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One of the questions Rasmussen Reports included on a 2018 survey of 2,500 likely voters asked...

One of the questions Rasmussen Reports included on a 2018 survey of 2,500 likely voters asked if the country is headed in right direction. Representative data are shown in the DATAfile named RightDirection. A response of Yes indicates that the respondent does think the country is headed in the right direction. A response of No indicates that the respondent does not think the country is headed in the right direction. Respondents may also give a response of Not Sure.

(a)What is the point estimate of the proportion of the population of respondents who do think that the country is headed in the right direction? (Round your answer to four decimal places.)

(b)At 95% confidence, what is the margin of error for the proportion of respondents who do think that the country is headed in the right direction? (Round your answer to four decimal places.)

(c)What is the 95% confidence interval for the proportion of respondents who do think that the country is headed in the right direction? (Round your answers to four decimal places.)

___to ___

(d)What is the 95% confidence interval for the proportion of respondents who do not think that the country is headed in the right direction? (Round your answers to four decimal places.)

____ to ____

(e)Which of the confidence intervals in parts (c) and (d) has the smaller margin of error? Why?

The confidence interval in part (c) has a (Smaller or Larger) margin of error than the confidence interval in part (d). This is because the sample proportion of respondents who do think that the country is headed in the right direction is  (closer to .5 / closer to 1 / farther from .5 / farther from 1) than the sample proportion of respondents who do not think that the country is headed in the right direction.

Dataset:

553 - No

70 - Not Sure

384 - Yes

Solutions

Expert Solution

Total response = 553+70+384 = 1007

(a) Point estimate of the proportion of the population of respondents who do think that the country is headed in the right direction = 384/1007 = 0.3813

(b) At α = 1-0.95 = 0.05, critical value, z_c = NORM.S.INV(0.05/2) = 1.960

margin of error for the proportion of respondents who do think that the country is headed in the right direction :

Margin of error, E = z*√( p̄ *(1- p̄ )/n) =1.96 *√(0.3813*0.6187/1007) = 0.0300

(c) 95% confidence interval for the proportion of respondents who do think that the country is headed in the right direction:

Lower Bound = p̄ - E = 0.3813- 0.0300 = 0.3513

Upper Bound = p̄ + E = 0.3813 + 0.0300 = 0.4113

0.3513 to 0.4113

(d) Point estimate for the proportion of respondents who do think that the country is headed in the right direction = 553/1007 = 0.5492

Margin of error, E = z*√( p̄ *(1- p̄ )/n) =1.96 *√(0.5492*0.4508/1007) = 0.0307

95% confidence interval for the proportion of respondents who do not think that the country is headed in the right direction:

Lower Bound = p̄ - E = 0.5492 - 0.0307 = 0.5185

Upper Bound = p̄ + E = 0.5492 + 0.0307 = 0.5799

0.5185 to 0.5799

(e) The confidence interval in part (c) has a Smaller margin of error than the confidence interval in part (d). This is because the sample proportion of respondents who do think that the country is headed in the right direction is farther from .5 than the sample proportion of respondents who do not think that the country is headed in the right direction.

milcah answered 8 months ago
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