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<p>The number of employees of a company, classified by department and gender, is shown below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 500px;"><colgroup><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"></colgroup><thead><tr style="height: 25px;"><th style="height: 25px; min-height: 25px; text-align: left; --cell-justify: flex-start"></th><th style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">Production</th><th style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">Development</th><th style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">Administration</th><th style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">Finance</th></tr></thead><tbody><tr style="height: 25px;"><td style="height: 25px; min-height: 25px; text-align: left; --cell-justify: flex-start">Male</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$2345$</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$1013$</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$237$</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$344$</td></tr><tr style="height: 25px;"><td style="height: 25px; min-height: 25px; text-align: left; --cell-justify: flex-start">Female</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$867$</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$679$</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$591$</td><td style="height: 25px; min-height: 25px; text-align: center; --cell-justify: center">$523$</td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-paragraph">The directors wish to survey a sample of $100$ of the employees. This sample is to be a stratified sample,based on department and gender.</div><ol class="abc-list"><li><div class="list-item-paragraph">How many males should be in the sample?<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">How many females from the Development department should be in the sample?<span class="inline-marks">[1]</span></div></li></ol></li></ol>
<p>The Managing Director knows that, some years ago, the mean age of employees was $37$ years. He believes that the mean age of employees now is less than $37$ years.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">State why the stratified sample from part <strong>(i)</strong> should not be used for a hypothesis test of the Managing Director's belief.<span class="inline-marks">[1]</span></div></li></ol>
<p>The Company Secretary obtains a suitable sample of $80$ employees in order to carry out a hypothesis test of the Managing Director’s belief that the mean age of the employees now is less than $37$ years. You are given that the population variance of the ages is $140$ years$^{2}$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Write down appropriate hypotheses to test the Managing Director’s belief. You are given that the result of the test, using a $5%$ significance level, is that the Managing Director’s belief should be accepted. Determine the set of possible values of the mean age of the sample of employees.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">You are given instead that the mean age of the sample of employees is $35.2$ years, and that the result of a test at the $\alpha %$ significance level is that the Managing Director’s belief should not be accepted. Find the set of possible values of $\alpha $.<span class="inline-marks">[3]</span></div></li></ol>

The number of employees of a company, classified by department and gender, is shown below.

The Managing Director knows that, some years ago, the mean age of employees was $37$ years. He believes that the mean age of employees now is less than $37$ years.

The Company Secretary obtains a suitable sample of $80$ employees in order to carry out a hypothesis test of the Managing Director’s belief that the mean age of the employees now is less than $37$ years. You are given that the population variance of the ages is $140$ years$^{2}$.

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