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PSLE

PSLE Foundation Mathematics

<p>A company claims that the mean weight of a certain type of packaged product is 500 grams. A quality control manager suspects that the company underestimates the mean weight of the packages. To test his hypothesis, the manager takes a random sample of $40$ packages from the production line. The weights of the packages, $x$ grams, are summarized as follows:\[\sum(x-500) = 211 \text{ and}\hspace{0.5em} \sum(x-500)^2 = 9479\]</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Calculate unbiased estimates of the population mean and variance for the weights of the packages.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">At the $1$% significance level, test whether the manager's suspicion is supported by the data.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain $p$-value in the context of the question.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>A new random sample of $80$ packages is taken from a different batch. The mean weight in this new sample is $k$ grams, and the population variance is known to be $225$ grams$^2$. A hypothesis test at the $2.5$% significance level gives sufficient evidence to conclude that the mean weight of the packages differs from $500$ grams.</p>
<ol class="abc-list" start="4"><li><div class="list-item-content"><span class="list-item-text">Find the range of possible values of $k$.</span><span class="inline-marks">[3]</span></div></li></ol>

A company claims that the mean weight of a certain type of packaged product is 500 grams. A quality control manager suspects that the company underestimates the mean weight of the packages. To test his hypothesis, the manager takes a random sample of $40$ packages from the production line. The weights of the packages, $x$ grams, are summarized as follows:\[\sum(x-500) = 211 \text{ and } \sum(x-500)^2 = 9479\]

Calculate unbiased estimates of the population mean and variance for the weights of the packages.

At the $1$% significance level, test whether the manager's suspicion is supported by the data.

Explain $p$-value in the context of the question.

A new random sample of $80$ packages is taken from a different batch. The mean weight in this new sample is $k$ grams, and the population variance is known to be $225$ grams$^2$. A hypothesis test at the $2.5$% significance level gives sufficient evidence to conclude that the mean weight of the packages differs from $500$ grams.

Find the range of possible values of $k$.

Calculating and Interpreting $p$-Values

Estimating Parameters from Summarised and Coded Data

One-Tailed Hypothesis Tests

Parameter Bounds from Hypothesis Tests

Two-Tailed Hypothesis Tests

Large-Sample Mean Tests with Estimated Variance

Large-Sample Mean Tests with Known Variance

2024 ASRJC H1 Q10

Hypothesis Testing

Sampling

<p>The branch manager of a bank claims that the mean waiting time for customers to be served by a bank consultant is 12.4 minutes. To investigate this claim, the branch manager plans to conduct a survey on a sample of customers an hour before the bank's closing time.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Explain why this method of sampling will not give a random sample for the purpose of the investigation.<span class="inline-marks">[1]</span></div></li></ol>
<p>After the survey was conducted, the branch manager realised that the waiting time for each customer before they were served at the counter was of a major concern. He then suspected that the waiting time per customer could be longer than 12.4 minutes. He decided to record the waiting time, $t$ minutes, for 50 customers at the branch.</p>
<p>The results are summarised by</p>
<p style="text-align: center">\[\sum (t - 12.4) = 5.5\ ,\ \sum (t - 12.4)^2 = 10.5\]</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Find the unbiased estimates of the population mean and variance.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Test, at the 5% significance level, whether this set of data supports the branch manager's suspicion<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Explain what is meant by the $p$-value obtained in part <strong>(c)</strong> in the context of the question.<span class="inline-marks">[1]</span></div></li></ol>
<p>The branch manager reviewed some processes for the counter service at his branch. A different random sample of 50 customers was chosen and it was found that the mean waiting time for each customer before they were served at the counter was $k$ minutes.</p>
<p>A significance test of the null hypothesis $\mu = 12.4$ against the alternative hypothesis $\mu \neq 12.4$ is carried out at 4% level of significance. It is assumed that the unbiased estimate of the population variance remains the same as in part <strong>(b)</strong>,</p>
<ol class="abc-list" start="5"><li><div class="list-item-paragraph">Find the set of possible values of the sample mean that would lead to the null hypothesis being rejected.<span class="inline-marks">[4]</span></div></li></ol>

The branch manager of a bank claims that the mean waiting time for customers to be served by a bank consultant is 12.4 minutes. To investigate this claim, the branch manager plans to conduct a survey on a sample of customers an hour before the bank's closing time.

