Question

1. In a poll, 191 out of a sample of 357 likely Connecticut voters indicate they intend to vote to legalize marijuana. Find
   1. A conservative 99% confidence interval for the proportion of all likely Connecticut voters who are in favor of this initiative.
   2. An approximate 95% confidence interval for the same proportion.
   3. An improved 90% confidence interval for the same proportion.

Answer 1

Solution :

Given that,

n = 357

x = 191

Point estimate = sample proportion = = x / n = 191 / 357 = 0.535

1 - = 1 - 0.535 = 0.465

a) At 99% confidence level

= 1 - 99%

=1 - 0.99 =0.01

/2 = 0.005

Z/2 = Z_{0.005 = 2.576}

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

= 2.576 (((0.535 * 0.465) / 357)

= 0.068

A 99% confidence interval for population proportion p is ,

± E   

= 0.535 ± 0.068

= (0.467, 0.603)

b) At 95% confidence level

= 1 - 95%

=1 - 0.95 =0.05

/2 = 0.025

Z/2 = Z_{0.025 = 1.960}

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

= 1.960 (((0.535 * 0.465) / 357)

= 0.052

A 95% confidence interval for population proportion p is ,

± E   

= 0.535 ± 0.052

= (0.483, 0.587)

c) At 90% confidence level

= 1 - 90%

=1 - 0.90 =0.10

/2 = 0.05

Z/2 = Z_{0.05 = 1.645}

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

= 1.645 (((0.535 * 0.465) / 357)

= 0.043

A 90% confidence interval for population proportion p is ,

± E   

= 0.535 ± 0.043

= (0.492, 0.578)