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PSLE

PSLE Foundation Mathematics

<p>A giant panda’s daily diet consists almost entirely of leaves, stems and shoots of various bamboo species. A panda foundation claims that giant pandas consume on average at most $25$ kg of bamboo every day to meet their energy needs. A random sample of $40$ pandas is taken and the mass, $x$ kilograms, of bamboo consumed per day by this group of pandas is summarised below.</p>
<p style="text-align: center">$\sum{\left( x-25 \right)=80.4}$, $\sum{{{\left( x-25 \right)}^{2}}=1893.5}$</p>
<ol class="abc-list"><li data-list-item-id="item-1760866778439-rlxe6anr4" data-list-item-path="item-1760866778439-rlxe6anr4"><div class="list-item-paragraph">Calculate the unbiased estimates of the population mean and variance for the mass of bamboo consumed per day by pandas.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760866779249-vsjw6yz4e" data-list-item-path="item-1760866779249-vsjw6yz4e"><div class="list-item-paragraph">State the hypotheses that can be used to test whether the panda foundation’s claim is valid. By finding the $p$-value, carry out the test at $2\%$ 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-1760866780007-j5mfs0lwb" data-list-item-path="item-1760866780007-j5mfs0lwb"><div class="list-item-paragraph">State, in context, the meaning of the $p$-value found in part <strong>(b)</strong>.<span class="inline-marks">[1]</span></div></li></ol>
<p>Another random sample of $n$ pandas is selected by zookeepers in a certain country and the sample mean was found to be $26.9$ kg per day. A test was carried out at $2\%$ level of significance.</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-1760866826326-avndpkr6a" data-list-item-path="item-1760866826326-avndpkr6a"><div class="list-item-paragraph">By using the unbiased estimate of the population variance found in <strong>(a)</strong>, find the least possible value of $n$ such that this new sample achieves a different conclusion from that in <strong>(b)</strong>.</div></li></ol>

A giant panda’s daily diet consists almost entirely of leaves, stems and shoots of various bamboo species. A panda foundation claims that giant pandas consume on average at most $25$ kg of bamboo every day to meet their energy needs. A random sample of $40$ pandas is taken and the mass, $x$ kilograms, of bamboo consumed per day by this group of pandas is summarised below.

$\sum{\left( x-25 \right)=80.4}$, $\sum{{{\left( x-25 \right)}^{2}}=1893.5}$

Another random sample of $n$ pandas is selected by zookeepers in a certain country and the sample mean was found to be $26.9$ kg per day. A test was carried out at $2\%$ level of significance.

Calculating and Interpreting $p$-Values

Estimating Parameters from Summarised and Coded Data

One-Tailed Hypothesis Tests

Unbiased Estimates of Population Parameters

Parameter Bounds from Hypothesis Tests

Hypothesis Tests Using Critical Regions and $p$-Values

Null and Alternative Hypotheses

Large-Sample Mean Tests with Estimated Variance

Interpreting Hypothesis Test Conclusions

2024 CJC P2 Q8

Hypothesis Testing

Sampling

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

Assumptions and Validity of Mean Tests

Central Limit Theorem

Mean Tests for Normal Populations with Known Variance

Two-Tailed Hypothesis Tests

Critical Values and Critical Regions

2024 RI P2 Q10

<p>For quality control, the production manager of a company wishes to take a random sample of a certain type of chocolate bar from his factory. He wants to check that the mean mass of the bars is $52$ grams, as stated on the packets.</p>
<ol class="abc-list"><li data-list-item-id="item-1760850761017-7e5flxaik" data-list-item-path="item-1760850761017-7e5flxaik"><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 masses, $x$ grams, of a random sample of $80$ chocolate bars are summarised as follows.</p>
<p style="text-align: center">${n = 80} \qquad \sum(x - 52) = -37 \qquad \sum(x - 52)^2 = 310.7$</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760850778522-476owwb43" data-list-item-path="item-1760850778522-476owwb43"><div class="list-item-paragraph">Calculate the unbiased estimates of the population mean and variance.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760850780480-bwbw010tn" data-list-item-path="item-1760850780480-bwbw010tn"><div class="list-item-paragraph">What do you understand by the term ‘unbiased estimate’?<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760850781162-b2709b40d" data-list-item-path="item-1760850781162-b2709b40d"><div class="list-item-paragraph">Carry out a suitable hypothesis test, explaining the choice between a $1$-tail test and a $2$-tail test.</div><div class="list-item-paragraph">You should state your hypotheses and define any symbols you use.</div><div class="list-item-paragraph">Referring to the $p$-value for your test, explain what it indicates.<span class="inline-marks">[6]</span></div></li><li data-list-item-id="item-1760850795277-w2prh92ix" data-list-item-path="item-1760850795277-w2prh92ix"><div class="list-item-paragraph">Explain whether there is a need for the manager to know anything about the population distribution of the masses of the chocolate bars.<span class="inline-marks">[2]</span></div></li></ol>

