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PSLE

PSLE Foundation Mathematics

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

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

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

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

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

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

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

Assumptions and Validity of Mean Tests

One-Tailed Hypothesis Tests

Parameter Bounds from Hypothesis Tests

Estimating Population Variance from Sample Variance

Mean Tests for Normal Populations with Known Variance

Assumptions for Sampling

Null and Alternative Hypotheses

Two-Tailed Hypothesis Tests

Large-Sample Mean Tests with Estimated Variance

Interpreting Hypothesis Test Conclusions

Critical Values and Critical Regions

2025 RI P2 Q9

Hypothesis Testing

Sampling

<p>A food producer claims that the mean mass of a can of beans it produces is $425$ g. Following customer feedback, the production manager wishes to test if the mean mass of a can of beans is indeed $425$ g.</p>
<p>The production manager took a random sample of size $50$ and the mass of each can, in $x$ g, is recorded and the results are shown below:</p>
<p style="text-align: center">\[\sum x = 21209, \sum(x - 424.18)^2 = 522\]</p>
<ol class="abc-list"><li data-list-item-id="item-6a" data-list-item-path="item-6a"><div class="list-item-paragraph">State what it means for a sample to be random in this context.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-6b" data-list-item-path="item-6b"><div class="list-item-paragraph">Find the unbiased estimates for the population mean and variance.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-6c" data-list-item-path="item-6c"><div class="list-item-paragraph">State the hypotheses for the manager's test, defining any parameters you use. Carry out the test at the $5\%$ level of significance, giving your conclusion in the context of the question.<span class="inline-marks">[5]</span></div></li></ol>
<p>The production manager wishes to test whether the mean mass of a can of beans has increased using the alternative manufacturing process. He finds that the mean mass of $55$ randomly chosen cans is $426.5$ g. He carries out a hypothesis test at $10\%$ level of significance.</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-6d" data-list-item-path="item-6d"><div class="list-item-paragraph">Explain, with justification, how the population standard deviation of the mass of a can produced under the alternative process will affect the conclusion made by the production manager.<span class="inline-marks">[3]</span></div></li></ol>

A food producer claims that the mean mass of a can of beans it produces is $425$ g. Following customer feedback, the production manager wishes to test if the mean mass of a can of beans is indeed $425$ g.

The production manager took a random sample of size $50$ and the mass of each can, in $x$ g, is recorded and the results are shown below:

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

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

Find the unbiased estimates for the population mean and variance.

The production manager wishes to test whether the mean mass of a can of beans has increased using the alternative manufacturing process. He finds that the mean mass of $55$ randomly chosen cans is $426.5$ g. He carries out a hypothesis test at $10\%$ level of significance.

Explain, with justification, how the population standard deviation of the mass of a can produced under the alternative process will affect the conclusion made by the production manager.

Estimating Parameters from Summarised and Coded Data

Effects of Sample Size and Variance on Mean Tests

Population and Random Samples

2025 HCI P2 Q8

<p>A machine is designed to make paper with mean thickness $56.8$ micrometres. The thickness, $T$ micrometres, of a piece of paper made by the machine is normally distributed with standard deviation $2.85$ micrometres. A random sample of $300$ pieces of the paper is taken and a test is carried out, at the $5\%$ significance level, to determine whether the machine is producing paper of the desired thickness.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State appropriate hypotheses for the test.<span class="inline-marks">[1]</span></div></li></ol>
<p>The sample mean thickness is denoted by $\bar{t}$ micrometres.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Use an algebraic method to calculate the range of values of $\bar{t}$ for which the null hypothesis would be rejected.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">State the conclusion of the test if&nbsp;$\bar{t}=56.9$.	<span class="inline-marks">[2]</span></div></li></ol>

A machine is designed to make paper with mean thickness $56.8$ micrometres. The thickness, $T$ micrometres, of a piece of paper made by the machine is normally distributed with standard deviation $2.85$ micrometres. A random sample of $300$ pieces of the paper is taken and a test is carried out, at the $5\%$ significance level, to determine whether the machine is producing paper of the desired thickness.

