# Central limit theorem problems

#### Lort

1-A box has a large number of items which have mean weight = 100 gm, and standard deviation = 30 gm
If 10 items are chosen at random, what's the probability that at least 2 items weigh more than 100 gms?

2-
A- The temperatures of summer weather this year are normally distributed with mean 38 degrees, and standard deviation = 4

If 50 days are selected at random what's the probability that the number of those days with temperatures more than 43.1 is at least 10 days?

#### romsek

MHF Helper
1) use the normal approximation to the binomial distribution. Whatever text you got the problem from should show you how to translate the parameters

2) first compute
$p=Pr[\text{temperature }>43.1]$. You can get this from the normal distribution given.

$P_{10+} = Pr[\text{at least 10 days out of 50 have temperatures }> 43.1] = 1 - P[\text{9 or less days out of 50 have temperatures }> 43.1]$

$P_{10+} = 1 - \sum \limits_{k=0}^9 ~\dbinom{50}{k} p^k (1-p)^{50-k}$

You can either compute that directly, or you can use the normal approximation

#### Lort

1. Not sure about that, I was only taught binomial approximation to normal, i'll just leave it.
2. Aha I see, so you considered it a binomial problem where success probability is the prob that temperature > 43.1. Thanks!

#### Jhevon

MHF Helper
1. I'm with @romsek here. It is very usually the case that we use the normal to approximate the binomial and not the other way around. If you think about it, this makes sense, since using the normal distribution table is a lot easier than doing the binomial calculations!

2. Yes, he used the normal distribution to find the probability of success in the binomial problem. The temperature on a particular day would be assumed to be independent here.

#### romsek

MHF Helper
1. Not sure about that, I was only taught binomial approximation to normal, i'll just leave it.
2. Aha I see, so you considered it a binomial problem where success probability is the prob that temperature > 43.1. Thanks!
This is an excellent 30 second tutorial on the normal approximation to the binomial distribution

Jhevon

#### romsek

MHF Helper
This is an excellent 30 second tutorial on the normal approximation to the binomial distribution
Actually looking at the problem a bit more carefully, it needs to be done identically to the second problem.
First find $p$ then use that and $n=10$ to make a binomial distribution and evaluate that as $1 - P[0]- P[1]$

Jhevon