For quality control, the production manager of a company wishes to take a random sample of a certain type of chocolate bar from his factory. He wants to check that the mean mass of the bars is $52$ grams, as stated on the packets.

The masses, $x$ grams, of a random sample of $80$ chocolate bars are summarised as follows.

${n = 80} \qquad \sum(x - 52) = -37 \qquad \sum(x - 52)^2 = 310.7$

Interpreting Unbiased Estimators

Population and Random Samples

2024 DHS P2 Q10

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

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

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

Significance Levels and $p$-Values

2024 NJC P2 Q9

<p>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.

Estimating Population Variance from Sample Variance

2024 JPJC P2 Q10

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

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

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

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

Assumptions for Sampling

2023 DHS 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.

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

Comparing One-Tailed and Two-Tailed Tests

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>Osoi is frequently late for school. His mean travelling time to school is $24.5$ minutes. The Year Head asks Osoi to take an alternative route to school and to record his travelling time, $t$ minutes, from home to school every morning on the alternative route. The results for&nbsp;$72$ randomly chosen mornings when Osoi took the alternative route are summarised as follows:</p>
<p style="text-align: center">${\sum(t - 20) = 215\text{ ;}} \qquad \sum(t - 20)^2 = 3234$</p>
<ol class="i-list"><li data-list-item-id="item-1760684338150-761yco7qa" data-list-item-path="item-1760684338150-761yco7qa"><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance of the travelling time for the alternative route.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760684338713-9l1b7mzt1" data-list-item-path="item-1760684338713-9l1b7mzt1"><div class="list-item-paragraph">Using a $5\%$ level of significance, test whether Osoi’s mean travelling time from home to school has shortened after taking the alternative route to school. You should state your hypotheses and define any symbols that you use.<span class="inline-marks">[5]</span></div></li></ol>
<p>Suppose the Year Head wants to investigate if the alternative route to school has made a difference to Osoi’s mean travelling time in the morning instead of whether it has shortened his mean travelling time in the morning, at the same level of significance.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760684364617-qmlgysq8p" data-list-item-path="item-1760684364617-qmlgysq8p"><div class="list-item-paragraph">Find the critical region for this test.<span class="inline-marks">[3]</span></div></li></ol>

Osoi is frequently late for school. His mean travelling time to school is $24.5$ minutes. The Year Head asks Osoi to take an alternative route to school and to record his travelling time, $t$ minutes, from home to school every morning on the alternative route. The results for $72$ randomly chosen mornings when Osoi took the alternative route are summarised as follows:

${\sum(t - 20) = 215\text{ ;}} \qquad \sum(t - 20)^2 = 3234$

Suppose the Year Head wants to investigate if the alternative route to school has made a difference to Osoi’s mean travelling time in the morning instead of whether it has shortened his mean travelling time in the morning, at the same level of significance.

2022 NJC P2 Q9

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

Large-Sample Mean Tests with Known Variance

2015 TJC P2 Q7 [Modified]

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

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

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

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

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

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

Solving Normal Probability Inequalities

Determining Sample Size from Probability Conditions

Normal Models for Sample Means and Averages

Sample Mean and Its Sampling Distribution

2012 AJC MYCT P2 Q4

Continuous Random Variables

<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>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>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>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.

