Complete Question
In 2015 as part of the General Social Survey, 1127 randomly selected American adults responded to this question: “Some countries are doing more to protect the environment than other countries. In general, do you think that America is doing more than enough, about the right amount, or too little?” Of the respondents, 514 replied that America is doing about the right amount. What is the 95 % confidence interval for the proportion of all American adults who feel that America is doing about the right amount to protect the environment.
Answer:
The 95% confidence interval is [tex] 0.427 < p < 0.4852 [/tex]
Step-by-step explanation:
From the question we are told that
The sample size is n = 1127
The number that America is doing the about the right amount is k = 514
Generally the sample proportion of the number that say america is doing about the right amount is mathematically represented as
[tex]\^ p = \frac{k}{n}[/tex]
=> [tex]\^ p = \frac{514 }{ 1127}[/tex]
=> [tex]\^ p = 0.4561[/tex]
From the question we are told the confidence level is 95% , hence the level of significance is
[tex]\alpha = (100 - 95 ) \%[/tex]
=> [tex]\alpha = 0.05[/tex]
Generally from the normal distribution table the critical value of [tex]\frac{\alpha }{2}[/tex] is
[tex]Z_{\frac{\alpha }{2} } = 1.96[/tex]
Generally the margin of error is mathematically represented as
[tex]E = Z_{\frac{\alpha }{2} } * \sqrt{\frac{\^ p (1- \^ p)}{n} } [/tex]
=> [tex]E = 1.96 * \sqrt{\frac{ 0.4561 (1- 0.4561)}{1127} } [/tex]
=> [tex]E = 0.0291 [/tex]
Generally 95% confidence interval is mathematically represented as
[tex]\^ p -E < p < \^ p +E[/tex]
=> [tex] 0.4561 -0.0291 < p < 0.4561 +0.0291 [/tex]
=> [tex] 0.427 < p < 0.4852 [/tex]
