approximate probability that average loss is more than $275

[Pensuave]

An insurance company sees that, in the entire population of homeowners, the mean loss from fire is mu = $250 and the standard deviation of the loss is sigma = $1000. The distribution of losses is strongly right-skewed: many policies have $0 loss, but a few have large losses. If the company sells 10,000 policies, what is the approximate probability that the average loss will be greater than $275?

I'm stuck on this problem and have no clue what to do for it. I can't make sense of the formula that is supplied in my text, and I am not able to find the proper number to look up on the "Table of Standard Normal Probabilities" that is in my text. If someone could explain it in detail, I'd greatly appreciate it.

I was told this by a classmate:

it's just finding the probability that a sample (rather than an individual) will have an average (rather than a value... since the value of an individual is the same as a average of that individual) above a certain amount. So... x = u where x is the mean of the sample chosen, and u is the mean of the population. however, instead of using sigma (standard deviation of the population) use s (standard deviation of the means of samples). s=sigma/(square root of N) where N = the number of individuals in the sample. so you find s, then find how many standard deviations (s) xf is from xo. Then, you can use Table A to find the probability that this event will occur.

But that step that says to find how many standard deviations xf is from xo doesn't make sense to me.

 

[galactus]

Are you sure the standard deviation is 1000?. Seems rather excessive.

 

[Pensuave]

yep, i copied it straight out of my text. it threw me off a bit as well. another problem that i am having which is similar to this one is:

The scores of students on the ACT college entrance examination in 2001 hasd a mean mu=21.0 and standard deviation sigma=4.7. The distribution of scores is only roughly normal.

a) what is the approximate probability that a single student randomly chosen from all those taking the test scores 23 or higher?
b)Now take an SRS of 50 students who took the test. What are the mean and standard deviation of the sample mean score x of the 50 students? What is the approximate probability that the mean score x of these students is 23 or higher?
c) Which of your two Normal probaility calculatons in (A) and (b) is more accurate? Why?

My classmates said that this one was easier to apply the formula to, but I'm still not able to solve it. I'm using N (mu, sigma/√n) but am not sure that I'm placing the right digits into it. I am getting N (21, 4.7).

Again, even if someone can supply me with the formula that I need to use to solve this or the process to solve this, I'd appreciate it.

 

[JakeD]

The point of these problems is to apply the Central Limit Theorem. It says that if you take samples of size n from a population with mean mu and standard deviation sigma, the sample mean will be approximately normally distributed with mean mu and standard deviation sigma/sqrt(n). The population distribution may be highly non-normal, but for large n the sample means will still be approximately normally distributed.