2021 ASRJC P2 Q11

<p>In a public swimming centre, the time spent by a randomly chosen user in using its facilities is $T$ minutes, is known to be normally distributed. The centre manager claims that its users spend an average of $50$ minutes to use its facilities. To check this claim, time spent by a random sample of $60$ users were recorded. The data recorded has an average of $47$ minutes and a standard deviation of $16.4$ minutes.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find an unbiased estimate of the population variance, giving your answer correct to $2$ decimal places.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Test, at the $5\%$ significance level, whether the centre manager overstated the average time spent.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Another sample of size $n\left( n>30 \right)$ that was collected independently is now used to test, at the $5\%$ significance level, whether the centre manager’s claim is valid. For this sample, the mean time taken is $46$ minutes. If the result of the test using this information and the unbiased estimate of the population variance in part <strong>(i)</strong> is that the null hypothesis is rejected, find the least possible value that $n$ can take.</div><p style="text-align: right;">[4]</p></li></ol>

In a public swimming centre, the time spent by a randomly chosen user in using its facilities is $T$ minutes, is known to be normally distributed. The centre manager claims that its users spend an average of $50$ minutes to use its facilities. To check this claim, time spent by a random sample of $60$ users were recorded. The data recorded has an average of $47$ minutes and a standard deviation of $16.4$ minutes.

2022 ASRJC P2 Q8

<p>Carbon steel is made by adding carbon to iron; this makes the iron stronger, though less flexible. A steel manufacturing company makes carbon steel in the form of round bars. The process is designed to manufacture bars in which the amount of carbon in each bar, by weight, is $1.5\%$. It is known that the percentage of carbon in the steel bars is distributed normally, and that the standard deviation is $0.09\%$.</p>
<p>After comments from customers, the production manager wishes to test, at the $5\%$ level of significance, if the percentage of carbon in the steel bars is, in fact, $1.5\%$. He examines a random sample of $15$ bars to determine the percentage of carbon in each bar.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the critical region for this test.</div><p style="text-align: right;">[4]</p></li></ol>
<p>The company recently launched a new line of flat bars made from mild steel. In mild steel the amount of carbon in each bar, by weight, is $0.25\%$.</p>
<p>Comments from customers suggests that these bars are not sufficiently flexible, which makes the production manager suspect that too much carbon has been added. He decides to perform a hypothesis test on a random sample of $40$ of the new flat bars to find out if this is the case.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Explain why the production manager takes a sample of $40$ flat bars for this test when he only took a sample of $15$ round bars in his earlier test.</div><p style="text-align: right;">[2]</p></li></ol>
<p>The amount of carbon, $x\%$, in a random sample of $40$ flat bars are summarised as follows.</p>
<p style="text-align: center">$n=40$       $\sum{x=10.16}$       $\sum{{{x}^{2}}=2.586342}$</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance for the percentage amount of carbon in the flat bars.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Test, at the $2\frac{1}{2}\%$ level of significance, whether the mean amount of carbon in the flat bars is more than $0.25\%$. You should state your hypotheses and define any symbols that you use.</div><p style="text-align: right;">[5]</p></li></ol>

Carbon steel is made by adding carbon to iron; this makes the iron stronger, though less flexible. A steel manufacturing company makes carbon steel in the form of round bars. The process is designed to manufacture bars in which the amount of carbon in each bar, by weight, is $1.5\%$. It is known that the percentage of carbon in the steel bars is distributed normally, and that the standard deviation is $0.09\%$.

After comments from customers, the production manager wishes to test, at the $5\%$ level of significance, if the percentage of carbon in the steel bars is, in fact, $1.5\%$. He examines a random sample of $15$ bars to determine the percentage of carbon in each bar.

The company recently launched a new line of flat bars made from mild steel. In mild steel the amount of carbon in each bar, by weight, is $0.25\%$.

Comments from customers suggests that these bars are not sufficiently flexible, which makes the production manager suspect that too much carbon has been added. He decides to perform a hypothesis test on a random sample of $40$ of the new flat bars to find out if this is the case.

The amount of carbon, $x\%$, in a random sample of $40$ flat bars are summarised as follows.

$n=40$       $\sum{x=10.16}$       $\sum{{{x}^{2}}=2.586342}$

N2020 P2 Q10

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAyNTE3Mzk1NjQtNWl4d3czcHlwZzUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg2ODQ1NiwiZXhwIjoxNzkyODY4NzU2fQ.zshGCAscpjueumS96LxihbvBB-QX-RJ-7UESZzN_olQ" 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.

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