Question

Suppose a simple random sample of size nequals1000 is obtained from a population whose size is Nequals1 comma 000 comma 000 and whose population proportion with a specified characteristic is p equals 0.44 . Complete parts​ (a) through​ (c) below. ​(a) Describe the sampling distribution of ModifyingAbove p with caret. A. Approximately​ normal, mu Subscript ModifyingAbove p with caretequals0.44 and sigma Subscript ModifyingAbove p with caretalmost equals0.0005 B. Approximately​ normal, mu Subscript ModifyingAbove p with caretequals0.44 and sigma Subscript ModifyingAbove p with caretalmost equals0.0002 C. Approximately​ normal, mu Subscript ModifyingAbove p with caretequals0.44 and sigma Subscript ModifyingAbove p with caretalmost equals0.0157 ​(b) What is the probability of obtaining xequals460 or more individuals with the​ characteristic? ​P(xgreater than or equals460​)equals nothing ​(Round to four decimal places as​ needed.) ​(c) What is the probability of obtaining xequals410 or fewer individuals with the​ characteristic? ​P(xless than or equals410​)equals nothing ​(Round to four decimal places as​ needed.)

Answer 1

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 = 0.44 . Complete parts​ (a) through​ (c) below.

​(a) Describe the sampling distribution of ModifyingAbove p with caret.

A.)Approximately​ normal, mu Subscript ModifyingAbove p with caretequals0.44 and sigma Subscript ModifyingAbove p with caretalmost equals0.0002

B.)Approximately​ normal, mu Subscript ModifyingAbove p with caretequals0.44 and sigma Subscript ModifyingAbove p with caretalmost equals0.0005

C.)Approximately​ normal, mu Subscript ModifyingAbove p with caretequals0.44 and sigma Subscript ModifyingAbove p with caretalmost equals0.0157

​(b) What is the probability of obtaining xequals480 or more individuals with the​ characteristic?

​P(xgreater than or equals480​)equals

nothing ​(Round to four decimal places as​ needed.)

​(c) What is the probability of obtaining xequals410 or fewer individuals with the​ characteristic?

​P(xless than or equals410​)equals

nothing ​(Round to four decimal places as​ needed.)

Answer:

a) Option C.

b) 0.1021

c) 0.0280

Explanation:

Given:

Sample size, n = 1000

p' = 0.44

a) up' = p' = 0.44

The sampling distribution will be:

`\sigma _p' = \sqrt{(p' (1 - p'))/(n)}`

`= \sqrt{(0.44 (1 - 0.44))/(1000)}`

`= \sqrt{(0.44*0.56)/(1000)} = 0.0157`

Option C is correct.

b) The probability when x ≥ 460

`P' = (x)/(n) = (460)/(1000) = 0.46`

p'(P ≥ 0.46)

`1 - P = ((p' - up'))/(\sigma _p') < (0.46 - 0.44)/(0.0157)`

`= 1-P( Z < 1.27)`

From the normal distribution table

NORMSDIST(1.27) = 0.898

1-0.8979 = 0.1021

Therefore, the probability = 0.102

c) x ≤ 410

`P' = (x)/(n) = (410)/(1000) = 0.41`

p'(P ≤ 0.41)

`P = ((p' - up'))/(\sigma _p') < (0.41 - 0.44)/(0.0157)`

`= P( Z < - 1.9108)`

From the normal distribution table

NORMSDIST(-1.9108) = 0.0280

Probability = 0.0280