# Central Limit Theorem

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• Jun 17th 2009, 07:56 PM
shinn
Central Limit Theorem
Hi,

I have a problem regarding CLT:

"The lengths of mass produced nails are normally distributed with a mean of \$\displaystyle 3cm \$ and variance of \$\displaystyle 0.01 cm^2\$. 16 nails are randomly chosen and laid end to end.

What is the probability that its length exceeds 48.5cm?"

This is my working out, I'm not sure what i am doing wrong:

Let \$\displaystyle T_{16} = X_1 + X_2 + .... + X_{16}\$, where \$\displaystyle X_i\$ are identically and independently distributed variables.

So, each of \$\displaystyle X_i \$~\$\displaystyle N(3,0.01^2) \$
Thus, \$\displaystyle T_n \$~ \$\displaystyle N(48, 0.04^2)\$

We need to find \$\displaystyle P(T_{16} > 48.5)\$. So
\$\displaystyle P(T_{16} > 48.5) = 1 - P( T_{16} < 48.5) = 1 - P( Z < (48.5 - 48) / (0.04) ) = 1 - P( Z < 12.5)\$

Then, I'm stuck here. Any help will be greatly appreciated thanks(Itwasntme)
• Jun 17th 2009, 08:02 PM
mr fantastic
Quote:

Originally Posted by shinn
Hi,

I have a problem regarding CLT:

"The lengths of mass produced nails are normally distributed with a mean of \$\displaystyle 3cm \$ and variance of \$\displaystyle 0.01 cm^2\$. 16 nails are randomly chosen and laid end to end.

What is the probability that its length exceeds 48.5cm?"

This is my working out, I'm not sure what i am doing wrong:

Let \$\displaystyle T_{16} = X_1 + X_2 + .... + X_{16}\$, where \$\displaystyle X_i\$ are identically and independently distributed variables.

So, each of \$\displaystyle X_i \$~\$\displaystyle N(3,0.01^2) \$
Thus, \$\displaystyle T_n \$~ \$\displaystyle N(48, 0.04^2)\$

We need to find \$\displaystyle P(T_{16} > 48.5)\$. So
\$\displaystyle P(T_{16} > 48.5) = 1 - P( T_{16} < 48.5) = 1 - P( Z < (48.5 - 48) / (0.04) ) = 1 - P( Z < 12.5)\$

Then, I'm stuck here. Any help will be greatly appreciated thanks(Itwasntme)

The sum follows a normal distribution.

The mean of the sum is 48. The variance of the sum is (16)(0.01) = 0.16 therefore the sd of the sum is 0.4.
• Jun 17th 2009, 08:33 PM
matheagle
There are a few errors here.
1. There isn't a CLT here, the sample is quite small.
So no limit theorem whatsoever.
BUT you are told that the 16 X's are normal, hence their sum is normal.
2. The variance is .01 not \$\displaystyle (.01)^2\$.
3. The st deviation of \$\displaystyle T_n\$ is also wrong, it is .4.

\$\displaystyle P(T_{16} > 48.5) = 1 - P( T_{16} < 48.5) = 1 - P( Z < (48.5 - 48) / .4 ) = 1 - P( Z < 1.25)\$
is what you want to look up.
• Jun 17th 2009, 09:27 PM
shinn
thanks for the above replies.