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PSLE

PSLE Foundation Mathematics

<p>A cereal producer states that the mean mass of cereal in its packets is $500$ g. A one-tailed test is carried out. At the $4\%$ significance level, it is found that there is significant evidence that the population mean mass exceeds $500$ g. 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.</p>
<ol class="abc-list"><li data-list-item-id="q1-statements-item-1" data-list-item-path="q1-statements-item-1"><div class="list-item-paragraph"><p>There is significant evidence at the $7\%$ significance level that the population mean mass exceeds $500$ g.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="q1-statements-item-2" data-list-item-path="q1-statements-item-2"><div class="list-item-paragraph"><p>There is significant evidence at the $2\%$ significance level that the population mean mass exceeds $500$ g.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1788345830691-s4mr2sp8p" data-list-item-path="item-1788345830691-s4mr2sp8p"><div class="list-item-paragraph"><p>Suppose instead a <strong>two-tailed test</strong> is carried out at the $4 \%$ significance level. There is significant evidence that the population mean mass is <strong>not equal to 500 g</strong>.</p><span class="inline-marks">[2]</span></div></li></ol>

A cereal producer states that the mean mass of cereal in its packets is $500$ g. A one-tailed test is carried out. At the $4\%$ significance level, it is found that there is significant evidence that the population mean mass exceeds $500$ g. 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.

Significance Levels and $p$-Values

Comparing One-Tailed and Two-Tailed Tests

Packet-mass conclusions at different significance levels

Hypothesis Testing

<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

One-Tailed Hypothesis Tests

Large-Sample Mean Tests with Estimated Variance

Sampling

<p>A mobile application states that the mean time taken to travel from Town A to Park B is $14.5$ minutes. A data analyst would like to test if the mean travel time is more than $14.5$ minutes. The data analyst gathers a random sample of $35$ travel times of various people. The travel times, $x$ minutes, are summarized below.</p>
<p style="text-align: center">$\sum{x}=539$        $\sum{{{x}^{2}}}=8647.4$</p>
<ol class="i-list"><li data-list-item-id="item-1760684654012-ke9mvneea" data-list-item-path="item-1760684654012-ke9mvneea"><div class="list-item-paragraph">Calculate the unbiased estimates of the population mean and variance of the travel time from Town A to Park B.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760684655204-nm2dc8g5g" data-list-item-path="item-1760684655204-nm2dc8g5g"><div class="list-item-paragraph">Test, at the $5\%$ level of significance, whether the mean travel time from Town A to Park B is more than $14.5$ minutes. You should state your hypotheses clearly and define any symbols used.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760684657117-whru1yw9o" data-list-item-path="item-1760684657117-whru1yw9o"><div class="list-item-paragraph">Explain, in the context of the question, the meaning of 'at the $5\%$ level of significance'.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760684658197-2pho65349" data-list-item-path="item-1760684658197-2pho65349"><div class="list-item-paragraph">Explain, without performing another test, whether the conclusion in part <strong>(ii)</strong> would have changed if instead the data analyst was testing if the mean travel time from Town A to Park B was not $14.5$ minutes.<span class="inline-marks">[1]</span></div></li></ol>

A mobile application states that the mean time taken to travel from Town A to Park B is $14.5$ minutes. A data analyst would like to test if the mean travel time is more than $14.5$ minutes. The data analyst gathers a random sample of $35$ travel times of various people. The travel times, $x$ minutes, are summarized below.

