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E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

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IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

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

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

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

Significance Levels and $p$-Values

One-Tailed Hypothesis Tests

Parameter Bounds from Hypothesis Tests

Estimating Population Variance from Sample Variance

Null and Alternative Hypotheses

Large-Sample Mean Tests with Estimated Variance

Population and Random Samples

Hypothesis Testing

Sampling

<p>A family business sells strings which are manually cut into stipulated lengths. A client placed an order for&nbsp;$10\,000$ strings of length $100$ centimetres. The business owner got the workers to cut out the $10\,000$ strings. To ensure quality, a random sample of $40$ strings is chosen from the $10\,000$ strings for measurement and a constant $a$ was subtracted from the lengths in centimetres to produce the following summary data.</p>
<p style="text-align: center">\[\sum (x-a) = 10.316 \quad \sum (x-a)^2 = 41.7\]</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Explain in the context of the question what it means for the sample to be random.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the unbiased estimates of the population mean and variance.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Using the data, a test was conducted at $10\%$ level of significance and found insufficient evidence to conclude that the mean is not $100$ centimetres. State the hypotheses for the test, defining any parameters that you use. State the critical region and use it to find the range of possible values for the sample mean, giving your answer correct to $2$ decimal places.</span><span class="inline-marks">[6]</span></div></li><li><div class="list-item-paragraph">Explain if the business owner needs to make any assumption about the distribution of the length of strings for the calculations in part <strong>(c)</strong> to be valid.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">The $p$-value found from the test in part <strong>(c)</strong> is $0.10286$. Write down the $p$-value if the test was conducted at $6\%$ level of significance on whether the strings are shorter than $100$ centimetres. State the conclusion of this test.<span class="inline-marks">[2]</span></div></li></ol>

A family business sells strings which are manually cut into stipulated lengths. A client placed an order for $10\,000$ strings of length $100$ centimetres. The business owner got the workers to cut out the $10\,000$ strings. To ensure quality, a random sample of $40$ strings is chosen from the $10\,000$ strings for measurement and a constant $a$ was subtracted from the lengths in centimetres to produce the following summary data.

\[\sum (x-a) = 10.316 \quad \sum (x-a)^2 = 41.7\]

Explain in the context of the question what it means for the sample to be random.

Find the unbiased estimates of the population mean and variance.

Using the data, a test was conducted at $10\%$ level of significance and found insufficient evidence to conclude that the mean is not $100$ centimetres. State the hypotheses for the test, defining any parameters that you use. State the critical region and use it to find the range of possible values for the sample mean, giving your answer correct to $2$ decimal places.

Estimating Parameters from Summarised and Coded Data

Central Limit Theorem

2025 RVHS P2 Q11

<p style="text-align: center">---</p>
<p>The production manager of a food container manufacturing company wishes to take a random sample of a certain type of lunch box from the thousands produced one day at his factory, for quality control purposes. He wishes to check that the mean height of the lunch boxes is $12.8$ cm.</p>
<ol class="i-list"><li data-list-item-id="item-q3-i" data-list-item-path="item-q3-i"><div class="list-item-paragraph">State what it means for a sample to be random in this context.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-q3-ii" data-list-item-path="item-q3-ii"><div class="list-item-paragraph">Explain why the manager should take a sample of at least $30$ lunch boxes to carry out a hypothesis test.<span class="inline-marks">[1]</span></div></li></ol>
<p>The heights, $x$ cm, of a random sample of $48$ lunch boxes are summarised as follows:\[\sum(x-13) = -3.6 \quad \text{and} \quad \sum(x-13)^2 = 12.02\]</p>
<ol class="i-list" start="3"><li data-list-item-id="item-q3-iii" data-list-item-path="item-q3-iii"><div class="list-item-paragraph">Calculate the unbiased estimates of the population mean and variance of the height of the lunch boxes.<span class="inline-marks">[2]</span></div></li></ol>
<p>The production manager claims that the mean height of the lunch boxes is $12.8$ cm.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-q3-iv" data-list-item-path="item-q3-iv"><div class="list-item-paragraph">Test his claim at the $10\%$ level of significance, stating the hypotheses clearly.<span class="inline-marks">[4]</span></div></li></ol>
<p>In the hope of keeping the mean height of the lunch boxes not to exceed $12.8$ cm, the production manager decides to replace the production line such that the population variance is reduced to $0.04$ cm$^2$. He then carries out a hypothesis test at $2\%$ level of significance using another random sample of $n$ lunch boxes.</p>
<ol class="i-list" start="5"><li data-list-item-id="item-q3-v" data-list-item-path="item-q3-v"><div class="list-item-paragraph">Assuming that&nbsp;$n$ is sufficiently large, find the critical region for this test in terms of $n$.<span class="inline-marks">[4]</span></div></li></ol>

