We have n = 100 many random variables Xi ’s, where the Xi ’s are independent and identically distributed Bernoulli random variables with p = 0.5 (E(Xi)=p and var(Xi)=p(1-p)). a. What distribution does Pn i=1 Xi follow exactly (sum of independent Bernoulli random variables)? State the type of distribution and what the parameter is. b. Using the central limit theorem, what approximate distribution does X¯ = 1 n Pn i=1 Xi follow? State i the type and the mean and variance. c. What is the approximate probability that X¯ = 1 n Pn i=1 Xi is greater than 0.5? Write down the integral using proper notation and use empirical rule or pnorm function in R to obtain numeric answer.

(a) The distribution of is a Binomial distribution.

(b) The sampling distribution of the sample mean will be approximately normal.

(c) The value of is 0.50.

Step-by-step explanation:

It is provided that random variables are independent and identically distributed Bernoulli random variables with p = 0.50.

The random sample selected is of size, n = 100.

(a)

Theorem:

Let be independent Bernoulli random variables, each with parameter p, then the sum of of thee random variables, is a Binomial random variable with parameter n and p.

Thus, the distribution of is a Binomial distribution.

(b)

According to the Central Limit Theorem if we have an unknown population with mean μ and standard deviation σ and appropriately huge random samples (n > 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. n = 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:

*Use a z-table.

Thus, the value of is 0.50.

14.3

step-by-step explanation:

add all the numbers together.

9 + 12 + 15 + 16 + 17 + 17 = 86

now, divide 86 by how many numbers we have: 6.

86 / 6 = 14.3