After the survey was conducted, the branch manager realised that the waiting time for each customer before they were served at the counter was of a major concern. He then suspected that the waiting time per customer could be longer than 12.4 minutes. He decided to record the waiting time, $t$ minutes, for 50 customers at the branch.

\[\sum (t - 12.4) = 5.5\ ,\ \sum (t - 12.4)^2 = 10.5\]

The branch manager reviewed some processes for the counter service at his branch. A different random sample of 50 customers was chosen and it was found that the mean waiting time for each customer before they were served at the counter was $k$ minutes.

A significance test of the null hypothesis $\mu = 12.4$ against the alternative hypothesis $\mu \neq 12.4$ is carried out at 4% level of significance. It is assumed that the unbiased estimate of the population variance remains the same as in part **(b)**,

Sampling Techniques and Bias

2024 SAJC H1 Q12

<p>Flour is packed in bags of $1.5$ kg, as stated on the packaging, by two separate machines $A$ and $B$ in a factory. During a quality control inspection, the production manager selects a random sample of $40$ bags filled by machine $A$ for checking. The masses, $x$ kg, of these $40$ bags are recorded as follows.</p>
<p style="text-align: center">\[\sum x = 60.32 \quad \sum x^2 = 90.993\]</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">State what it means for a sample to be random in this context.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find unbiased estimates of the population mean and variance of the mass of the filled bags.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Test, at the $5$% level of significance, whether machine $A$ produces overweight bags. You should state your hypotheses and define any symbols you use.</span><span class="inline-marks">[5]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain in the context of the question, the meaning of '$p$-value'.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">The manager also wants to test whether the mean mass of the bags filled by machine $B$ differs from $1.5$ kg. It is known that the mass of the bags filled by machine $B$ is normally distributed with variance $0.00198$ kg${}^2$. A sample of size $n$ is randomly selected and its mean is found to be $1.489$ kg. Find the range of values that&nbsp;$n$ can take if there is sufficient evidence to conclude that the mean mass of the filled bags differs from $1.5$ kg, at the $5$% level of significance.<span class="inline-marks">[3]</span></div></li></ol>

Flour is packed in bags of $1.5$ kg, as stated on the packaging, by two separate machines $A$ and $B$ in a factory. During a quality control inspection, the production manager selects a random sample of $40$ bags filled by machine $A$ for checking. The masses, $x$ kg, of these $40$ bags are recorded as follows.

\[\sum x = 60.32 \quad \sum x^2 = 90.993\]

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

Find unbiased estimates of the population mean and variance of the mass of the filled bags.

Test, at the $5$% level of significance, whether machine $A$ produces overweight bags. You should state your hypotheses and define any symbols you use.

Explain in the context of the question, the meaning of '$p$-value'.

Mean Tests for Normal Populations with Known Variance

Null and Alternative Hypotheses

Population and Random Samples

2025 ASRJC P2 Q12

<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.

Unbiased Estimates of Population Parameters

Hypothesis Tests Using Critical Regions and $p$-Values

Interpreting Hypothesis Test Conclusions

2024 CJC P2 Q8

<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.

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

Assumptions and Validity of Mean Tests

Interpreting Unbiased Estimators

Central Limit Theorem

2024 DHS P2 Q10

<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.

Significance Levels and $p$-Values

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

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

2022 JPJC P2 Q10

<p>A gaming company claims that gamers spend an average of $m$ hours daily on a new game that they have launched. A random sample of $30$ gamers was surveyed and the time, $x$ hours, spent daily on the game was recorded. The results are summarized by</p>
<p style="text-align: center">$\sum{x}=126$ , $\sum{{{x}^{2}}}=550.1525$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Calculate the unbiased estimates of the population mean and variance of the number of hours a gamer spends daily on the game.</div><p style="text-align: right;">[2]</p></li></ol>
<p>A test is carried out, at the $5\%$ significance level, to determine whether the average time spent daily on the game has been overstated by the company.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">State appropriate hypotheses for the test.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the range of values of $m$ for which the null hypothesis is not rejected.</div><p style="text-align: right;">[3]</p></li></ol>
<p>A second sample of $50$ gamers is surveyed and the time, $y$ hours, spent daily on the game was recorded, and the results are summarized by</p>
<p style="text-align: center">$\sum{y}=215$,$\sum{{{y}^{2}}}=964.19$.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">The two samples are combined into a single sample and a test is carried out at the $5\%$ level of significance level to determine whether the company’s claim is valid. Given that the value of $m$ is $4.5$, find $p$-value of this test.</div><p style="text-align: right;">[4]</p></li></ol>

A gaming company claims that gamers spend an average of $m$ hours daily on a new game that they have launched. A random sample of $30$ gamers was surveyed and the time, $x$ hours, spent daily on the game was recorded. The results are summarized by

$\sum{x}=126$ , $\sum{{{x}^{2}}}=550.1525$.

