Question

According to a 2010 study conducted by the Toronto-based social media analytics firm Sysomos, 71% of all tweets get no reaction. That is, these are tweets that are not replied to or retweeted (Sysomos website, January 5, 2015). Suppose we randomly select 100 tweets.

a.What is the expected number of these tweets with no reaction?
b.What are the variance and standard deviation for the number of these tweets with no reaction?

Answer 1

a)
`E(X) = 71`

b)

Variance:
`V(X) = 20.59`

Standard deviation:
`√(V(X)) = 4.58`

Explanation:

For each tweet, there are only two possible outcomes. Either it got a reaction, or it did not. So we use the binomial probability distribution to solve this problem.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

`E(X) = np`

The variance of the binomial distribution is:

`V(X) = np(1-p)`

The standard deviation of the binomial distribution is:

`√(V(X)) = √(np(1-p))`

In this problem, we have that:

`n = 100, p = 0.71`

a.What is the expected number of these tweets with no reaction?

`E(X) = np = 100*0.71 = 71`

b.What are the variance and standard deviation for the number of these tweets with no reaction?

Variance:

`V(X) = np(1-p) = 100*0.71*0.29 = 20.59`

Standard deviation:

`√(V(X)) = √(np(1-p)) = √(100*0.71*0.29) = 4.58`