The production manager of a food container manufacturing company wishes to take a random sample of a certain type of lunch box from the thousands produced one day at his factory, for quality control purposes. He wishes to check that the mean height of the lunch boxes is $12.8$ cm.

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

Explain why the manager should take a sample of at least $30$ lunch boxes to carry out a hypothesis test.

The heights, $x$ cm, of a random sample of $48$ lunch boxes are summarised as follows:\[\sum(x-13) = -3.6 \quad \text{and} \quad \sum(x-13)^2 = 12.02\]

Calculate the unbiased estimates of the population mean and variance of the height of the lunch boxes.

The production manager claims that the mean height of the lunch boxes is $12.8$ cm.

In the hope of keeping the mean height of the lunch boxes not to exceed $12.8$ cm, the production manager decides to replace the production line such that the population variance is reduced to $0.04$ cm$^2$. He then carries out a hypothesis test at $2\%$ level of significance using another random sample of $n$ lunch boxes.

Assuming that $n$ is sufficiently large, find the critical region for this test in terms of $n$.

Assumptions and Validity of Mean Tests

Two-Tailed Hypothesis Tests

Large-Sample Mean Tests with Known Variance

Critical Values and Critical Regions

2025 NJC P2 Q10

<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:

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

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.

Effects of Sample Size and Variance on Mean Tests

Interpreting Hypothesis Test Conclusions

2025 HCI P2 Q8

<p>The basketball club in a college has&nbsp;$5$ centers, $8$ forwards and $7$ guards. With the National School Games approaching, the coach wishes to find out the opinions of members of the club about the training programme. He gives a questionnaire to all the members of the club and receives replies from everyone.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Explain whether the $20$ members form a sample or a population.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>The coach then decides to select teams to play in the matches for National School Games. A basketball team to play in a match consists of $1$ center, $2$ forwards and $2$ guards.</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Explain an advantage for choosing a random sample in each category of members for the match.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">How many different teams can be formed?</span><span class="inline-marks">[1]</span></div></li></ol>
<p>In the club, one particular forward is the classmate of one particular guard. Both classmates are injured and cannot participate in a particular match. The coach decides that one of the remaining guards can play either as a guard or as a forward.</p>
<ol class="abc-list" start="4"><li><div class="list-item-content"><span class="list-item-text">How many different teams can now be formed?</span><span class="inline-marks">[3]</span></div></li></ol>

The basketball club in a college has $5$ centers, $8$ forwards and $7$ guards. With the National School Games approaching, the coach wishes to find out the opinions of members of the club about the training programme. He gives a questionnaire to all the members of the club and receives replies from everyone.

Explain whether the $20$ members form a sample or a population.

The coach then decides to select teams to play in the matches for National School Games. A basketball team to play in a match consists of $1$ center, $2$ forwards and $2$ guards.

Explain an advantage for choosing a random sample in each category of members for the match.

In the club, one particular forward is the classmate of one particular guard. Both classmates are injured and cannot participate in a particular match. The coach decides that one of the remaining guards can play either as a guard or as a forward.

How many different teams can now be formed?

Sampling Techniques and Bias

Selections with Restrictions

2025 CJC P2 Q5

Permutation and Combination

<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.

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

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'.

