Question

7. Suppose that we are given that there exists a random variable X whose distribution is binomial with parameters are n = 500 and p= 0.300, and here X represents the number of desired outcomes of the random experiment and n-X is the number of undesired outcomes obtained from a random experiment of n independent trials. From this random experiment   p̂ sample proportion is found as X/n. (Round your answers to 3 decimal places in all parts.)

1. What is the expected value of this statistic?

Expected value:

2. Within what limits will 99% of the sample proportion occur?

Value of sample proportion is between …................ and …................

3. What is the probability that sample proportion exceed 0.353

Probability:

4. What is the probability that sample proportion less than 0.247

Probability:

Answer 1

Sampling distribution of p̂ is approximately normal if np >=10 and n (1-p) >= 10
n * p = 500 * 0.3 = 150
n * (1 - p ) = 500 * (1 - 0.3) = 350
Mean = = p = 0.3
Standard deviation =    = 0.020494

Part a)

Expected value = = p = 0.3

Part b)

X ~ N ( µ = 0.3 , σ = 0.020494 )
P ( a < X < b ) = 0.99
Dividing the area 0.99 in two parts we get 0.99/2 = 0.495
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.495
Area above the mean is b = 0.5 + 0.495
Looking for the probability 0.005 in standard normal table to calculate Z score = -2.58
Looking for the probability 0.995 in standard normal table to calculate Z score = 2.58
Z = ( X - µ ) / σ
-2.58 = ( X - 0.3 ) / 0.020494
a = 0.247
2.58 = ( X - 0.3 ) / 0.020494
b = 0.353

P ( 0.247 < X < 0.353 ) = 0.99

Part c)

X ~ N ( µ = 0.3 , σ = 0.020494 )
P ( X > 0.353 ) = 1 - P ( X < 0.353 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 0.353 - 0.3 ) / 0.020494
Z = 2.59
P ( ( X - µ ) / σ ) > ( 0.353 - 0.3 ) / 0.020494 )
P ( Z > 2.59 )
P ( X > 0.353 ) = 1 - P ( Z < 2.59 )
P ( X > 0.353 ) = 1 - 0.9952
P ( X > 0.353 ) = 0.005

Part d)

X ~ N ( µ = 0.3 , σ = 0.020494 )
P ( X < 0.247 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 0.247 - 0.3 ) / 0.020494
Z = -2.59
P ( ( X - µ ) / σ ) < ( 0.247 - 0.3 ) / 0.020494 )
P ( X < 0.247 ) = P ( Z < -2.59 )
P ( X < 0.247 ) = 0.005