-time is not $5$ hours.</div></li></ol>

A random sample of $90$ batteries, used in a particular model of mobile phone, is tested and the 'standby-time', $x$ hours, is measured. The results are summarised by $\sum x = 3040.8$ and $\sum x^2 = 115773.66$. Test, at the $1\%$ significance level, whether the mean standby-time is less than $36.0$ hours.

State, with a reason, whether in using the above test, it is necessary to assume that the 'standby-time' has a normal distribution.

In a test at the $5\%$ significance level, it is found that there is significant evidence that the population mean talk-time is less than $5$ hours. **Using only this information, and giving a reason in each case**, state whether each of the following statements is (i) necessarily true, (ii) necessarily false, or (iii) neither necessarily true nor false.

Assumptions and Validity of Mean Tests

Estimating Parameters from Summarised and Coded Data

Central Limit Theorem

One-Tailed Hypothesis Tests

Large-Sample Mean Tests with Estimated Variance

Sampling

<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 $\sum x = 527.1$.

Parameter Bounds from Hypothesis Tests

Null and Alternative Hypotheses

Two-Tailed Hypothesis Tests

Large-Sample Mean Tests with Known Variance

<p>A hand sanitiser company, Cleanser, claims that the mean volume of each bottle of their hand sanitiser is $15$ ml. The consumer association took a random sample of $80$ bottles and measured the volume of each bottle. It was found that the sample mean and the sample variance were $\overline{x}$ and $2$ ml$^2$ respectively. A test was conducted at the $5\%$ level of significance to test if Cleanser has overstated the mean volume of its hand sanitiser.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Explain, in the context of the question, the meaning of a random sample.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Explain, in the context of the question, the meaning of 'at the $5\%$ level of significance'.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Find the unbiased estimate of the population variance.<span class="inline-marks">[1]</span></div></li></ol>
<p>It was found that there was insufficient evidence to conclude that Cleanser had overstated the mean volume of their hand sanitiser.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">By defining any parameters used and stating the hypotheses clearly, find the range of values of $\overline{x}$, correct to $2$ decimal places.<span class="inline-marks">[5]</span></div></li></ol>

A hand sanitiser company, Cleanser, claims that the mean volume of each bottle of their hand sanitiser is $15$ ml. The consumer association took a random sample of $80$ bottles and measured the volume of each bottle. It was found that the sample mean and the sample variance were $\overline{x}$ and $2$ ml$^2$ respectively. A test was conducted at the $5\%$ level of significance to test if Cleanser has overstated the mean volume of its hand sanitiser.

It was found that there was insufficient evidence to conclude that Cleanser had overstated the mean volume of their hand sanitiser.

Estimating Population Variance from Sample Variance

Population and Random Samples

<p>In a factory, salt is packed into bags of $25$ grams by a filling machine.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">The quality control manager selects a random sample of $50$ bags of salt. The masses, $x$ grams, are summarized by\[\sum x = 995.7 \text{ and}\hspace{0.5em} \sum x^2 = 26884.37\]Test, at the $5\%$ level of significance, whether the filling machine produces underweight bags of salt.<span class="inline-marks">[7]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Following a slight adjustment to the filling machine and assuming that the mass of salt in a bag follows a normal distribution, the quality control manager takes a new random sample of $50$ bags to check if the mean mass of salt per bag is $25$ grams. The mean and standard deviation of the sample is found to be $22$ grams and $12.0$ grams respectively. A test at the $\alpha\%$ significance level gives insufficient evidence to conclude that the mean mass of salt in each bag differs from $25$ grams. Find the range of possible values of $\alpha$.</span><span class="inline-marks">[4]</span></div></li></ol>

In a factory, salt is packed into bags of $25$ grams by a filling machine.

Following a slight adjustment to the filling machine and assuming that the mass of salt in a bag follows a normal distribution, the quality control manager takes a new random sample of $50$ bags to check if the mean mass of salt per bag is $25$ grams. The mean and standard deviation of the sample is found to be $22$ grams and $12.0$ grams respectively. A test at the $\alpha\%$ significance level gives insufficient evidence to conclude that the mean mass of salt in each bag differs from $25$ grams. Find the range of possible values of $\alpha$.

2024 TJC H1 Q11

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

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

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

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

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

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

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

$\sum{x=110200}$ and $\sum{{{x}^{2}}}=180512000$<br></br>

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

Worked Example 5

<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>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.