Calculating and Interpreting $p$-Values

Mean Tests for Normal Populations with Known Variance

2025 ASRJC P2 Q12

<p>A store owner receives a shipment of stationery items, including notebooks, pens, and correction tapes. Historical data indicates that&nbsp;$1\%$ of the notebooks, $2\%$ of the pens, and $4\%$ of the correction tapes are defective. The quality of notebooks, pens and correction tapes is independent of one another. The store owner decides to sell the stationery in $180$ packets, each containing one notebook, two pens and one correction tape. A packet is deemed unsatisfactory if any of the four items is defective.</p>
<ol class="i-list"><li data-list-item-id="item-1761026739916-r8mf46f1p" data-list-item-path="item-1761026739916-r8mf46f1p"><div class="list-item-paragraph">Show that the probability that a randomly selected packet is unsatisfactory is $0.0872$, correct to $3$ significant figures.<span class="inline-marks">[2]</span></div></li></ol>
<p>The number of packets that are unsatisfactory is denoted by $X$. You may assume that&nbsp;$X$ can be modelled by a binomial distribution</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1761026778901-9zeslxr7m" data-list-item-path="item-1761026778901-9zeslxr7m"><div class="list-item-paragraph">Find the probability that there are at least $5$ but less than $10$ packets that are unsatisfactory.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761026779456-k1lne558d" data-list-item-path="item-1761026779456-k1lne558d"><div class="list-item-paragraph">Find the least value of $r$ such that the probability that there are more than $r$ packets that are unsatisfactory is at most $0.12$.<span class="inline-marks">[2]</span></div></li></ol>
<p>Before selling the packets of stationery, he decided to select a sample of $9$ packets to check for unsatisfactory packets.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-1761026803727-iw5hgizpy" data-list-item-path="item-1761026803727-iw5hgizpy"><div class="list-item-paragraph">How should the packets be selected? Give a reason for this method of selection.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761026804701-qwibu9mih" data-list-item-path="item-1761026804701-qwibu9mih"><div class="list-item-paragraph">Find the probability that the ninth packet is the third unsatisfactory packet selected.<span class="inline-marks">[2]</span></div></li></ol>

A store owner receives a shipment of stationery items, including notebooks, pens, and correction tapes. Historical data indicates that $1\%$ of the notebooks, $2\%$ of the pens, and $4\%$ of the correction tapes are defective. The quality of notebooks, pens and correction tapes is independent of one another. The store owner decides to sell the stationery in $180$ packets, each containing one notebook, two pens and one correction tape. A packet is deemed unsatisfactory if any of the four items is defective.

The number of packets that are unsatisfactory is denoted by $X$. You may assume that $X$ can be modelled by a binomial distribution

Before selling the packets of stationery, he decided to select a sample of $9$ packets to check for unsatisfactory packets.

Complementary Events

Binomial Probabilities Using Technology

Probabilities of Successive Events

Binomial Probabilities Using the Probability Mass Function

Waiting-Time and Stopping-Rule Probabilities

2024 JPJC P2 Q8

Probability

Discrete Random Variables (DRV)