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

Unbiased Estimates of Population Parameters

Hypothesis Tests Using Critical Regions and $p$-Values

Null and Alternative Hypotheses

2022 RVHS P2 Q10

<p>In a large busy restaurant, the mean time taken for a server to clear a table and set it for the next guest is $4.5$ minutes. In order to improve the quality of service and maintain a clean environment, the restaurant manager introduced a new routine to clear tables. A random sample of $30$ servers is taken and the time taken by each server to clear a table, $x$ minutes, is recorded. The data are summarised as follows:</p>
<p style="text-align: center">$\sum{x=132}$ and $\sum{{{x}^{2}}=\mathrm{583}\mathrm{.96}}$.</p>
<ol class="i-list"><li data-list-item-id="item-1760358318404-6mot9auhx" data-list-item-path="item-1760358318404-6mot9auhx"><div class="list-item-paragraph">Find unbiased estimates of the population mean and variance.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760358320144-siphcm6rd" data-list-item-path="item-1760358320144-siphcm6rd"><div class="list-item-paragraph">Test, at the $10\%$ significance level, whether the mean time taken for a server to clear a table has changed.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-1760358345021-v9ibabikb" data-list-item-path="item-1760358345021-v9ibabikb"><div class="list-item-paragraph">Using the results in part <strong>(ii)</strong>, state the range of values of $\alpha $, where&nbsp;$\alpha \%$ is the level of significance, at which a test would lead the manager to conclude that the mean time has not changed.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760358346011-1kxquuqf3" data-list-item-path="item-1760358346011-1kxquuqf3"><div class="list-item-paragraph">Without carrying out another test, state with reasons, the conclusion if the test at $5\%$ significance level was to determine whether the mean time taken to clear a table is less than $4.5$ minutes.<span class="inline-marks">[2]</span></div></li></ol>
<p>After a few weeks, based on feedback from servers and guests, the manager modifies the routine. Another random sample of $30$ servers is taken and their time taken to clear a table has mean $\bar{x}$ minutes and variance $1.01$ minutes$^{2}$.</p>
<ol class="i-list" start="5"><li data-list-item-id="item-1760358379988-udt6gpuft" data-list-item-path="item-1760358379988-udt6gpuft"><div class="list-item-paragraph">Find the set of values of $\bar{x}$ for which a test at $1\%$ significance level concludes that the mean time taken to clear a table is greater than $4.5$ minutes, giving your answer to $3$ decimal places.<span class="inline-marks">[4]</span></div></li></ol>

In a large busy restaurant, the mean time taken for a server to clear a table and set it for the next guest is $4.5$ minutes. In order to improve the quality of service and maintain a clean environment, the restaurant manager introduced a new routine to clear tables. A random sample of $30$ servers is taken and the time taken by each server to clear a table, $x$ minutes, is recorded. The data are summarised as follows:

$\sum{x=132}$ and $\sum{{{x}^{2}}=\mathrm{583}\mathrm{.96}}$.

After a few weeks, based on feedback from servers and guests, the manager modifies the routine. Another random sample of $30$ servers is taken and their time taken to clear a table has mean $\bar{x}$ minutes and variance $1.01$ minutes$^{2}$.

Parameter Bounds from Hypothesis Tests

Estimating Population Variance from Sample Variance

Two-Tailed Hypothesis Tests

2018 CJC P2 Q10

<p>A marine biologist claims that the mean mass of an adult sockeye salmon is at least $4.2$kg. The mass, $x$kg, of each adult sockeye salmon in a random sample of $100$ sockeye salmon is weighed. The results are summarised as follows.</p>
<p style="text-align: center">$\sum{\left( x-4.2 \right)}=-7.84$ $\sum{{{\left( x-4.2 \right)}^{2}}=\mathrm{32}\mathrm{.186}}$</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find unbiased estimates of the population mean and variance of the mass of adult sockeye salmons.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Test, at the $5\%$ level of significance, whether the marine biologist’s claim is valid.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Explain, in the context of the question, the meaning of ‘at the $5\%$ level of significance’.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Explain, without further calculation, whether the conclusion of the test in part <strong>(ii)</strong> will remain the same if the test were conducted with the same data but with only the following changes.</div><ol class="abc-list"><li><div class="list-item-paragraph">The marine biologist claims that the mean mass of an adult sockeye salmon is $4.2$ kg.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">The population variance is known and is smaller than the unbiased estimate of the population variance found in part <strong>(i)</strong>.</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

A marine biologist claims that the mean mass of an adult sockeye salmon is at least $4.2$kg. The mass, $x$kg, of each adult sockeye salmon in a random sample of $100$ sockeye salmon is weighed. The results are summarised as follows.

