A population proportion is 02 A sample of size 250 will be t

A population proportion is 0.2. A sample of size 250 will be taken and the sample proportion will be used to estimate the population proportion. Round your answers to four decimal places. a. What is the probability that the sample proportion will be within ±0.03 of the population proportion? b. What is the probability that the sample proportion will be within ±0.07 of the population proportion?

Solution

Normal Distribution
Proportion ( P ) =0.2
Standard Deviation ( sd )= Sqrt (P*Q /n) = Sqrt(0.2*0.8/250)
Normal Distribution = Z= X- u / sd ~ N(0,1)                  

a)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 0.17) = (0.17-0.2)/0.0253
= -0.03/0.0253 = -1.1858
= P ( Z <-1.1858) From Standard Normal Table
= 0.11786
P(X < 0.23) = (0.23-0.2)/0.0253
= 0.03/0.0253 = 1.1858
= P ( Z <1.1858) From Standard Normal Table
= 0.88214
P(0.17 < X < 0.23) = 0.88214-0.11786 = 0.7643                  
              
b)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 0.13) = (0.13-0.2)/0.0253
= -0.07/0.0253 = -2.7668
= P ( Z <-2.7668) From Standard Normal Table
= 0.00283
P(X < 0.27) = (0.27-0.2)/0.0253
= 0.07/0.0253 = 2.7668
= P ( Z <2.7668) From Standard Normal Table
= 0.99717
P(0.13 < X < 0.27) = 0.99717-0.00283 = 0.9943