<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

Unbiased Estimates of Population Parameters

Hypothesis Tests Using Critical Regions and $p$-Values

2024 DHS P2 Q10

<ol class="abc-list"><li><div class="list-item-paragraph">In a certain university, there are $110$ Co-Curricular Activities (CCAs) clustered into $4$ categories. There are $22$ Arts and Culture CCAs, $16$ Community Service CCAs, $38$ Physical Sports CCAs and $34$ Special Interest CCAs.</div><ol class="i-list"><li><div class="list-item-paragraph">Albert wishes to find out about approaches to training of the Physical Sports CCAs, so he sends a questionnaire to $22$ Physical Sports CCAs. Explain whether these $22$ Physical Sports CCAs form a sample or a population.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Benedict wishes to investigate the level of student engagement in CCAs, but does not want to obtain the detailed information necessary from all $110$ CCAs. Explain how he should carry out his investigation, and why he should do the investigation in this way.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the number of different possible samples of $16$ CCAs, with $4$ CCAs chosen from each category.<span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-paragraph">The probability of events $A$, $B$ and $C$ occurring are given by $\mathrm{P}\left( A \right)=\frac{8}{25}$, $\mathrm{P}\left( B \right)=\frac{61}{100}$ and $\mathrm{P}\left( C \right)=\frac{7}{20}$ respectively. It is also known that&nbsp;$\mathrm{P}\left( A\cap C \right)=\frac{11}{100}$, $\mathrm{P}\left( \left. B \right|C \right)=\frac{13}{35}$ and $\mathrm{P}\left( A\cap B\cap C \right)=\frac{7}{100}$.</div><ol class="i-list"><li><div class="list-item-paragraph">Find $\mathrm{P}\left( B\cap C \right)$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">If events $A$ and $B$ are independent, find $\mathrm{P}\left( A\cap B\cap C' \right)$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">If events $A$ and $B$ are not known to be independent, find the greatest and least possible values of $\mathrm{P}\left( A\cap B\cap C' \right)$.<span class="inline-marks">[4]</span></div></li></ol></li></ol>

Independent Events

Conditional Probability

Probability Bounds and Inequalities

2024 CJC 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>A large company is organised into $4$ departments: Administration, Production, Marketing and Staffing.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">The company owner wishes to find out about the qualifications of the employees in Staffing, so she gives a questionnaire to all $75$ employees in this department. She receives replies from $53$ of these employees. Explain whether these $53$ employees form a population or a sample.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">The company owner also wishes to investigate the views of employees about the refreshment facilities but does not have time to ask every employee. Explain how she should carry out her investigation, and why she should do the investigation this way.<span class="inline-marks">[2]</span></div></li></ol>
<p>The company operates with a system of Team Leaders. There are $7$ Team Leaders in Administration, $6$ in Production, $4$ in Marketing and $3$ in Staffing.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Find the number of ways in which a working party of $8$ Team Leaders can be formed so that there is at least one Team Leader from each of the $4$ departments AND there are more Team Leaders from Administration in the working party from any other single department.<span class="inline-marks">[5]</span></div></li></ol>

A large company is organised into $4$ departments: Administration, Production, Marketing and Staffing.

The company operates with a system of Team Leaders. There are $7$ Team Leaders in Administration, $6$ in Production, $4$ in Marketing and $3$ in Staffing.

N2022 P2 Q7

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

Assumptions for Sampling

2022 CJC P2 Q12

<p>On average, $7.8\%$ of markers produced by a factory are faulty. Each day, the factory’s quality manager picks a random sample of n markers and inspects the number of faulty markers found in the sample. The number of faulty markers found in the sample is the random variable $X$ and whether the markers are faulty are independent of one another.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Explain what is meant by a random sample in this context.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Given that the probability that more than $\left( n-4 \right)$ markers in the random sample are found to be non-faulty is less than $0.3$, find the least possible value of $n$.<span class="inline-marks">[2]</span></div></li></ol>
<p>Assume for the remainder of this question that&nbsp;$n=10$.</p>
<p>The samples in $50$ randomly chosen days are collected and the number of faulty markers in each sample is recorded.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find the probability that exactly $35$ of the samples each contains at least one faulty marker, given that none of the $50$ samples contain more than $2$ faulty markers each.<span class="inline-marks">[4]</span></div></li></ol>

On average, $7.8\%$ of markers produced by a factory are faulty. Each day, the factory’s quality manager picks a random sample of n markers and inspects the number of faulty markers found in the sample. The number of faulty markers found in the sample is the random variable $X$ and whether the markers are faulty are independent of one another.

Assume for the remainder of this question that $n=10$.

The samples in $50$ randomly chosen days are collected and the number of faulty markers in each sample is recorded.