$\sum{\left( x-4.2 \right)}=-7.84$ $\sum{{{\left( x-4.2 \right)}^{2}}=\mathrm{32}\mathrm{.186}}$

Effects of Sample Size and Variance on Mean Tests

2022 MI P2 Q10

<p>An airline which charter various flights within the United States of America has a flight route that travels from New Orleans to Miami. The airline claims that the average flight time is $115$ minutes. Brandon, a regular commuter of this flight route, thinks that his journey takes a shorter time on average.</p>
<ol class="i-list"><li data-list-item-id="item-1760357693932-7bur7gn59" data-list-item-path="item-1760357693932-7bur7gn59"><div class="list-item-paragraph">Explain, whether Brandon should carry out a $1$-tail test or a $2$-tail test. State the hypotheses for the test, defining any symbols you use.<span class="inline-marks">[2]</span></div></li></ol>
<p>Brandon records the flight time (in minutes) of $8$ randomly selected flights as shown below.</p>
<p style="text-align: center">$113.0$      $112.4$     $110.0$     $113.8$     $111.3$     $115.2$     $114.2$     $115.5$</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1760357724584-v20oj7d7l" data-list-item-path="item-1760357724584-v20oj7d7l"><div class="list-item-paragraph">Assuming that the flight time from New Orleans to Miami follows a normal distribution that has a variance of $11.3$ minutes$^{2}$, carry out the test for Brandon at $10\%$ level of significance. Give your conclusion in the context of the question.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760357725349-dhjtaag1r" data-list-item-path="item-1760357725349-dhjtaag1r"><div class="list-item-paragraph">Explain what you understand by the phrase “at $10\%$ level of significance” in the context of the question.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760357725987-2qaas3bcq" data-list-item-path="item-1760357725987-2qaas3bcq"><div class="list-item-paragraph">The same sample of $8$ flights is now used to carry out a test at the $10\%$ significance level, to test whether the mean flight times is different from what the airline claims. Using the answer in <strong>(ii)</strong>, state the conclusion of this test.<span class="inline-marks">[2]</span></div></li></ol>
<p>Another flight route travels from Portland to Los Angeles. Due to bad weather conditions, the flights sometimes take a longer time to arrive at Los Angeles. The airline claims that the mean flight time is $k$ minutes. The operation manager of the airline believes that this number is understated and wants to test this claim. Sixty flights were randomly selected, and the mean flight time of $181$ minutes and standard deviation of $13.6$ minutes were recorded.</p>
<ol class="i-list" start="5"><li data-list-item-id="item-1760357762780-5vgrdu4gn" data-list-item-path="item-1760357762780-5vgrdu4gn"><div class="list-item-paragraph">Find the range of values of $k$ for which the claim by the airline is supported at $4\%$ level of significance. Give your answer to the nearest minute.<span class="inline-marks">[4]</span></div></li></ol>

An airline which charter various flights within the United States of America has a flight route that travels from New Orleans to Miami. The airline claims that the average flight time is $115$ minutes. Brandon, a regular commuter of this flight route, thinks that his journey takes a shorter time on average.

Brandon records the flight time (in minutes) of $8$ randomly selected flights as shown below.

$113.0$      $112.4$     $110.0$     $113.8$     $111.3$     $115.2$     $114.2$     $115.5$

Another flight route travels from Portland to Los Angeles. Due to bad weather conditions, the flights sometimes take a longer time to arrive at Los Angeles. The airline claims that the mean flight time is $k$ minutes. The operation manager of the airline believes that this number is understated and wants to test this claim. Sixty flights were randomly selected, and the mean flight time of $181$ minutes and standard deviation of $13.6$ minutes were recorded.

