<p>Mr Dough’s factory produces packets of flour for sale in local supermarkets packed in \(2\) sizes – Regular-sized packets and Large-sized packets. Automated machines weigh and pack the flour. A recent major power outage resulted in all his machines being needed to be recalibrated upon restarting.</p>
<p>Regular-sized packets are supposed to have a mean mass of \(1\) kg. Mr Dough suspects that his machines now pack Regular-sized packets that are underweight. He took a sample of his Regular-sized packets and measured their masses, \(x\) g (correct to nearest \(5\) g). The results are recorded as follows:</p>
<div class="content-block content-block--table"><div class="table-container"><table><thead><tr><th style="text-align: left; --cell-justify: flex-start">Mass, \(x\) (g)</th><th style="text-align: left; --cell-justify: flex-start">No. of Regular-sized packets</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">\(980\)</td><td style="text-align: left; --cell-justify: flex-start">\(5\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(985\)</td><td style="text-align: left; --cell-justify: flex-start">\(6\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(990\)</td><td style="text-align: left; --cell-justify: flex-start">\(7\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(995\)</td><td style="text-align: left; --cell-justify: flex-start">\(7\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(1000\)</td><td style="text-align: left; --cell-justify: flex-start">\(9\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(1005\)</td><td style="text-align: left; --cell-justify: flex-start">\(6\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(1010\)</td><td style="text-align: left; --cell-justify: flex-start">\(5\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(1015\)</td><td style="text-align: left; --cell-justify: flex-start">\(3\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(1020\)</td><td style="text-align: left; --cell-justify: flex-start">\(2\)</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Determine the unbiased estimate of the population variance of the mass of Regular-sized packets.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain whether Mr Dough should conduct a one-tail or a two-tail test.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Carry out the test at \(5\)% level of significance and give your conclusion in context. Explain whether it is necessary to assume that the mass of Regular-sized packets of flour follow a normal distribution.</span><span class="inline-marks">[5]</span></div></li></ol>
<p>Large-sized packets of flour are known to have a normal distribution with a mean mass of \(2\) kg. Mr Dough also suspects that his machines are now not packing the Large-sized packets according to the required mass correctly. A sample of \(15\) Large-sized packets of flour gave a total mass of \(30075\) g.</p>
<p>Assuming Large-sized packets of flour have a population variance of \(\sigma^2\) \(\mathrm{g}^2\), an appropriate test was carried out which gave a \(p\)-value of \(0.0529\).</p>
<ol class="abc-list" start="4"><li><div class="list-item-content"><span class="list-item-text">Showing all necessary workings, prove that \(\sigma=10.0\) correct to \(3\) significant figures.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>Another packet of Large-sized packet of flour with mass \(h\) g is then added to the sample of \(15\). The combined sample of Large-sized packets resulted in a rejection of the claim that the mean mass of a Large-sized packet of flour is \(2\) kg at \(4\)% level of significance.</p>
<ol class="abc-list" start="5"><li><div class="list-item-content"><span class="list-item-text">Using \(\sigma=10.0\), find the possible range of values of \(h\), giving your answer to the nearest gram.</span><span class="inline-marks">[3]</span></div></li></ol>

Mr Dough’s factory produces packets of flour for sale in local supermarkets packed in \(2\) sizes – Regular-sized packets and Large-sized packets. Automated machines weigh and pack the flour. A recent major power outage resulted in all his machines being needed to be recalibrated upon restarting.

Regular-sized packets are supposed to have a mean mass of \(1\) kg. Mr Dough suspects that his machines now pack Regular-sized packets that are underweight. He took a sample of his Regular-sized packets and measured their masses, \(x\) g (correct to nearest \(5\) g). The results are recorded as follows:

Determine the unbiased estimate of the population variance of the mass of Regular-sized packets.

Explain whether Mr Dough should conduct a one-tail or a two-tail test.

Carry out the test at \(5\)% level of significance and give your conclusion in context. Explain whether it is necessary to assume that the mass of Regular-sized packets of flour follow a normal distribution.

Large-sized packets of flour are known to have a normal distribution with a mean mass of \(2\) kg. Mr Dough also suspects that his machines are now not packing the Large-sized packets according to the required mass correctly. A sample of \(15\) Large-sized packets of flour gave a total mass of \(30075\) g.

Assuming Large-sized packets of flour have a population variance of \(\sigma^2\) \(\mathrm{g}^2\), an appropriate test was carried out which gave a \(p\)-value of \(0.0529\).

Showing all necessary workings, prove that \(\sigma=10.0\) correct to \(3\) significant figures.

Another packet of Large-sized packet of flour with mass \(h\) g is then added to the sample of \(15\). The combined sample of Large-sized packets resulted in a rejection of the claim that the mean mass of a Large-sized packet of flour is \(2\) kg at \(4\)% level of significance.

Using \(\sigma=10.0\), find the possible range of values of \(h\), giving your answer to the nearest gram.

2025 DHS P2 Q11

2025 DHS P2 Q11