Minimum Number of Binomial Trials

Repeated Binomial Experiments

2022 NJC P2 Q7

<p>A company manufactures a wide variety of components for use in domestic appliances.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">The company introduces a new type of light fitting for refrigerators. Each day a supervisor selects a sample of $100$ of the light fittings for testing.</div><ol class="i-list"><li><div class="list-item-paragraph">How should the light fittings be selected? Give a reason for this method of selection.</div><p style="text-align: right;">[2]</p><div class="list-item-paragraph">The supervisor records the number of faulty light fittings found on each of the $150$ working days.</div><div class="list-item-paragraph">Her results are shown in the table.</div><div class="list-item-image"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760338294891-ydesdmm9oj.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAzMzgyOTQ4OTEteWRlc2RtbTlvai5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTIwNDY1LCJleHAiOjE3OTM1MjA3NjV9.7khgVcvnlgyqQhnRfNkgQhwt4uQ4Ws235pwoqp9sdVM" alt="Sub-item image" /></div></li><li><div class="list-item-paragraph">Use the information in the table to estimate $p$, the probability that a light fitting is faulty.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Assuming that the number of faulty fittings found each day follows the binomial distribution $\mathrm{B}\left( 100,\mathrm{ }p \right)$, find the expected number of days on which $3$ faulty fittings are found in a period of $150$ days.</div><p style="text-align: right;">[2]</p></li></ol></li><li><div class="list-item-paragraph">The company also makes heating elements for electric ovens. A fixed number of randomly chosen heating elements are tested each day and the number found to be faulty is denoted by $X$.</div><ol class="i-list"><li><div class="list-item-paragraph">State, in context, two assumptions needed for $X$ to be well modelled by a binomial distribution.</div><p style="text-align: right;">[2]</p><div class="list-item-paragraph">Assume now that $X$ has the distribution $\mathrm{B}\left( 80,\mathrm{ }0.02 \right)$.</div></li><li><div class="list-item-paragraph">Find the probability that, on a randomly chosen day, the number of elements found to be faulty is between $1$ and $4$ inclusive.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the probability that, in a randomly chosen $5$-day working week, more than $3$ elements are found to be faulty on at least $2$ days.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the probability that, in a randomly chosen $5$-day working week, no more than $8$ faulty elements are found in total.</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

A company manufactures a wide variety of components for use in domestic appliances.

Binomial Models and Assumptions

Determining an Unknown Binomial Success Probability

Mean and Variance of a Binomial Distribution

N2021 P2 Q11

<ol class="abc-list"><li data-list-item-id="item-1760254424641-v3pg7gdel" data-list-item-path="item-1760254424641-v3pg7gdel"><ol class="i-list"><li data-list-item-id="subitem-1760254426673-4bp48robu" data-list-item-path="item-1760254424641-v3pg7gdel/subitem-1760254426673-4bp48robu"><div class="list-item-paragraph">It is known that the probability of a customer using e-payment at a hawker stall is $0.25$. A group of customers is chosen at random, find the probability that the ${{7}^{\mathrm{th}}}$ chosen customer is the ${{5}^{\mathrm{th}}}$ customer using e-payment at the hawker stall.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760254427566-3norv1x3a" data-list-item-path="item-1760254424641-v3pg7gdel/subitem-1760254427566-3norv1x3a"><div class="list-item-paragraph">A sample of $40$ customers were randomly chosen from the hawker stall each day. In a month of $30$ days, find the probability that there is at least $15$ days with at most $10$ customers per sample making e-payment.<span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-1760254431759-an5cbjxw9" data-list-item-path="item-1760254431759-an5cbjxw9"><div class="list-item-paragraph">The probability that a hawker uses the online delivery platform Foodgowhere is $p$. A random sample of $~n$ hawkers is taken and the random variable $X$ denotes the number of hawkers in the sample that uses Foodgowhere.</div><ol class="i-list"><li data-list-item-id="subitem-1760254456096-ajuc8x69g" data-list-item-path="item-1760254431759-an5cbjxw9/subitem-1760254456096-ajuc8x69g"><div class="list-item-paragraph">Explain what is meant by a random sample in this context.<span class="inline-marks">[1]</span></div><div class="list-item-paragraph">Assuming that&nbsp;$X$ follows a binomial distribution.</div></li><li data-list-item-id="subitem-1760254456613-hlb5mic08" data-list-item-path="item-1760254431759-an5cbjxw9/subitem-1760254456613-hlb5mic08"><div class="list-item-paragraph">It is given that&nbsp;$\mathrm{P}\left( X\le 1 \right)=\mathrm{0}\mathrm{.05303}$ and the expected number of hawkers using Foodgowhere is $3.96$. </div><div class="list-item-paragraph">Write down two equations satisfied by $p$ and $~n$. Hence find the value of $p$ and $~n$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760254457664-ksry1c6x4" data-list-item-path="item-1760254431759-an5cbjxw9/subitem-1760254457664-ksry1c6x4"><div class="list-item-paragraph">Given that&nbsp;$n=15$, find the set of values of $p$ so that that the most likely number of hawkers in the sample who uses Foodgowehere is $5$.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

