# Central Limit Theorem

• Oct 3rd 2008, 04:01 PM
MathGeek06
Central Limit Theorem
A website posts movie ratings which are computed by taking the average of the ratings posted by users on the website. Each user rates a particular movie on a scale of 1 to 10. Assume that the ratings by the users are random variables which are identically and independently distributed. The variance of this distribution is known to be 1.5. Each movie is typically rated by about 100,000 users. The latest blockbuster comes out on Friday and the website wants to post the rating for the movie as soon as possible and does not want to wait till all its users post their ratings.

a) What is the minimum number of ratings that it needs so that the average calculated from this sample will be within 0.2 of the actual mean with 95% probability?

b) You are told that for this movie, the mean of the individual random variables is 8.5. What is the number of samples you will take so that the average rating of the sample is above 8.1 with 80% probability ?
• Oct 4th 2008, 12:36 AM
CaptainBlack
Quote:

Originally Posted by MathGeek06
A website posts movie ratings which are computed by taking the average of the ratings posted by users on the website. Each user rates a particular movie on a scale of 1 to 10. Assume that the ratings by the users are random variables which are identically and independently distributed. The variance of this distribution is known to be 1.5. Each movie is typically rated by about 100,000 users. The latest blockbuster comes out on Friday and the website wants to post the rating for the movie as soon as possible and does not want to wait till all its users post their ratings.

a) What is the minimum number of ratings that it needs so that the average calculated from this sample will be within 0.2 of the actual mean with 95% probability?

b) You are told that for this movie, the mean of the individual random variables is 8.5. What is the number of samples you will take so that the average rating of the sample is above 8.1 with 80% probability ?

Central limit theorem (Hand-Waving version): if $Z=(X_1, ..., X_n)/n$ is a random variable with the $X_i$'s independent identicaly distributed random variables (iirv) with mean $\mu$ and variance $\sigma^2$, then:

1. $E(Z)=\mu$

2. ${\rm{Var}}(Z)=\sigma^2/n$

3. As $n$ becomes large the distribution of $Z$ becomes more and more nearly normal (defined in some appropriate sense)

So in your problem you are going to assume that you have enough individual ratings so that the distribution of the average rating may be treated as normal. So now you have enough information to represent the distribution of the average rating when it is comounded from $n$ individual ratings.

RonL