Question

1. While working for a polling company at C-Span, you are asked  to determine the percentage of the population that feel that the president has done a favorable job in handling the Pandemic.  You will take a SRS of adults in the USA.  We would like to be 95 % confident with a margin of error of 3 points.

1. If the last estimate done by FOX News was known to be 58%, what size of sample would be needed?   
2. Since you believe that the responses will be between 30% and 70% and you are not sure that previous estimate was accurate what estimate should you use? And what sample size would be needed?
3. If you would like to be 99% confident with a margin of error of 3 points and unsure of the last estimate, what sample size would be needed?
4. Comment about the difference between answers of b and c.  Also why would a polling company use 95% rather than 99%?

Write a report to your boss at C-Span, explaining to her what you are doing.

Answer 1

z value for 95% confidence is 1.96

Margin of error E = 3/100 = 0.03

Sample size =

= 1040 (Rounding to nearest integer)

If we do not know the proportion estimate, we assume p = 0.5 so as to get the conservative estimate of sample size. This is because the standard error (and margin error) is maximum at p = 0.5 for a given sample size.

Sample size =

= 1067 (Rounding to nearest integer)

For 99% confidence level, z = 2.576

Sample size =

= 1843 (Rounding to nearest integer)

We require larger sample size for 99% confidence level as compared with 95% confidence level. The polling company use 95% to get a smaller confidence interval length which will provide more precise estimate than 99% confidence interval.

If we know the proportion estimate, we need 1040 samples to estimate interval within 3 points with 95% confidence. If we do not know the proportion estimate, we need 1067 samples to estimate interval within 3 points with 95% confidence. If we need to increase the confidence level to 99%, we require a large sample size of 1843.