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

Mean Tests for Normal Populations with Known Variance

Hypothesis Tests Using Critical Regions and $p$-Values

Critical Values and Critical Regions

2024 NJC P2 Q9

<p>In a drinks factory, a machine is programmed to dispense $500$ ml of green tea into empty bottles.</p>
<ol class="abc-list"><li data-list-item-id="item-1760849155823-cwfmudd7t" data-list-item-path="item-1760849155823-cwfmudd7t"><div class="list-item-paragraph">After a routine check of the machine, the production manager suspects that the machine is dispensing more green tea than expected. A random sample of $50$ bottles is taken for a hypothesis test at $\alpha \%$ level of significance and the data is as follows:</div><div class="list-item-paragraph">Taking $\alpha =2$, test the production manager’s suspicion and state the meaning of the $p$-value in the context of the question.<span class="inline-marks">[5]</span></div><div class="table-container"><table style="table-layout: fixed; width: 530px;"><colgroup><col style="width: 250px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"></colgroup><thead><tr><th style="text-align: left; --cell-justify: flex-start"><p>Volume of green tea (correct to nearest ml)</p></th><th style="text-align: center; --cell-justify: center"><p>$498$</p></th><th style="text-align: center; --cell-justify: center"><p>$499$</p></th><th style="text-align: center; --cell-justify: center"><p>$500$</p></th><th style="text-align: center; --cell-justify: center"><p>$501$</p></th><th style="text-align: center; --cell-justify: center"><p>$502$</p></th><th style="text-align: center; --cell-justify: center"><p>$503$</p></th><th style="text-align: center; --cell-justify: center"><p>$504$</p></th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start"><p>Number of bottles</p></td><td style="text-align: center; --cell-justify: center"><p>$5$</p></td><td style="text-align: center; --cell-justify: center"><p>$11$</p></td><td style="text-align: center; --cell-justify: center"><p>$10$</p></td><td style="text-align: center; --cell-justify: center"><p>$11$</p></td><td style="text-align: center; --cell-justify: center"><p>$10$</p></td><td style="text-align: center; --cell-justify: center"><p>$1$</p></td><td style="text-align: center; --cell-justify: center"><p>$2$</p></td></tr></tbody></table></div><ol class="i-list"><li data-list-item-id="subitem-1760849179335-gzzo4fifa" data-list-item-path="item-1760849155823-cwfmudd7t/subitem-1760849179335-gzzo4fifa"><div class="list-item-paragraph">Determine the set of values of $\alpha $ for which the production manager’s suspicion is valid.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1760849181660-e5nlmtzad" data-list-item-path="item-1760849155823-cwfmudd7t/subitem-1760849181660-e5nlmtzad"><div class="list-item-paragraph">Explain why the production manager is able to carry out the hypothesis test without knowing anything about the distribution of the volume of green tea dispensed by the machine.<span class="inline-marks">[1]</span></div></li></ol></li><li data-list-item-id="item-1760849182808-rbkv3l3rx" data-list-item-path="item-1760849182808-rbkv3l3rx"><div class="list-item-paragraph">The machine is being recalibrated to dispense $500$ ml of green tea. Another random sample of $50$ bottles of green tea is taken and the mean and standard deviation of this sample are $502$ ml and $k$ ml respectively. A hypothesis test, at the $5\%$ significance level, concluded that the recalibration is done accurately. Find the set of values that&nbsp;$k$ can take.<span class="inline-marks">[5]</span></div></li></ol>

In a drinks factory, a machine is programmed to dispense $500$ ml of green tea into empty bottles.

