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E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

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

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

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

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

Assumptions and Validity of Mean Tests

Estimating Parameters from Summarised and Coded Data

Central Limit Theorem

Significance Levels and $p$-Values

One-Tailed Hypothesis Tests

Comparing One-Tailed and Two-Tailed Tests

Large-Sample Mean Tests with Estimated Variance

Hypothesis Testing

Sampling

<p>A refrigerator manufacturer claims that the mean lifespan of refrigerators of a particular model is $12$ years. A consumers association representative suspects that the mean lifespan of the refrigerators is actually less than $12$ years.</p>
<p>The durations $x$, in years, of a random sample of $45$ refrigerators are summarised as follows.</p>
<p style="text-align: center">\[\sum(x-12) = -4.3 \qquad \sum(x-12)^2 = 17.08\]</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance of the lifespan of the refrigerators.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">State hypotheses that can be used to test if the mean lifespan of the refrigerators is less than $12$ years, defining any parameters you use. Test, at the $5\%$ level of significance, whether the mean lifespan of the refrigerators is less than $12$ years.<span class="inline-marks">[4]</span></div></li></ol>
<p>The manufacturer switches to a different coolant and decides to test whether the mean lifespan of the refrigerators has changed from $12$ years. He records the durations of a large random sample of $n$ refrigerators and finds that their mean lifespan is $12.4$ years and variance is $4.1$ square years.</p>
<p>A test at $5\%$ significance level, is carried out on the new random sample. The test shows that there is sufficient evidence that the mean lifespan of refrigerators has changed.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find the range of possible values of $n$.<span class="inline-marks">[4]</span></div></li></ol>

A refrigerator manufacturer claims that the mean lifespan of refrigerators of a particular model is $12$ years. A consumers association representative suspects that the mean lifespan of the refrigerators is actually less than $12$ years.

The durations $x$, in years, of a random sample of $45$ refrigerators are summarised as follows.

\[\sum(x-12) = -4.3 \qquad \sum(x-12)^2 = 17.08\]

The manufacturer switches to a different coolant and decides to test whether the mean lifespan of the refrigerators has changed from $12$ years. He records the durations of a large random sample of $n$ refrigerators and finds that their mean lifespan is $12.4$ years and variance is $4.1$ square years.

A test at $5\%$ significance level, is carried out on the new random sample. The test shows that there is sufficient evidence that the mean lifespan of refrigerators has changed.

Parameter Bounds from Hypothesis Tests

Estimating Population Variance from Sample Variance

Null and Alternative Hypotheses

Two-Tailed Hypothesis Tests

2024 ACJC P2 Q8

<p>A private-hire vehicle firm, Euber, employs drivers to operate a fleet of cars. The mileage covered by an Euber driver on a weekday is denoted by $M$ km. A random sample of $60$ Euber drivers is taken and their mileage can be summarised by</p>
<p style="text-align: center">\[\sum(m-250) = 540 \text{ and}\hspace{0.5em} \sum(m-259)^2 = 8176.\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Calculate unbiased estimates of the population mean and variance.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>Assume that the population mean and variance follow the answers calculated in part <strong>(i)</strong>.</p>
<p>Random samples of $n$ Euber drivers are taken, where&nbsp;$n$ is large.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Over a long period of time, it is found that&nbsp;$25\%$ of the samples have a mean mileage of at least $260$ km. Find the value of $n$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain whether it is necessary to assume that&nbsp;$M$ follows a normal distribution in order for your calculations in part <strong>(ii)</strong> to hold.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">It is now given that&nbsp;$M$ is normally distributed with population variance $5^2$. Estimate the sample size needed so that Euber can be $95\%$ certain that the sample mean mileage will differ from the true mean by less than $500$m.</span><span class="inline-marks">[2]</span></div></li></ol>

A private-hire vehicle firm, Euber, employs drivers to operate a fleet of cars. The mileage covered by an Euber driver on a weekday is denoted by $M$ km. A random sample of $60$ Euber drivers is taken and their mileage can be summarised by

\[\sum(m-250) = 540 \text{ and } \sum(m-259)^2 = 8176.\]

Calculate unbiased estimates of the population mean and variance.

Assume that the population mean and variance follow the answers calculated in part **(i)**.

Random samples of $n$ Euber drivers are taken, where $n$ is large.

Over a long period of time, it is found that $25\%$ of the samples have a mean mileage of at least $260$ km. Find the value of $n$.

Explain whether it is necessary to assume that $M$ follows a normal distribution in order for your calculations in part **(ii)** to hold.