A test is carried out, at the $5\%$ significance level, to determine whether the average time spent daily on the game has been overstated by the company.

A second sample of $50$ gamers is surveyed and the time, $y$ hours, spent daily on the game was recorded, and the results are summarized by

Combining Samples for Parameter Estimation

2008 MJC MYE P2 Q11

<p>A consumer association is testing the lifetime of a particular type of battery that is claimed to have a lifetime of $150$ hours. A random sample of $70$ batteries of this type is tested and the lifetime, $x$ hours, of each battery is measured. The results are summarized by</p>
<p style="text-align: center">$\sum{x}=10317$ , $\sum{{{x}^{2}}}=1540231$.</p>
<p>The population mean lifetime is denoted by $\mu $ hours. The null hypothesis $\mu =150$ is to be tested against the alternative hypothesis $\mu <150$. Find the $p$-value of the test and state the meaning of this $p$-value in the context of the question.</p>
<p>A second random sample of $50$ batteries of this type is tested and the lifetime, $y$ hours, of each battery is measured, with results summarized by</p>
<p style="text-align: center">$\sum{y}=7331$ , $\sum{{{y}^{2}}}=1100565$.</p>
<p>Combining the two samples into a single sample, carry out a test, at the $10\%$ significance level, of the same null and alternative hypotheses.</p>

A consumer association is testing the lifetime of a particular type of battery that is claimed to have a lifetime of $150$ hours. A random sample of $70$ batteries of this type is tested and the lifetime, $x$ hours, of each battery is measured. The results are summarized by

$\sum{x}=10317$ , $\sum{{{x}^{2}}}=1540231$.

The population mean lifetime is denoted by $\mu $ hours. The null hypothesis $\mu =150$ is to be tested against the alternative hypothesis $\mu <150$. Find the $p$-value of the test and state the meaning of this $p$-value in the context of the question.

A second random sample of $50$ batteries of this type is tested and the lifetime, $y$ hours, of each battery is measured, with results summarized by

$\sum{y}=7331$ , $\sum{{{y}^{2}}}=1100565$.

Combining the two samples into a single sample, carry out a test, at the $10\%$ significance level, of the same null and alternative hypotheses.

2008 H1 Q10

<p>Based on past records, shoppers spend an average of $1.5$ hours at MJ mall per visit. After a revamp and expansion to include more stores, the mall management wishes to conduct a hypothesis test to check if the duration that shoppers spend in the mall per visit has increased. A random sample of $50$ shoppers were surveyed and the duration of time, $x$, in hours per visit is recorded. The sample sum is $82.5$ hours and the sample variance is $0.425$ hours$^{2}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State appropriate hypotheses for the test and find the unbiased estimates of the population mean and variance.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Explain why the mall management is able to carry out a hypothesis test without making any assumptions about the distribution of the duration shoppers spend at MJ mall per visit.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Calculate the $p$-value and state its meaning in context of the question. Hence state the set of values of the level of significance $\alpha \%$ for which the management’s hypothesis is valid.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">The mall management believes that the mean spending per shopper per visit is $\$150$. A test at $5\%$ level of significance found that there is significant evidence that the population mean spending is less than $\$150$. Using only this information, explain if the following statements are necessarily true, necessarily false, or neither necessarily true nor necessarily false.</div><ol class="abc-list"><li><div class="list-item-paragraph">There is significant evidence at the $10\%$ level of significance that the population mean spending is less than $\$150$.</div></li><li><div class="list-item-paragraph">There is significant evidence at the $5\%$ level of significance that the population mean spending differs from $\$150$.</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

Based on past records, shoppers spend an average of $1.5$ hours at MJ mall per visit. After a revamp and expansion to include more stores, the mall management wishes to conduct a hypothesis test to check if the duration that shoppers spend in the mall per visit has increased. A random sample of $50$ shoppers were surveyed and the duration of time, $x$, in hours per visit is recorded. The sample sum is $82.5$ hours and the sample variance is $0.425$ hours$^{2}$.

