Question

2years ago a political party AS received 9:8% of the votes in an election. To study the current political preferences a statistical research institute plans to organise a poll by the end of the present year. In this study n voters will be interviewed about the political party they prefer.Below, p, denotes the proportion of voters that would vote ASR if the elections were held now. Furthermore, bp denotes the random sample proportion of the AS voters.a) Suppose that n=500.Determine an interval that with probability 0:90 will contain the not yet observed realisation of bp if p were the same 2 years ago.b)The interval in part b will be the starting point to create at the end of the current year when the data are observed an interval that will probably contain the proportion p. How large should the sample size be to Obtain an interval width about 0:02?

Answer 1

a) $90\%CI=(p-z_{0.05}\sqrt{\frac{p(1-p)}{n}},p+z_{0.05}\sqrt{\frac{p(1-p)}{n}})=$

$=(0.098-1.645\sqrt{\frac{0.098(1-0.098)}{500}},0.098+1.645\sqrt{\frac{0.098(1-0.098)}{500}})=$

$=(0.0761,0.1199).$

b) $E=z_{0.05}\sqrt{\frac{p(1-p)}{n}}.$

$n=(\frac{z_{0.05}}{E})^2p(1-p)=(\frac{1.645}{0.01})^20.098(1-0.098)=2393.$