It is now given that $M$ is normally distributed with population variance $5^2$. Estimate the sample size needed so that Euber can be $95\%$ certain that the sample mean mileage will differ from the true mean by less than $500$m.

Solving Normal Probability Inequalities

Determining Sample Size from Probability Conditions

Normal Models for Sample Means and Averages

Sample Mean and Its Sampling Distribution

2019 EJC BT2 P2 Q8 [Modified]

Continuous Random Variables

<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

Large-Sample Mean Tests with Known Variance

2024 ASRJC H1 Q10

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

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

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

2024 TJC H1 Q11

<p>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>A company advertises that its rechargeable batteries can run non-stop for an average of $505$ minutes after a full charge. $100$ batteries are randomly selected, and their running time after a full charge, $x$ minutes, are measured. The data obtained is summarised by\[\sum (x - 500) = 320, \quad \sum (x - 500)^2 = 8416.\]</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Suggest a reason why, in this context, the given data is summarised in terms of $(x - 500)$ rather than $x$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance of the running time of the batteries.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph"><p>A hypothesis test is carried out at the $5\%$ level of significance. Determine whether the company is overestimating the average running time of the batteries.</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">State, giving a reason, whether it is necessary to assume a normal distribution for the test to be valid.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-e" data-list-item-path="item-e"><div class="list-item-paragraph"><p>After a change is made in the manufacturing process, another test at the $5\%$ significance level is carried out using a new random sample of size $50$. The sample standard deviation is found to be $8.5$ minutes. Find the range of values of the sample mean in order to conclude that the average running time of the batteries after a full charge is more than $505$ minutes.</p><span class="inline-marks">[4]</span></div></li></ol>

A company advertises that its rechargeable batteries can run non-stop for an average of $505$ minutes after a full charge. $100$ batteries are randomly selected, and their running time after a full charge, $x$ minutes, are measured. The data obtained is summarised by\[\sum (x - 500) = 320, \quad \sum (x - 500)^2 = 8416.\]

Suggest a reason why, in this context, the given data is summarised in terms of $(x - 500)$ rather than $x$.

Calculate unbiased estimates of the population mean and variance of the running time of the batteries.

State, giving a reason, whether it is necessary to assume a normal distribution for the test to be valid.

2025 YIJC P2 Q10

<p>A coffee shop owner claims that its signature coffee has a mean caffeine content of $120$ mg per cup. To investigate this claim, a random sample of $60$ cups of coffee is selected and the caffeine content, $x$ mg per cup, is summarised as follows.\[\sum x = 6960, \quad \sum x^2 = 822465\]</p>
<ol class="abc-list"><li data-list-item-id="item-a3" data-list-item-path="item-a3"><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance for the caffeine content per cup.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b3" data-list-item-path="item-b3"><div class="list-item-paragraph"><p>Test, at the $5\%$ level of significance, whether the coffee shop owner’s claim is supported by the data. You should state your hypotheses and define any symbols that you use.</p><span class="inline-marks">[4]</span></div></li></ol>
<p>The coffee shop also sells premium coffee, where its caffeine content is normally distributed with population variance $200$ mg${}^2$. The coffee shop owner now claims that its premium coffee has a mean caffeine content less than $120$ mg per cup. Another large random sample of $n$ cups of premium coffee is selected and the sample mean caffeine content per cup is found to be $116.6$ mg. A test is carried out at the $2\frac{1}{2}\%$ level of significance and the result supports the owner’s claim.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c3" data-list-item-path="item-c3"><div class="list-item-paragraph">Find the set of values that&nbsp;$n$ can take.<span class="inline-marks">[4]</span></div></li></ol>

A coffee shop owner claims that its signature coffee has a mean caffeine content of $120$ mg per cup. To investigate this claim, a random sample of $60$ cups of coffee is selected and the caffeine content, $x$ mg per cup, is summarised as follows.\[\sum x = 6960, \quad \sum x^2 = 822465\]

Calculate unbiased estimates of the population mean and variance for the caffeine content per cup.

The coffee shop also sells premium coffee, where its caffeine content is normally distributed with population variance $200$ mg${}^2$. The coffee shop owner now claims that its premium coffee has a mean caffeine content less than $120$ mg per cup. Another large random sample of $n$ cups of premium coffee is selected and the sample mean caffeine content per cup is found to be $116.6$ mg. A test is carried out at the $2\frac{1}{2}\%$ level of significance and the result supports the owner’s claim.

Find the set of values that $n$ can take.