Mean Tests for Normal Populations with Known Variance

Interpreting Hypothesis Test Conclusions

2021 ASRJC P2 Q11

<p>The batteries, used in a particular model of mobile phone, is advertised as having a ‘talk-time’ of $5$ hours. In a test at the $5$% level of significance, it is found 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 <strong>(i)</strong> necessarily true, <strong>(ii)</strong> necessarily false, or <strong>(iii)</strong> neither necessarily true or necessarily false.</p>
<ol class="abc-list"><li><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><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>

The batteries, used in a particular model of mobile phone, is advertised as having a ‘talk-time’ of $5$ hours. In a test at the $5$% level of significance, it is found 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 or necessarily false.

N2001 P2 Q10

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

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

Calculating and Interpreting $p$-Values

2018 MJC P2 Q9

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

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

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

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

Large-Sample Mean Tests with Known Variance

Population and Random Samples

2021 EJC P2 Q9

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

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

2019 CJC P2 Q9 [Modified]

<p>The time spent per day, in minutes, by Eunice on social media Toktik is normally distributed with mean $20.2$ and standard deviation $4.5$. Due to an unfortunate event, Eunice is confined to home for two months. During this period, the time spent per day by Eunice on Toktik is recorded for $20$ randomly chosen days, which averages $22.4$ minutes.</p>
<p>It is assumed that the population variance remains unchanged throughout.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Test, at $1\%$ significance level, if there has been a change in the average time spent per day by Eunice on Toktik.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Without carrying out another test, find the range of $\alpha $ for which a test at $\alpha \%$ level of significance suggests an increase in the average time spent per day by Eunice on Toktik.</div><p style="text-align: right;">[2]</p></li></ol>
<p>Another set of $15$ timings is taken. Using only this set of $15$ timings, find the range of values within which the mean of this set of readings must lie such that the claim that there has been a change in the average time spent by Eunice on Toktik is supported at $5\%$ level of significance.</p><p style="text-align: right;">[3]</p>

The time spent per day, in minutes, by Eunice on social media Toktik is normally distributed with mean $20.2$ and standard deviation $4.5$. Due to an unfortunate event, Eunice is confined to home for two months. During this period, the time spent per day by Eunice on Toktik is recorded for $20$ randomly chosen days, which averages $22.4$ minutes.

It is assumed that the population variance remains unchanged throughout.

Another set of $15$ timings is taken. Using only this set of $15$ timings, find the range of values within which the mean of this set of readings must lie such that the claim that there has been a change in the average time spent by Eunice on Toktik is supported at $5\%$ level of significance.

2020 EJC P2 Q8

<p>A teacher of Peaceful Childcare Centre claims that a child at the centre spends an average of 4 hours on afternoon nap. A survey was conducted and the time, $x$ hours, spent by $50$ randomly chosen children on afternoon naps is as follows:</p>
<p style="text-align: center">$\sum{x}=185$, $\sum{{{\left( x-\bar{x} \right)}^{2}}}=39.2$</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find an unbiased estimate of the population mean and variance.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Carry out a test at $2\%$ level of significance to determine if the teacher’s claim is correct.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Suppose the teacher now claims that a randomly chosen child spends less than $4$ hours on afternoon nap. Without carrying out another test, state, with a reason, whether the conclusion in part <strong>(ii)</strong> would remain the same.</div><p style="text-align: right;">[2]</p></li></ol>
<p>The principal of Peaceful Childcare Centre now claims that a randomly chosen child spends ${{\mu }_{0}}$ hours on afternoon nap. Find the set of values of ${{\mu }_{0}}$ such that the principal is confident, at $5\%$ level of significance, that he did not understate the mean time that the children took for afternoon naps.</p><p style="text-align: right;">[4]</p>

A teacher of Peaceful Childcare Centre claims that a child at the centre spends an average of 4 hours on afternoon nap. A survey was conducted and the time, $x$ hours, spent by $50$ randomly chosen children on afternoon naps is as follows:

$\sum{x}=185$, $\sum{{{\left( x-\bar{x} \right)}^{2}}}=39.2$

The principal of Peaceful Childcare Centre now claims that a randomly chosen child spends ${{\mu }_{0}}$ hours on afternoon nap. Find the set of values of ${{\mu }_{0}}$ such that the principal is confident, at $5\%$ level of significance, that he did not understate the mean time that the children took for afternoon naps.

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