Question $1.$ In$-$Person Office Work Practices $[20$ marks$]$

Federal government labour unions such as$,$ the Public Service Alliance of Canada $($PSAC$),$ the

Professional Institute of the Public Service of Canada $($PIPSC$),$ the Canadian Association of

Professional Employees $($CAPE$),$ and the Association of Canadian Financial Officers $($ACFO$),$

as well as other labour unions representing the federal government public servants in different

federal ministries and agencies are claiming that the federal government mandate in $2024$ that

most public servants return in office three$-$days a week in$-$person is not in line with current

practices in public servants working for provincial governments in different ministries and

agencies across Canada. Suppose that the federal public service unions have commissioned a

national survey to compare teleworking policies between similar federal government and

provincial government practice in this matter. The survey looked at two independent random

samples of $($a$)100$ federal government employers in various lines of functions; and $($b$)120$

provincial government ministry employers by similar functions. They found that $71$ federal

government employers impose a $3-$day per week presence in office compared to $68$ for

provincial government sector employers.

a$)[12$ marks$]$ Test whether employers in these two, federal and provincial government

sectors$,$ are equally likely to impose a three days per week presence in office. Use the

critical value and p$-$value approach and a $5\%$ level of significance. Make sure to include all

the steps of hypotheses $($hypotheses$,$ decision zone, critical value and p$-$value, conclusion

with justification$).$ Show your manual calculations and provide a StatCrunch output

confirming your results. $[$Notation suggestion: Let $p_1$ be the proportion of federal public

service and $p_2$ the proportion of provincial service employers who impose a three$-$day per

week presence in office$].($Manual calculations and StatCrunch$)$

b$)[6$ marks$]$ Calculate and interpret an appropriate $95\%$ confidence interval for the true

difference between the proportions of federal and provincial government public service

employers who impose a three days per week presence in office. Show your Excel or

StatCrunch output for verification. $($Manual calculations and StatCrunch$)$

c$)[2$ marks$]$ Explain how your conclusions in parts a$)$ and b$)$ above are consistent.