Mean Tests for Normal Populations with Known Variance

2025 TMJC P2 Q7

<p>At a school canteen, the average queuing time (in minutes) for a student to be served at a stall is known to be $4.5$ minutes. The school operation manager claims that with a recent change in students’ timetable, there is an improvement in the average queuing time.</p>
<p>A random sample of $10$ students’ queuing time is observed after the change in students’ timetable and the data is shown below.</p>
<p style="text-align: center">\[4.2, \quad 5.2, \quad 3.1, \quad 3.5, \quad 4.6, \quad 4.8, \quad 3.8, \quad 3.7, \quad 4.5, \quad 5.0\]</p>
<p>Given that the population variance of queuing time is known to be $0.48$ minutes$^2$, a one-tailed test is carried out to determine whether the manager’s claim is justified.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">State an assumption needed to carry out the hypothesis test.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Test whether the manager’s claim is justified at $5$% level of significance.</span><span class="inline-marks">[5]</span></div></li></ol>
<p>A student leader Jason suspected that with the change in students’ timetable, the average queuing time for a student to be served at a stall is in fact more than $\lambda$ minutes and also that the population variance of queuing time is no longer $0.48$ minutes$^2$. To investigate this, Jason recorded the queuing time of the first $50$ students who visited the stall from $12$ pm to $1$ pm.</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Give a reason why the sample collected is not a random sample.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>Jason decided to collect another set of data to do his investigation. He recorded the queuing time, $x$, in minutes, of a random sample of $50$ students. The summarized data is as shown below.</p>
<p style="text-align: center">\[\bar{x} = 4.6 \quad \text{and} \quad \sum x^2 = 1068\]</p>
<ol class="abc-list" start="4"><li><div class="list-item-content"><span class="list-item-text">Find an unbiased estimate of the population variance.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>Jason carried out a hypothesis test at $2$% level of significance with an alternative hypothesis of $\mu &gt; \lambda$ where&nbsp;$\mu$ is the population mean queuing time.</p>
<ol class="abc-list" start="5"><li><div class="list-item-content"><span class="list-item-text">Explain why the assumption in (a) is no longer necessary.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">By finding the critical region in terms of $z$-values, find the range of values of $\lambda$ such that the Jason’s suspicion is justified.</span><span class="inline-marks">[3]</span></div></li></ol>

At a school canteen, the average queuing time (in minutes) for a student to be served at a stall is known to be $4.5$ minutes. The school operation manager claims that with a recent change in students’ timetable, there is an improvement in the average queuing time.

A random sample of $10$ students’ queuing time is observed after the change in students’ timetable and the data is shown below.

\[4.2, \quad 5.2, \quad 3.1, \quad 3.5, \quad 4.6, \quad 4.8, \quad 3.8, \quad 3.7, \quad 4.5, \quad 5.0\]

Given that the population variance of queuing time is known to be $0.48$ minutes$^2$, a one-tailed test is carried out to determine whether the manager’s claim is justified.

State an assumption needed to carry out the hypothesis test.

Test whether the manager’s claim is justified at $5$% level of significance.

A student leader Jason suspected that with the change in students’ timetable, the average queuing time for a student to be served at a stall is in fact more than $\lambda$ minutes and also that the population variance of queuing time is no longer $0.48$ minutes$^2$. To investigate this, Jason recorded the queuing time of the first $50$ students who visited the stall from $12$ pm to $1$ pm.

Give a reason why the sample collected is not a random sample.

Jason decided to collect another set of data to do his investigation. He recorded the queuing time, $x$, in minutes, of a random sample of $50$ students. The summarized data is as shown below.

\[\bar{x} = 4.6 \quad \text{and} \quad \sum x^2 = 1068\]

Find an unbiased estimate of the population variance.

Jason carried out a hypothesis test at $2$% level of significance with an alternative hypothesis of $\mu > \lambda$ where $\mu$ is the population mean queuing time.

Explain why the assumption in (a) is no longer necessary.

By finding the critical region in terms of $z$-values, find the range of values of $\lambda$ such that the Jason’s suspicion is justified.