Calculating and Interpreting $p$-Values

2024 JPJC P2 Q10

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

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

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

Unbiased Estimates of Population Parameters

2022 RVHS P2 Q10

<p>From a school-wide survey done in 2022, it was found that the students in a certain school spend an average of $45.5$ hours per week on social media platforms. In 2023, the school’s Student Well-being team decides to carry out a survey on a group of $50$ randomly chosen students.</p>
<ol class="abc-list"><li data-list-item-id="item-1760684452937-xysm00fag" data-list-item-path="item-1760684452937-xysm00fag"><div class="list-item-paragraph">State what it means for a sample to be random in this context.<span class="inline-marks">[1]</span></div></li></ol>
<p>The time, $x$ hours, spent per week on social media platforms by the $50$ students are summarised below.</p>
<p style="text-align: center">$\sum{\left( x-45.4 \right)}=55$                             ${{\sum{\left( x-45.4 \right)}}^{2}}=\mathrm{1494}\mathrm{.74}$</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760684470202-rrxs6w0ni" data-list-item-path="item-1760684470202-rrxs6w0ni"><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance of the time spent by students per week on social media platforms.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760684470848-563dfjb4y" data-list-item-path="item-1760684470848-563dfjb4y"><div class="list-item-paragraph">State hypotheses that can be used to test if the mean time spent by students per week differs from $45.5$ hours. Work out the critical region in this case, and use it to carry out the test at the $5\%$ level of significance, giving your conclusion in the context of the question.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760684492282-0bdzmhofl" data-list-item-path="item-1760684492282-0bdzmhofl"><div class="list-item-paragraph">Explain why it is not necessary to assume that the time spent by students per week on social media platforms follows a normal distribution when performing the test in part <strong>(c)</strong>.<span class="inline-marks">[1]</span></div></li></ol>
<p>It is now given that the time spent by students per week on social media platforms follows a normal distribution with a variance of $25$ hours$^{2}$. Based on observations, the Head of Student Well-being, Ms Tan, believes that the mean time spent by students per week on social media platforms has in fact increased.</p>
<ol class="abc-list" start="5"><li data-list-item-id="item-1760684510702-pgqmc86s3" data-list-item-path="item-1760684510702-pgqmc86s3"><div class="list-item-paragraph">By carrying out the test at $\alpha %$ level of significance using data collected from $50$ randomly chosen students with a mean of $47$ hours, determine the set of values of $\alpha $ for which Ms Tan’s belief should be accepted.<span class="inline-marks">[4]</span></div></li></ol>

From a school-wide survey done in 2022, it was found that the students in a certain school spend an average of $45.5$ hours per week on social media platforms. In 2023, the school’s Student Well-being team decides to carry out a survey on a group of $50$ randomly chosen students.

The time, $x$ hours, spent per week on social media platforms by the $50$ students are summarised below.

$\sum{\left( x-45.4 \right)}=55$                             ${{\sum{\left( x-45.4 \right)}}^{2}}=\mathrm{1494}\mathrm{.74}$

It is now given that the time spent by students per week on social media platforms follows a normal distribution with a variance of $25$ hours$^{2}$. Based on observations, the Head of Student Well-being, Ms Tan, believes that the mean time spent by students per week on social media platforms has in fact increased.

2023 RI P2 Q8

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

2015 TJC P2 Q7 [Modified]

<p>In a large busy restaurant, the mean time taken for a server to clear a table and set it for the next guest is $4.5$ minutes. In order to improve the quality of service and maintain a clean environment, the restaurant manager introduced a new routine to clear tables. A random sample of $30$ servers is taken and the time taken by each server to clear a table, $x$ minutes, is recorded. The data are summarised as follows:</p>
<p style="text-align: center">$\sum{x=132}$ and $\sum{{{x}^{2}}=\mathrm{583}\mathrm{.96}}$.</p>
<ol class="i-list"><li data-list-item-id="item-1760358318404-6mot9auhx" data-list-item-path="item-1760358318404-6mot9auhx"><div class="list-item-paragraph">Find unbiased estimates of the population mean and variance.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760358320144-siphcm6rd" data-list-item-path="item-1760358320144-siphcm6rd"><div class="list-item-paragraph">Test, at the $10\%$ significance level, whether the mean time taken for a server to clear a table has changed.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-1760358345021-v9ibabikb" data-list-item-path="item-1760358345021-v9ibabikb"><div class="list-item-paragraph">Using the results in part <strong>(ii)</strong>, state the range of values of $\alpha $, where&nbsp;$\alpha \%$ is the level of significance, at which a test would lead the manager to conclude that the mean time has not changed.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760358346011-1kxquuqf3" data-list-item-path="item-1760358346011-1kxquuqf3"><div class="list-item-paragraph">Without carrying out another test, state with reasons, the conclusion if the test at $5\%$ significance level was to determine whether the mean time taken to clear a table is less than $4.5$ minutes.<span class="inline-marks">[2]</span></div></li></ol>
<p>After a few weeks, based on feedback from servers and guests, the manager modifies the routine. Another random sample of $30$ servers is taken and their time taken to clear a table has mean $\bar{x}$ minutes and variance $1.01$ minutes$^{2}$.</p>
<ol class="i-list" start="5"><li data-list-item-id="item-1760358379988-udt6gpuft" data-list-item-path="item-1760358379988-udt6gpuft"><div class="list-item-paragraph">Find the set of values of $\bar{x}$ for which a test at $1\%$ significance level concludes that the mean time taken to clear a table is greater than $4.5$ minutes, giving your answer to $3$ decimal places.<span class="inline-marks">[4]</span></div></li></ol>

In a large busy restaurant, the mean time taken for a server to clear a table and set it for the next guest is $4.5$ minutes. In order to improve the quality of service and maintain a clean environment, the restaurant manager introduced a new routine to clear tables. A random sample of $30$ servers is taken and the time taken by each server to clear a table, $x$ minutes, is recorded. The data are summarised as follows:

$\sum{x=132}$ and $\sum{{{x}^{2}}=\mathrm{583}\mathrm{.96}}$.

