Question

A randomized telephone survey of adults in the United States determines that a certain number out of 504 respondents have one or more pets. They use p^ to construct a 95% confidence interval to estimate the proportion p of American adults who have pets to be (0.56137,0.60529).
How many respondents said they have one or more pets?
283
294
305
290

Answer 1

Answer:

$\hat p=\frac{0.56137+0.60529}{2}= 0.58333$

The critical value for 95% of confidence is $z_{\alpha/2}=1.96$

And since we know the total and the proportion is defined as:

$\hat p=\frac{X}{n}$

If we solve for X (number of people who say yes) we got:

$X= 0.58333* 504= 293.998 \approx 294$

And the best answer would be:

294

Step-by-step explanation:

We know that the confidence interval for the proportion is given by:

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

The margin of error for the proportion interval is given by this formula:  

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

The margin of error can be calculated with this formula:

$ME=\frac{0.60529- 0.56137}{2}= 0.02196$

And the estimation for the true proportion is:

$\hat p=\frac{0.56137+0.60529}{2}= 0.58333$

The critical value for 95% of confidence is $z_{\alpha/2}=1.96$

And since we know the total and the proportion is defined as:

$\hat p=\frac{X}{n}$

If we solve for X (number of people who say yes) we got:

$X= 0.58333* 504= 293.998 \approx 294$

And the best answer would be:

294