Assumptions for Sampling

Critical Values and Critical Regions

2025 TJC P2 Q9

<p>A beverage company produces bottled orange juice, claiming that the mean vitamin $C$ content in each $250$ ml bottle is $50$ mg. It is known that the vitamin $C$ content is normally distributed with a standard deviation of $2$ mg</p>
<p>Due to feedback from consumers, the quality control manager wants to test, at the $4$% level of significance, whether the mean vitamin $C$ content is, in fact, $50$ mg. He randomly samples $12$ bottles and measures the vitamin $C$ content.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State the hypotheses and find the critical region for this test.<span class="inline-marks">[3]</span></div></li></ol>
<p>Recently, the company launched a new "Vitamin Boost" orange juice line, advertised to contain an average of $60$ mg of vitamin $C$ content per $250$ ml bottle. After some complaints that the juice is too sour, the manager suspects that the vitamin $C$ content might be higher than advertised. He decides to conduct a hypothesis test to investigate. This time, he randomly samples $30$ bottles from the new juice line.</p>
<p><strong>(ii)</strong> Explain why the manager takes a sample of $30$ bottles for this test when he only took a sample of $12$ bottles in his earlier test.<span class="inline-marks">[2]</span></p>
<p>The vitamin $C$ content, $y$ mg, in the sample of $30$ bottles is summarized as follows:</p>
<p>\[\sum y = 1865, \quad \sum y^2 = 116275\]</p>
<p><strong>(iii)</strong> Calculate unbiased estimates of the population mean and variance for the vitamin $C$ content in the bottles of the new juice line.<span class="inline-marks">[2]</span></p>
<p><strong>(iv)</strong> At the $5$% level of significance, test whether the mean vitamin $C$ content in the bottles of the new juice line is greater than $60$ mg. Clearly state your hypotheses and define any symbols you use.<span class="inline-marks">[4]</span></p>

A beverage company produces bottled orange juice, claiming that the mean vitamin $C$ content in each $250$ ml bottle is $50$ mg. It is known that the vitamin $C$ content is normally distributed with a standard deviation of $2$ mg

Due to feedback from consumers, the quality control manager wants to test, at the $4$% level of significance, whether the mean vitamin $C$ content is, in fact, $50$ mg. He randomly samples $12$ bottles and measures the vitamin $C$ content.

Recently, the company launched a new "Vitamin Boost" orange juice line, advertised to contain an average of $60$ mg of vitamin $C$ content per $250$ ml bottle. After some complaints that the juice is too sour, the manager suspects that the vitamin $C$ content might be higher than advertised. He decides to conduct a hypothesis test to investigate. This time, he randomly samples $30$ bottles from the new juice line.

**(ii)** Explain why the manager takes a sample of $30$ bottles for this test when he only took a sample of $12$ bottles in his earlier test.

The vitamin $C$ content, $y$ mg, in the sample of $30$ bottles is summarized as follows:

\[\sum y = 1865, \quad \sum y^2 = 116275\]

**(iii)** Calculate unbiased estimates of the population mean and variance for the vitamin $C$ content in the bottles of the new juice line.

**(iv)** At the $5$% level of significance, test whether the mean vitamin $C$ content in the bottles of the new juice line is greater than $60$ mg. Clearly state your hypotheses and define any symbols you use.

2025 SAJC P2 Q11

<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.

Population and Random Samples

2025 RVHS P2 Q11

<p>Helen is a conservationist who monitors the health of fish in a river. She is investigating whether a new water-treatment plant has affected the mass of adult trout in the river. Previously, the mean mass of adult trout in the river has been $w$ kg. Helen carries out a test, at the $10$% level of significance to find out whether there has been a change in the mean mass of adult trout in the river.</p>
<p>Helen catches $45$ adult trout and measures the mass, $X$, kilograms, of each fish. Her summarised data are as follows.</p>
<p style="text-align: center">\[n=45 \quad \sum x=80.1 \quad \sum (x-\bar{x})^2=24.29\]</p>
<ol class="abc-list"><li data-list-item-id="item-4a" data-list-item-path="item-4a"><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance of the mass of adult trout.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-4b" data-list-item-path="item-4b"><div class="list-item-paragraph">Use an algebraic method to calculate the set of values of $w$ for which there is insufficient evidence to conclude that there has been a change in the mean mass of adult trout in the river. You should state your hypotheses and define any symbols you use.<span class="inline-marks">[6]</span></div></li><li data-list-item-id="item-4c" data-list-item-path="item-4c"><div class="list-item-paragraph">Explain why there is no need for Helen to know anything about the population distribution of the mass of adult trout.<span class="inline-marks">[2]</span></div></li></ol>

Helen is a conservationist who monitors the health of fish in a river. She is investigating whether a new water-treatment plant has affected the mass of adult trout in the river. Previously, the mean mass of adult trout in the river has been $w$ kg. Helen carries out a test, at the $10$% level of significance to find out whether there has been a change in the mean mass of adult trout in the river.