The sample mean thickness is denoted by $\bar{t}$ micrometres.

2013 JJC P2 Q6

<p>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

Unbiased Estimates of Population Parameters

Hypothesis Tests Using Critical Regions and $p$-Values

2024 CJC P2 Q8

<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>

Central Limit Theorem

2024 RI 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.

Significance Levels and $p$-Values

Comparing One-Tailed and Two-Tailed Tests

2021 ASRJC P2 Q11

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

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

Large-Sample Mean Tests with Known Variance

2013 RI P2 Q8

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

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

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

2022 ACJC P2 Q10

<p>The security guard of a particular school claims that the average speed of the cars in the school compound is greater than the speed limit of $25$ km/h. To investigate the security guard’s claim, the traffic police randomly selected $50$ cars and the speed was recorded. The total speed and the standard deviation of the $50$ cars are found to be $1325$ km/h and $7.75$ km/h respectively.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the unbiased estimates of the population mean and variance.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Test at $5$% level of significance whether there is sufficient evidence to support the security guard’s claim.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Without any further test, state the conclusion if instead the traffic police is testing that the average speed of the cars is not $25$ km/h at $20$% level of significance</div><p style="text-align: right;">[2]</p></li></ol>

The security guard of a particular school claims that the average speed of the cars in the school compound is greater than the speed limit of $25$ km/h. To investigate the security guard’s claim, the traffic police randomly selected $50$ cars and the speed was recorded. The total speed and the standard deviation of the $50$ cars are found to be $1325$ km/h and $7.75$ km/h respectively.

2019 CJC P2 Q9 [Modified]

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

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

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

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

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

2022 HCI P2 Q7

<p>The recruitment manager of the private car hire company, I-ber, claims that the mean weekly earnings of a full-time driver is $\$980$. The managing director suspects that the mean weekly earnings is less than $\$980$ and he instructs the recruitment manager to carry out a hypothesis test on a sample of drivers. It is given that the population standard deviation of the weekly earnings is $\$88$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State suitable hypotheses for the test, defining any symbols that you use.<span class="inline-marks">[2]</span></div></li></ol>
<p>The recruitment manager takes a random sample of $10$ drivers. He finds that the weekly earnings in dollars, are as follows.</p>
<p style="text-align: center">$942 \quad 950 \quad 905 \quad 1003 \quad 883 \quad 987 \quad 924 \quad 920 \quad 913 \quad 968$</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the mean weekly earnings of the sample of these $10$ drivers. Carry out the test, at $5\%$ level of significance, for the recruitment manager. Give your conclusion in context and state a necessary assumption for the test to be valid.<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">Find the smallest level of significance at which the test would result in rejection of the null hypothesis, giving your answer correct to $1$ decimal place.<span class="inline-marks">[1]</span></div></li></ol>

The recruitment manager of the private car hire company, I-ber, claims that the mean weekly earnings of a full-time driver is $\$980$. The managing director suspects that the mean weekly earnings is less than $\$980$ and he instructs the recruitment manager to carry out a hypothesis test on a sample of drivers. It is given that the population standard deviation of the weekly earnings is $\$88$.

State suitable hypotheses for the test, defining any symbols that you use.

The recruitment manager takes a random sample of $10$ drivers. He finds that the weekly earnings in dollars, are as follows.

$942 \quad 950 \quad 905 \quad 1003 \quad 883 \quad 987 \quad 924 \quad 920 \quad 913 \quad 968$

Find the mean weekly earnings of the sample of these $10$ drivers. Carry out the test, at $5\%$ level of significance, for the recruitment manager. Give your conclusion in context and state a necessary assumption for the test to be valid.

Find the smallest level of significance at which the test would result in rejection of the null hypothesis, giving your answer correct to $1$ decimal place.

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