Answer:
(a) The distribution of  is a Binomial distribution.
 is a Binomial distribution.
(b) The sampling distribution of the sample mean will be approximately normal.
(c) The value of  is 0.50.
 is 0.50.
Step-by-step explanation:
It is provided that random variables  are independent and identically distributed Bernoulli random variables with <em>p</em> = 0.50.
 are independent and identically distributed Bernoulli random variables with <em>p</em> = 0.50.
The random sample selected is of size, <em>n</em> = 100.
(a)
Theorem:
Let  be independent Bernoulli random variables, each with parameter <em>p</em>, then the sum of of thee random variables,
 be independent Bernoulli random variables, each with parameter <em>p</em>, then the sum of of thee random variables,  is a Binomial random variable with parameter <em>n</em> and <em>p</em>.
 is a Binomial random variable with parameter <em>n</em> and <em>p</em>.
Thus, the distribution of  is a Binomial distribution.
 is a Binomial distribution.
(b)
According to the Central Limit Theorem if we have an unknown population with mean <em>μ</em> and standard deviation <em>σ</em> and appropriately huge random samples (<em>n</em> > 30) are selected from the population with replacement, then the distribution of the sample mean will be approximately normally distributed.  
The sample size is large, i.e. <em>n</em> = 100 > 30.
So, the sampling distribution of the sample mean will be approximately normal.
The mean of the distribution of sample mean is given by,
 
And the standard deviation of the distribution of sample mean is given by,

(c)
Compute the value of  as follows:
 as follows:

                     
*Use a <em>z</em>-table.
Thus, the value of  is 0.50.
 is 0.50.