Mode of a Binomial Distribution

2022 ACJC P2 Q11

<p>The manufacturer of a particular toy drone claims that its average flying time (AFT) is $12$ minutes. The quality control manager suspects that the AFT is different from what the manufacturer has claimed and decides to carry out a hypothesis test on a sample of these drones.</p>
<p>The flight durations, $x$ minutes, of a random sample of $100$ drones are summarised as follows:</p>
<p style="text-align: center">$\sum{x}=1171$ $\sum{{{x}^{2}}}=14023$</p>
<ol class="abc-list"><li><div class="list-item-paragraph">State what it means for a sample to be random in this context.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance of the AFT of the toy drones.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Test, at the $5\%$ level of significance, the manufacturer’s claim that the AFT of toy drones is $12$ minutes. You should state your hypotheses and define any symbols you use.</div><p style="text-align: right;">[5]</p></li></ol>
<p>Over a long period of time, it is established that the AFT of the toy drone is normally distributed with mean and standard deviation of $12$ minutes and $2$ minutes respectively. However, changes in the manufacturing process due to an upgrade of the drone’s rechargeable battery, is expected to improve the AFT of the drone. To verify this, a random sample of $n$ drones is taken with sample mean AFT of $m$. If $m$ is greater than some critical value, it will be accepted that the AFT of the toy drone has in fact increased.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find in terms of $n$, the critical value for a $5\%$ significance level.</div><p style="text-align: right;">[3]</p></li></ol>

The manufacturer of a particular toy drone claims that its average flying time (AFT) is $12$ minutes. The quality control manager suspects that the AFT is different from what the manufacturer has claimed and decides to carry out a hypothesis test on a sample of these drones.

The flight durations, $x$ minutes, of a random sample of $100$ drones are summarised as follows:

Over a long period of time, it is established that the AFT of the toy drone is normally distributed with mean and standard deviation of $12$ minutes and $2$ minutes respectively. However, changes in the manufacturing process due to an upgrade of the drone’s rechargeable battery, is expected to improve the AFT of the drone. To verify this, a random sample of $n$ drones is taken with sample mean AFT of $m$. If $m$ is greater than some critical value, it will be accepted that the AFT of the toy drone has in fact increased.