After a few weeks, based on feedback from servers and guests, the manager modifies the routine. Another random sample of $30$ servers is taken and their time taken to clear a table has mean $\bar{x}$ minutes and variance $1.01$ minutes$^{2}$.

2018 CJC P2 Q10

<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.

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

Effects of Sample Size and Variance on Mean Tests

2022 MI P2 Q10

<p>An airline which charter various flights within the United States of America has a flight route that travels from New Orleans to Miami. The airline claims that the average flight time is $115$ minutes. Brandon, a regular commuter of this flight route, thinks that his journey takes a shorter time on average.</p>
<ol class="i-list"><li data-list-item-id="item-1760357693932-7bur7gn59" data-list-item-path="item-1760357693932-7bur7gn59"><div class="list-item-paragraph">Explain, whether Brandon should carry out a $1$-tail test or a $2$-tail test. State the hypotheses for the test, defining any symbols you use.<span class="inline-marks">[2]</span></div></li></ol>
<p>Brandon records the flight time (in minutes) of $8$ randomly selected flights as shown below.</p>
<p style="text-align: center">$113.0$      $112.4$     $110.0$     $113.8$     $111.3$     $115.2$     $114.2$     $115.5$</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1760357724584-v20oj7d7l" data-list-item-path="item-1760357724584-v20oj7d7l"><div class="list-item-paragraph">Assuming that the flight time from New Orleans to Miami follows a normal distribution that has a variance of $11.3$ minutes$^{2}$, carry out the test for Brandon at $10\%$ level of significance. Give your conclusion in the context of the question.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760357725349-dhjtaag1r" data-list-item-path="item-1760357725349-dhjtaag1r"><div class="list-item-paragraph">Explain what you understand by the phrase “at $10\%$ level of significance” in the context of the question.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760357725987-2qaas3bcq" data-list-item-path="item-1760357725987-2qaas3bcq"><div class="list-item-paragraph">The same sample of $8$ flights is now used to carry out a test at the $10\%$ significance level, to test whether the mean flight times is different from what the airline claims. Using the answer in <strong>(ii)</strong>, state the conclusion of this test.<span class="inline-marks">[2]</span></div></li></ol>
<p>Another flight route travels from Portland to Los Angeles. Due to bad weather conditions, the flights sometimes take a longer time to arrive at Los Angeles. The airline claims that the mean flight time is $k$ minutes. The operation manager of the airline believes that this number is understated and wants to test this claim. Sixty flights were randomly selected, and the mean flight time of $181$ minutes and standard deviation of $13.6$ minutes were recorded.</p>
<ol class="i-list" start="5"><li data-list-item-id="item-1760357762780-5vgrdu4gn" data-list-item-path="item-1760357762780-5vgrdu4gn"><div class="list-item-paragraph">Find the range of values of $k$ for which the claim by the airline is supported at $4\%$ level of significance. Give your answer to the nearest minute.<span class="inline-marks">[4]</span></div></li></ol>

An airline which charter various flights within the United States of America has a flight route that travels from New Orleans to Miami. The airline claims that the average flight time is $115$ minutes. Brandon, a regular commuter of this flight route, thinks that his journey takes a shorter time on average.

Brandon records the flight time (in minutes) of $8$ randomly selected flights as shown below.

$113.0$      $112.4$     $110.0$     $113.8$     $111.3$     $115.2$     $114.2$     $115.5$

Another flight route travels from Portland to Los Angeles. Due to bad weather conditions, the flights sometimes take a longer time to arrive at Los Angeles. The airline claims that the mean flight time is $k$ minutes. The operation manager of the airline believes that this number is understated and wants to test this claim. Sixty flights were randomly selected, and the mean flight time of $181$ minutes and standard deviation of $13.6$ minutes were recorded.

Interpreting Hypothesis Test Conclusions

2021 ASRJC P2 Q11

<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>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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAyNDY0NTU3NTctOHRuNGx0N2RoaTIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg2ODM5MSwiZXhwIjoxNzkyODY4NjkxfQ.y96-SAgq3-bKP23JgxbAG7_dXbbffV36SoMomrl3_wU" 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.

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