Helen catches $45$ adult trout and measures the mass, $X$, kilograms, of each fish. Her summarised data are as follows.

\[n=45 \quad \sum x=80.1 \quad \sum (x-\bar{x})^2=24.29\]

Calculate unbiased estimates of the population mean and variance of the mass of adult trout.

Use an algebraic method to calculate the set of values of $w$ for which there is insufficient evidence to conclude that there has been a change in the mean mass of adult trout in the river. You should state your hypotheses and define any symbols you use.

Explain why there is no need for Helen to know anything about the population distribution of the mass of adult trout.

2025 NYJC P2 Q8

<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 $n$.

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>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'.

2025 ASRJC P2 Q12

<p>An IP school has&nbsp;$2000$ students – there are $200$ students in each level from Secondary 1-4, $600$ in JC1 and $600$ in JC2. The Student Council is interested in finding $x$ hours, the amount of time each student spends in a week doing homework. The president of the Student Council thus decides to survey $50$ students. The data gathered is summarised as follows:</p>
<p style="text-align: center">$\sum{x}=600$;	$\sum{{{x}^{2}}}=7300$</p>
<ol class="i-list"><li><div class="list-item-paragraph">Calculate the unbiased estimates of the population mean and variance.	</div></li><li><div class="list-item-paragraph">Given that&nbsp;$\mu $=12, find the probability that the sample mean exceeds $12.1$.</div></li></ol>

An IP school has $2000$ students – there are $200$ students in each level from Secondary 1-4, $600$ in JC1 and $600$ in JC2. The Student Council is interested in finding $x$ hours, the amount of time each student spends in a week doing homework. The president of the Student Council thus decides to survey $50$ students. The data gathered is summarised as follows:

2012 NJC Q10 [Modified]

<p>A manufacturer of candles claimed that it produced birthday candles with a mean burning time of $6$ minutes. A random sample of $150$ birthday candles was tested and the burning times, $X$ minutes, were summarized by $\sum{(x-5)=120}$ and $\sum{{{(x-5)}^{2}}=638}$ . Calculate the unbiased estimates for the mean $\mu $ and variance${{\sigma }^{2}}$ .</p>

A manufacturer of candles claimed that it produced birthday candles with a mean burning time of $6$ minutes. A random sample of $150$ birthday candles was tested and the burning times, $X$ minutes, were summarized by $\sum{(x-5)=120}$ and $\sum{{{(x-5)}^{2}}=638}$ . Calculate the unbiased estimates for the mean $\mu $ and variance${{\sigma }^{2}}$ .

2005 SAJC P2 Q28

<p style="margin-bottom: 0.5rem">The thirty students in a class each took a Biology test. The marks obtained are shown in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 400px;"><colgroup><col style="width: 100px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Marks</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$4$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$5$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$6$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$7$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$8$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$10$</p></th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Frequency</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$2$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$5$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$11$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$8$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$3$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$1$</p></td></tr></tbody></table></div></div>
<p>Calculate the unbiased estimates of the population mean and variance.  </p>

The thirty students in a class each took a Biology test. The marks obtained are shown in the table below.

Calculate the unbiased estimates of the population mean and variance.  

Statistical Estimates of Biology Test Marks

<p>An aptitude test for deep-sea divers produces scores which are normally distributed on a scale from $0$ to $100$. A random sample of $160$ divers were assessed, and each of their individual test scores, $x$, was recorded. The results are summarised by</p>
<p style="text-align: center">$\sum x = 9120, \qquad\qquad\qquad \sum (x-\bar{x})^{2} = 35775$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance.</div></li><li><div class="list-item-paragraph">Assuming that the unbiased estimate of the population mean obtained above is equal to the population mean, find the value of $c$ such that there is a probability of $0.05$ that a sample of $160$ scores has a sample mean that differs from its mean score by more than $c$.</div></li><li><div class="list-item-paragraph">Comment on the validity of the calculations if the scores had not been known to be normally distributed.</div></li></ol>

An aptitude test for deep-sea divers produces scores which are normally distributed on a scale from $0$ to $100$. A random sample of $160$ divers were assessed, and each of their individual test scores, $x$, was recorded. The results are summarised by

$\sum x = 9120, \qquad\qquad\qquad \sum (x-\bar{x})^{2} = 35775$.

Assumptions For Sampling

<p>Sample size = 150, $\sum(x-170)=-2250$, $\sum(x-154)^{2}=11000$</p>

Sample size = 150, $\sum(x-170)=-2250$, $\sum(x-154)^{2}=11000$

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