Answer:
1537 voters should be polled.
Explanation:
In a sample with a number n of people surveyed with a probability of a success of
, and a confidence level of
, we have the following confidence interval of proportions.
![\pi \pm z\sqrt{(\pi(1-\pi))/(n)}](https://img.qammunity.org/2022/formulas/mathematics/college/xaspnvwmqbzby128e94p45buy526l3lzrv.png)
In which
z is the zscore that has a pvalue of
.
The margin of error is:
![M = z\sqrt{(\pi(1-\pi))/(n)}](https://img.qammunity.org/2022/formulas/mathematics/college/nqm1cetumuawgnf21cjwekd4pqalhffs6t.png)
95% confidence level
So
, z is the value of Z that has a pvalue of
, so
.
20% of voters are in favor of certain legislation.
This means that
![\pi = 0.2](https://img.qammunity.org/2022/formulas/mathematics/college/55ah7xygmetuvs2xc4zy2qtyaodwxtm0h5.png)
How many voters should be polled in order that the probability is at least .95 that fA(n) differs from 0.20 by less than 0.02
This is n for which M = 0.02. So
![M = z\sqrt{(\pi(1-\pi))/(n)}](https://img.qammunity.org/2022/formulas/mathematics/college/nqm1cetumuawgnf21cjwekd4pqalhffs6t.png)
![0.02 = 1.96\sqrt{(0.2*0.8)/(n)}](https://img.qammunity.org/2022/formulas/mathematics/college/swa5mgxzjw8u315pay4vprxo6wrop7mi0w.png)
![0.02√(n) = 1.96√(0.2*0.98)](https://img.qammunity.org/2022/formulas/mathematics/college/w2bok7ti85lzb45o8d7b9zftwoquc7izj0.png)
![√(n) = (1.96√(0.2*0.98))/(0.02)](https://img.qammunity.org/2022/formulas/mathematics/college/92gh81m65atmlxc5zend39a9l4kw00iybz.png)
![(√(n))^2 = ((1.96√(0.2*0.98))/(0.02))^2](https://img.qammunity.org/2022/formulas/mathematics/college/r5ooque3wb036q6k1eofaj1y3pgv0q0rih.png)
![n = 1536.6](https://img.qammunity.org/2022/formulas/mathematics/college/wdh3ml9pxge8o8zqj7to5mw00xckscg2ac.png)
Rounding up
1537 voters should be polled.