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

The best point of estimate is given by:

`\hat p =(X)/(n)= (735)/(1255)=0.586`

`0.586 - 1.96\sqrt{(0.586(1-0.586))/(1255)}=0.559`

`0.586 + 1.96\sqrt{(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

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 =(X)/(n)= (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{(\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{(0.586(1-0.586))/(1255)}=0.559`

`0.586 + 1.96\sqrt{(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

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).

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{(\pi(1-\pi))/(n)}`

In which

z is the zscore that has a pvalue of
`1 - (\alpha)/(2)`.

For this problem, we have that:

`n = 1255, \pi = (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 - (0.05)/(2) = 0.975`, so
`Z = 1.96`.

The lower limit of this interval is:

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

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.5857 + 1.96\sqrt{(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).