Question

Two years ago, a political party ASR 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 p^ denotes the (random) sample proportion of the ASR voters.a)Suppose that n = 500. Determine an interval that with probability 0.90 will contain the (not yet observed) realisation of p^ if p were the same as two years ago.b)Find random bounds (depending on p) that will include the proportion p with probability 0.95.Express the width of the accompanying interval in terms of p and n.

Answer 1

a) We need to construct the $90\%$ confidence interval for the population proportion. We have been provided with the following information about the sample proportion:

$\begin{matrix} Sample\ proportion & \hat{p}=0.098 \\ Sample\ Size & n=500 \end{matrix}$

The critical value for $\alpha=0.05$ is $z_c=z_{1-\alpha/2}=1.645.$ The corresponding confidence interval is computed as shown below:

$$CI(Proportion)=$$

$$=\big(\hat{p}-z_c\sqrt{{\hat{p}(1-\hat{p}) \over n}},\hat{p}+z_c\sqrt{{\hat{p}(1-\hat{p}) \over n}} \big)$$

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

$$=(0.076,0.120)$$

b) We compute the standard errors using the formula: 

$$SE=\sqrt{{p(1-p) \over n}}$$

$$width=p+z_c\sqrt{{p(1-p) \over n}}-(p-z_c\sqrt{{p(1-p) \over n}})=$$

$$=2z_c\sqrt{{p(1-p) \over n}}=2z_c\times SE$$

$$z_c=z_{1-\alpha/2}=1.96$$

$$width=3.92\sqrt{{p(1-p) \over n}}$$