Question

A hot topic in the news is the possible replacement of the Brent Spence Bridge which connects northern Kentucky to Cincinnati. If the bridge is replaced, it will likely become a toll bridge. A 2013 survey of 1619 northern Kentucky AAA members was asked their opinion on instituting a toll on the Brent Spence Bridge. Of those surveyed, 955 were opposed to tolls.

• With 96% confidence, we estimate between % and % of all northern Kentucky residents are opposed to tolls on the Brent Spence Bridge. (After converting to percentages, round the limits of your interval to two decimal places.)

• Does the interval above provide evidence that more than 60% of northern Kentucky residents are opposed to tolls?
  • Yes because both limits of the interval are greater than 60%.
  • No because both limits of the interval are below 60%.
  • No because the upper limit of the interval is greater than 60% while the lower limit is less than 60%.
  • Yes because the upper limit of the interval is greater than 60%.

• State legislators would like to conduct a new survey to update the estimate for 2016. They would like to increase the level of confidence to 98% with a margin of error of 1.97%. If they were to use the estimate in 2013 as a starting point, how many people should be included in the 2015 sample?
  • n =

Answer 1

Solution :

Given that,

n = 1619

x = 955

Point estimate = sample proportion = = x / n = 955 / 1619 = 0.590

1 - = 1 - 0.590 = 0.410

At 96% confidence level

= 1 - 96%

=1 - 0.96 =0.04

/2 = 0.02

Z/2 = Z_{0.02 = 2.054}

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 2.054 (((0.590 * 0.410) / 1619)

= 0.0251

A 96% confidence interval for population proportion p is ,

± E

= 0.590 ± 0.0251

= ( 0.5649, 0.6151 )

= ( 56.49%, 61.51% )

No because the upper limit of the interval is greater than 60% while the lower limit is less than 60%.

b) margin of error = E = 1.97% = 0.0197

At 98% confidence level

= 1 - 98%

=1 - 0.98 =0.02

/2 = 0.01

Z/2 = 2.326

sample size = n = (Z _{/ 2} / E )² * * (1 - )

= (2.326 / 0.0197)² * 0.590 * 0.410

= 3372.27

sample size = n = 3373