2023 NYJC P2 Q9

<p>It is found that certain kinds of meat lose weight as a result of being cooked. A restaurant chef is prepared to accept up to $10\%$ loss but suspects that the recent consignments have a higher percentage weight loss. She decides to carry out a hypothesis test on a random sample of steaks.</p>
<ol class="i-list"><li data-list-item-id="item-1760242787976-hvdi0d192" data-list-item-path="item-1760242787976-hvdi0d192"><div class="list-item-paragraph">Explain the meaning of ‘a random sample’ in the context of the question and why the chef should sample a large number of steaks.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760242788554-bwp5oqbne" data-list-item-path="item-1760242788554-bwp5oqbne"><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 chef takes a random sample of $40$ steaks, and calculate the percentage weight loss of each steak, $x$ (in percent). The mean and variance of the percentage weight loss of the $40$ steaks are $10.48\%$ with variance $3.37\%$ respectively.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760242816244-54660qzba" data-list-item-path="item-1760242816244-54660qzba"><div class="list-item-paragraph">Test, at the $5\%$ significance level, to determine whether the sample supports the chef’s suspicion.<span class="inline-marks">[3]</span></div></li></ol>
<p>The chef carries out another test using $100$ readings obtained from chefs of other restaurants. The percentage weight loss of each steak, $y$ (in percent), is summarised by:</p>
<p style="text-align: center">$\sum{y=\mathrm{1055}\mathrm{.2}}$, $\sum{{{y}^{2}}=k}$.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-1760242854472-ui9ke0ika" data-list-item-path="item-1760242854472-ui9ke0ika"><div class="list-item-paragraph">Find the range of values of $k$ such that the chef’s suspicion is not valid at the $5\%$ level of significance, giving your answer correct to $2$ decimal places.<span class="inline-marks">[5]</span></div></li></ol>

It is found that certain kinds of meat lose weight as a result of being cooked. A restaurant chef is prepared to accept up to $10\%$ loss but suspects that the recent consignments have a higher percentage weight loss. She decides to carry out a hypothesis test on a random sample of steaks.

The chef takes a random sample of $40$ steaks, and calculate the percentage weight loss of each steak, $x$ (in percent). The mean and variance of the percentage weight loss of the $40$ steaks are $10.48\%$ with variance $3.37\%$ respectively.

The chef carries out another test using $100$ readings obtained from chefs of other restaurants. The percentage weight loss of each steak, $y$ (in percent), is summarised by:

$\sum{y=\mathrm{1055}\mathrm{.2}}$, $\sum{{{y}^{2}}=k}$.

2020 TJC P2 Q11

<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.

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

Comparing One-Tailed and Two-Tailed Tests

2021 EJC P2 Q9

<p>The Particle Filtration Efficiency (PFE) of a mask is a measure of how well a mask filters airborne particles such as pollen or dust. A mask with higher PFE is deemed to be of better efficiency as it filters more particles. A mask with a PFE of $95\%$ would have met the requirement for surgical masks.</p>
<p>A factory manufactures Brand BEY surgical masks that is known to have expected PFE of ${95.8\%}$. During a routine check of the manufacturing process, the quality control manager suspects that the efficiency of the Brand BEY surgical masks produced is compromised such that the mean PFE is reduced. The PFE, ${x\%}$, of a random sample of ${50}$ masks is taken and the summarised results are as follows.</p>
<p style="text-align: center">${\sum(x - 90) = 289} \qquad \sum(x - 90)^2 = 1670.56$</p>
<p></p>
<ol class="i-list"><li data-list-item-id="item-1760082550839-x8oqc5mab" data-list-item-path="item-1760082550839-x8oqc5mab"><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 manager carries out a hypothesis test at $1\%$ level of significance.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1760082573203-tz93hopff" data-list-item-path="item-1760082573203-tz93hopff"><div class="list-item-paragraph">Explain whether there is a need for the manager to make any assumption about the population distribution of the PFE of the masks.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760082574103-7zfvyn7id" data-list-item-path="item-1760082574103-7zfvyn7id"><div class="list-item-paragraph">State the appropriate hypotheses, defining any symbols that you use. Test whether the manager’s suspicion is justified at $1\%$ level of significance.<span class="inline-marks">[5]</span></div></li></ol>
<p>The manager discovered an error in the data collection process of the sample and discarded $10$ out of the $50$ readings. From the remaining random sample of $40$ masks, the manager found that the mean PFE is $95.5\%$ and the standard deviation is $k\%$. Given that the manager concludes that there is insufficient evidence to justify his suspicion at $1\%$ level of significance, find the range of possible values of $k$ used in calculating the test statistic.<span class="inline-marks">[3]</span></p>

The Particle Filtration Efficiency (PFE) of a mask is a measure of how well a mask filters airborne particles such as pollen or dust. A mask with higher PFE is deemed to be of better efficiency as it filters more particles. A mask with a PFE of $95\%$ would have met the requirement for surgical masks.

