Question

Is the environment a major issue with Americans? To answer that question, a researcher conducts a survey of 1255 randomly selected Americans. Suppose 735 of the sampled people replied that the environment is a major issue with them. Construct a 95% confidence interval to estimate the proportion of Americans who feel that the environment is a major issue with them. What is the point estimate of this proportion?

Answer 1

Answer:

The best point of estimate is given by:

$\hat p =\frac{X}{n}= \frac{735}{1255}=0.586$

$0.586 - 1.96\sqrt{\frac{0.586(1-0.586)}{1255}}=0.559$

$0.586 + 1.96\sqrt{\frac{0.586(1-0.586)}{1255}}=0.613$

And we have 95% of confidence that the true proportion of Americans who feel that the environment is a major issue with them is between 0.559 and 0.613

Step-by-step explanation:

The interest is the real proportion of Americans who feel that the environment is a major issue with them. The best point of estimate is given by:

$\hat p =\frac{X}{n}= \frac{735}{1255}=0.586$

Confidence interval

The confidence interval for the true proportion of interest is given by this:  

$\hat p \pm z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}$

We want an interval at 95% of confidence, so then the significance level is $\alpha=1-0.95=0.05$ and $\alpha/2 =0.025$. And the critical value for this case are:

$z_{\alpha/2}=-1.96, z_{1-\alpha/2}=1.96$

Replacing into the confidence interval formula we got:

$0.586 - 1.96\sqrt{\frac{0.586(1-0.586)}{1255}}=0.559$

$0.586 + 1.96\sqrt{\frac{0.586(1-0.586)}{1255}}=0.613$

And we have 95% of confidence that the true proportion of Americans who feel that the environment is a major issue with them is between 0.559 and 0.613

Answer 2

Answer:

The point estimate of this proportion is $\pi = 0.5857$

The 95% confidence interval to estimate the proportion of Americans who feel that the environment is a major issue with them is (0.5584, 0.6130).

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of $\pi$, and a confidence level of $1-\alpha$, we have the following confidence interval of proportions.

$\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}$

In which

z is the zscore that has a pvalue of $1 - \frac{\alpha}{2}$.

For this problem, we have that:

$n = 1255, \pi = \frac{735}{1255} = 0.5857$

The point estimate of this proportion is $\pi = 0.5857$

95% confidence level

So $\alpha = 0.05$, z is the value of Z that has a pvalue of $1 - \frac{0.05}{2} = 0.975$, so $Z = 1.96$.

The lower limit of this interval is:

$\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.5857 - 1.96\sqrt{\frac{0.5857*0.4143}{1255}} = 0.5584$

The upper limit of this interval is:

$\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.5857 + 1.96\sqrt{\frac{0.5857*0.4143}{1255}} = 0.6130$

The 95% confidence interval to estimate the proportion of Americans who feel that the environment is a major issue with them is (0.5584, 0.6130).