Question

In 2010 as part of the General Social Survey, 1295 randomly selected American adults responded to this question: "Some countries are doing more to protect the environment than other countries are. In general, do you think that America is doing more than enough, about the right amount, or too little?" Of the respondents, 473 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 1

Answer: $(0.2838,\ 0.3342)$

Step-by-step explanation:

The confidence interval for proportion (p) is given by :-

$\hat{p}\pm z^*\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$

, where$\hat{p}$ = Sample proportion

n= sample size.

z* = Critical z-value.

Let p be the true proportion of all American adults who feel that America is doing about the right amount to protect the environment.

Given : n= 1295

x= 473

$\hat{p}=\dfrac{x}{n}=\dfrac{400}{1295}\approx0.309$

Confidence interval = 95%

The critical value(two-tailed) for 95% confidence interval : z*=1.96

Then, 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. :-

$0.309\pm (1.96)\sqrt{\dfrac{0.309(1-0.309)}{1295}}\\\\=0.05\pm (1.96)(0.01284054269)\\\\\approx0.309\pm0.0252\\\\=(0.309-0.0252,\ 0.309+0.0252)\\\\=(0.2838,\ 0.3342)$

Hence, the required 95% confidence interval for the proportion : $(0.2838,\ 0.3342)$