<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><div class="list-item-content"><span class="list-item-text">Calculate unbiased estimates of the population mean and variance of the mass of adult trout.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">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><span class="inline-marks">[6]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain why there is no need for Helen to know anything about the population distribution of the mass of adult trout.</span><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

2025 NYJC P2 Q8