Question

Suppose a simple random sample of size

n=1000

is obtained from a population whose size is

N=1,000,000

and whose population proportion with a specified characteristic is

p=0.22.

Complete parts​ (a) through​ (c) below.

​(a) Describe the sampling distribution of

p. Awnser all the questions correctly awnser part A part B and part C

$8:45 PM Suppose a simple random sample of size n = 1000 is obtained from a population whose size is N = 1,000,000 and whose population proportion with characteristic is p = 0.22. Complete parts (a) through (c) below. (a) Describe the sampling distribution of P. O A. Approximately normal, Hg = 0.22 and 0.0131 OB. Approximately normal, pp = 0.22 and a 20.0002 OC. Approximately normal, = 0.22 and GA 0.0004 (b) What is the probability of obtaining x= 240 or more individuals with the characteristic? P(x2 240) = (Round to four decimal places as needed.) (c) What is the probability of obtaining x = 180 or fewer individuals with the characteristic? P(xs 180) = (Round to four decimal places as needed.) Click to select your answer(s). Δ O

Answer 1

Solution

Given that,

p = 0.22

1 - p = 1 - 0.22 = 0.78

n = 1000

a)  $\mu$_{$\hat p$} = p = 0.22

$\sigma$_{$\hat p$} =  $\sqrt$[p( 1 - p ) / n] = $\sqrt$ [(0.22 * 0.78) / 1000 ] = 0.0131

correct option is = A

b) P( $\hat p$   $\geq$ 0.24) = 1 - P( $\hat p$ $\leq$ 0.24 )

= 1 - P(( $\hat p$ - $\mu$ _{$\hat p$} ) / $\sigma$ _{$\hat p$}  $\leq$ (0.24 - 0.22) / 0.0131)

= 1 - P(z $\leq$ 1.53)

= 1 - 0.9370

= 0.0630

c) P( $\hat p$ $\leq$ 0.18 )

= P[( $\hat p$ - $\mu$ _{$\hat p$} ) / $\sigma$ _{$\hat p$}$\leq$  (0.18 - 0.22) / 0.0131]

= P(z $\leq$ -3.05)

Using z table,

= 0.0011