A factory manufactures Brand BEY surgical masks that is known to have expected PFE of ${95.8\%}$. During a routine check of the manufacturing process, the quality control manager suspects that the efficiency of the Brand BEY surgical masks produced is compromised such that the mean PFE is reduced. The PFE, ${x\%}$, of a random sample of ${50}$ masks is taken and the summarised results are as follows.

${\sum(x - 90) = 289} \qquad \sum(x - 90)^2 = 1670.56$

The manager carries out a hypothesis test at $1\%$ level of significance.

The manager discovered an error in the data collection process of the sample and discarded $10$ out of the $50$ readings. From the remaining random sample of $40$ masks, the manager found that the mean PFE is $95.5\%$ and the standard deviation is $k\%$. Given that the manager concludes that there is insufficient evidence to justify his suspicion at $1\%$ level of significance, find the range of possible values of $k$ used in calculating the test statistic.

2021 ACJC P2 Q9

<p>A company sells a certain type of fitness tracker batteries. The company claims that the mean lifetime of the batteries is at least $2000$ hours. The production manager of the company wishes to take a random sample of batteries from the thousands produced in a day at his factory, for quality control purposes.</p>
<ol class="i-list"><li data-list-item-id="item-1759920878341-7zg9yw9xk" data-list-item-path="item-1759920878341-7zg9yw9xk"><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>A random sample of $200$ batteries is taken and it is found that the mean lifetime was $1995$ hours and the standard deviation was $25.5$ hours.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1759920899414-8cd6c7odw" data-list-item-path="item-1759920899414-8cd6c7odw"><div class="list-item-paragraph">Test, at the $1\%$ significance level, whether there is any evidence to doubt the company’s claim.<span class="inline-marks">[6]</span></div></li><li data-list-item-id="item-1759920909596-imv44khzz" data-list-item-path="item-1759920909596-imv44khzz"><div class="list-item-paragraph">State two assumptions or approximations that are involved when carrying out the significance test using the above sample data.<span class="inline-marks">[2]</span></div></li></ol>
<p>The company wishes to test whether the average lifetime of a second type of battery is longer than $2000$ hours. It can be assumed that the lifetime of the second type of battery has a normal distribution with variance ${{\sigma }^{2}}$. $8$ randomly chosen batteries of the second type $2$ are tested and it is found that their lifetimes, in hours, are as follows.</p>
<p style="text-align: center">${2001.5, \qquad 1997.3, \qquad 1989.0, \qquad 2018.7, \qquad 2030.7, \qquad 2023.0, \qquad 2027.5, \qquad 2013.3}$</p>
<p>The company carries out a hypothesis test at the $5\%$ level of significance.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-1759920947141-uqs9wcv8w" data-list-item-path="item-1759920947141-uqs9wcv8w"><div class="list-item-paragraph">Explain, with justification, how the population variance of the lifetimes, ${{\sigma }^{2}}$, will affect the conclusion.<span class="inline-marks">[4]</span></div></li></ol>

A company sells a certain type of fitness tracker batteries. The company claims that the mean lifetime of the batteries is at least $2000$ hours. The production manager of the company wishes to take a random sample of batteries from the thousands produced in a day at his factory, for quality control purposes.

A random sample of $200$ batteries is taken and it is found that the mean lifetime was $1995$ hours and the standard deviation was $25.5$ hours.

The company wishes to test whether the average lifetime of a second type of battery is longer than $2000$ hours. It can be assumed that the lifetime of the second type of battery has a normal distribution with variance ${{\sigma }^{2}}$. $8$ randomly chosen batteries of the second type $2$ are tested and it is found that their lifetimes, in hours, are as follows.

${2001.5, \qquad 1997.3, \qquad 1989.0, \qquad 2018.7, \qquad 2030.7, \qquad 2023.0, \qquad 2027.5, \qquad 2013.3}$

The company carries out a hypothesis test at the $5\%$ level of significance.

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