Question

From a random sample of 1,201 Americans, it was discovered that 1,139 of them lived in neighborhoods with acceptable levels of carbon monoxide. Given the value of p-hat and LaTeX: \sigma σ σp-hat found in the above two questions, and with the knowledge that zLaTeX: _{\frac{\alpha}{2}} α 2 α 2=2.575 for LaTeX: \alpha α α=.005, Construct a 99% confidence interval for the proportion of Americans who live in neighborhoods with acceptable levels of carbon monoxide. Choose the best answer. Group of answer choices ( .897 , .999 ) ( .936 , .960 ) ( .938 , .958 )

( .933 , .963 )

Answer 1

Solution:

Let denotes the sample proportion.

     = x/n = 1139/1201 =  0.948

Our aim is to construct 99% confidence interval.

c = 0.99

= 1 - c = 1- 0.99 = 0.01

  /2 = 0.005 and 1- /2 = 0.995

Search the probability 0.995 in the Z table and see corresponding z value

  =  2.575

Now , the margin of error is given by

E =   *   = 2.575  * [0.948 *(1 - 0.948)/1201] = 0.015

Now the confidence interval is given by

( - E)   ( + E)

(0.948 - 0.015)   (0.948 + 0.015)

0.933   0.963

Required 99% Confidence Interval is

(0.933 , 0.963)