Comparing One-Tailed and Two-Tailed Tests

2018 MJC P2 Q9

<p>A company produces light bulbs with a brightness rating of $470$ lumens. It is known that the brightness of these light bulbs is normally distributed with a standard deviation of $10$ lumens. A production manager wishes to test whether the mean brightness is less than $470$ lumens. He collects a random sample of $40$ light bulbs and finds that their mean brightness is $466.8$ lumens.</p>
<ol class="i-list"><li data-list-item-id="item-1760087559360-n75m5hbqz" data-list-item-path="item-1760087559360-n75m5hbqz"><div class="list-item-paragraph">Explain what is meant by the term “random sample” in the context of this question.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760087559915-b0kby9n7f" data-list-item-path="item-1760087559915-b0kby9n7f"><div class="list-item-paragraph">Carry out a test at the $5\%$ level of significance for the production manager, clearly stating the $p$-value of this test. Explain what $p$-value means in the context of this question.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-1760087561183-u0t5wfajw" data-list-item-path="item-1760087561183-u0t5wfajw"><div class="list-item-paragraph">Suppose that the production manager had tested if the mean brightness differed from $470$ lumens at the $5\%$ level of significance. Without performing another hypothesis test, determine whether your conclusion in part <strong>(ii)</strong> would be affected.<span class="inline-marks">[2]</span></div></li></ol>
<p>The company decides to produce a new line of ‘lowlight’ bulbs with a brightness rating of $350$ lumens. A random sample of $50$ ‘lowlight’ bulbs is taken and the brightness level, $y$ lumens, are summarised as follows.</p>
<p style="text-align: center">$n=50$          $\sum{\left( y-350 \right)}=-125$           $\sum{{{\left( y-347.5 \right)}^{2}}=4704}$</p>
<ol class="i-list" start="4"><li data-list-item-id="item-1760087609190-fr27w0ef4" data-list-item-path="item-1760087609190-fr27w0ef4"><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance for the brightness of the ‘lowlight’ bulbs.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760087619600-s7kwkdnuv" data-list-item-path="item-1760087619600-s7kwkdnuv"><div class="list-item-paragraph">Given instead that the standard deviation of the brightness of the ‘lowlight’ bulbs is known to be $\sigma $, the company found insufficient evidence from this sample to conclude that the mean brightness was less than $\mathrm{350}$ lumens at the $1\%$ level of significance. Determine the range of values of $\sigma $.<span class="inline-marks">[2]</span></div></li></ol>

A company produces light bulbs with a brightness rating of $470$ lumens. It is known that the brightness of these light bulbs is normally distributed with a standard deviation of $10$ lumens. A production manager wishes to test whether the mean brightness is less than $470$ lumens. He collects a random sample of $40$ light bulbs and finds that their mean brightness is $466.8$ lumens.

The company decides to produce a new line of ‘lowlight’ bulbs with a brightness rating of $350$ lumens. A random sample of $50$ ‘lowlight’ bulbs is taken and the brightness level, $y$ lumens, are summarised as follows.

$n=50$          $\sum{\left( y-350 \right)}=-125$           $\sum{{{\left( y-347.5 \right)}^{2}}=4704}$

2021 EJC P2 Q9

<p>The time taken, $T$ (in minutes), for a $17$-year-old student to complete a 5km run is random vairbale with mean $30$. After a new training programme is introduced for these students, a random sample of $n$ students is taken. The mean time and standard deviation for the sample are found to be $28.9$ minutes and $4.0$ minutes respectively.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the unbiased estimate of the population variance in terms of $n$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Using $n=40$,</div><ol class="i-list"><li><div class="list-item-paragraph">carry out a test at the 10% significance level to determine if the mean time taken has changed. State appropriate hypotheses for the test and define any symbols you use.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">State what it means by the $p$-value in this context.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Give a reason why no assumptions about the population are needed in order for the test to be valid.</div><p style="text-align: right;">[1]</p></li></ol></li><li><div class="list-item-paragraph">The trainer claims instead that the new training programme is able to improve the mean of $T$, $30$ minutes, by at least $5$%. The school wants to test his claim.</div><ol class="i-list"><li><div class="list-item-paragraph">Write down the null and alternative hypothesis.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Using the existing sample, the school carried out a test at $1$% significance level and found that there was sufficient evidence to reject the trainer’s claim. Find the set of values that $n$ can take, stating any necessary assumption(s) needed to carry out the test.</div><p style="text-align: right;">[4]</p></li></ol></li></ol>

The time taken, $T$ (in minutes), for a $17$-year-old student to complete a 5km run is random vairbale with mean $30$. After a new training programme is introduced for these students, a random sample of $n$ students is taken. The mean time and standard deviation for the sample are found to be $28.9$ minutes and $4.0$ minutes respectively.

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