Question

Suppose a simple random sample of size nequals200 is obtained from a population whose size is Upper N equals 15 comma 000 and whose population proportion with a specified characteristic is p equals 0.6 . ​(a) Describe the sampling distribution of ModifyingAbove p with caret. Choose the phrase that best describes the shape of the sampling distribution below. A. Approximately normal because n less than or equals 0.05 Upper N and np left parenthesis 1 minus p right parenthesis less than 10. B. Approximately normal because n less than or equals 0.05 Upper N and np left parenthesis 1 minus p right parenthesis greater than or equals 10. C. Not normal because n less than or equals 0.05 Upper N and np left parenthesis 1 minus p right parenthesis greater than or equals 10. D. Not normal because n less than or equals 0.05 Upper N and np left parenthesis 1 minus p right parenthesis less than 10. Determine the mean of the sampling distribution of ModifyingAbove p with caret. mu Subscript ModifyingAbove p with caret Baseline equals nothing ​(Round to one decimal place as​ needed.) Determine the standard deviation of the sampling distribution of ModifyingAbove p with caret. sigma Subscript ModifyingAbove p with caretequals nothing ​(Round to six decimal places as​ needed.) ​(b) What is the probability of obtaining xequals128 or more individuals with the​ characteristic? That​ is, what is ​P(ModifyingAbove p with caretgreater than or equals0.64​)? ​P(ModifyingAbove p with caretgreater than or equals0.64​)equals nothing ​(Round to four decimal places as​ needed.) ​(c) What is the probability of obtaining xequals106 or fewer individuals with the​ characteristic? That​ is, what is ​P(ModifyingAbove p with caretless than or equals0.53​)? ​P(ModifyingAbove p with caretless than or equals0.53​)equals nothing ​(Round to four decimal places as​ needed.)

Answer 1

a) Option - B) Approximately normal because n < 0.05N and np(1 - p) > 10.

= p = 0.6

= sqrt(p(1 - p)/n)

     = sqrt(0.6 * 0.4/200)

     = 0.034641

b) P(> 0.64)

= P(( - )/> (0.64 - )/)

= P(Z > (0.64 - 0.6)/0.034641)

= P(Z > 1.15)

= 1 - P(Z < 1.15)

= 1 - 0.8749

= 0.1251

c) P(< 0.53)

= P(( - )/< (0.53 - )/)

= P(Z < (0.53 - 0.6)/0.034641)

= P(Z < 2.02)

= 0.0217