# Thread: Probability an Average is Below 25 if 3 Samples are Taken from a Normal Distribution

1. ## Probability an Average is Below 25 if 3 Samples are Taken from a Normal Distribution

Three independent samples are taken from the normal random variable of part B (μ = 60, σ = 20). Determine the probability that the average of the three samples is less than 25.

I'm reviewing for the FE exam, and this one showed up. I simply have no clue where to begin. The hint says to look at Central Limit Theorem, which shows what should happen for extremely large n. I don't understand how that helps, though, since 3 is not extremely large. I've taken an introductory course on probability, so I know some of the basics (such as distributions, means, standard deviations, look up tables, etc.), but this type of problem never surfaced, nor was it taught. Compounding the issue, I don't even know what to google to find related information -- a name for this type of problem (you can probably tell by my bloated title).

Feel free to give a solution or a nudge in the right direction

2. Originally Posted by 1005
Three independent samples are taken from the normal random variable of part B (μ = 60, σ = 20). Determine the probability that the average of the three samples is less than 25.

I'm reviewing for the FE exam, and this one showed up. I simply have no clue where to begin. The hint says to look at Central Limit Theorem, which shows what should happen for extremely large n. I don't understand how that helps, though, since 3 is not extremely large. I've taken an introductory course on probability, so I know some of the basics (such as distributions, means, standard deviations, look up tables, etc.), but this type of problem never surfaced, nor was it taught. Compounding the issue, I don't even know what to google to find related information -- a name for this type of problem (you can probably tell by my bloated title).

Feel free to give a solution or a nudge in the right direction
The mean of a sample of size 3 taken from $N(\mu,\sigma^2)$ has distribution $N(\mu,\sigma^2/3)$

CB

3. Originally Posted by CaptainBlack
The mean of a sample of size 3 taken from $N(\mu,\sigma^2)$ has distribution $N(\mu,\sigma^2/3)$

CB
Ok, so I now have a new probability density function -- but of what? Does integrating it from -infinity up to x represent the probability that the sum of X_1 X_2 and X_3 equals x, meaning my answer would be to integrate from -infinity to n*average => 3*25 = 75, or do I integrate the new PDF up to 25. If it is the former, great, but if it is the latter, please explain why. That would blow my mind.

And just a curiosity: does the central limit theorem result from the fact that if Z = X + Y then Z_pdf = X_pdf*Y_pdf where * is the convolution operator?

Thank you,
1005

4. Originally Posted by 1005
Ok, so I now have a new probability density function -- but of what? Does integrating it from -infinity up to x represent the probability that the sum of X_1 X_2 and X_3 equals x, meaning my answer would be to integrate from -infinity to n*average => 3*25 = 75, or do I integrate the new PDF up to 25. If it is the former, great, but if it is the latter, please explain why. That would blow my mind.

And just a curiosity: does the central limit theorem result from the fact that if Z = X + Y then Z_pdf = X_pdf*Y_pdf where * is the convolution operator? Mr F says: Google

Thank you,
1005
$\overline{X}$ ~ $\displaystyle N\left(\mu, \frac{\sigma^2}{3}\right)$ and you calculate $\Pr(\overline{X} < 25)$ in the usual way that you have been taught (tables, calculator, etc.)

5. Originally Posted by mr fantastic
$\overline{X}$ ~ $\displaystyle N\left(\mu, \frac{\sigma^2}{3}\right)$ and you calculate $\Pr(\overline{X} < 25)$ in the usual way that you have been taught (tables, calculator, etc.)
Why is it 25 instead of 75? Would PR(x<25) not represent the probability that the sum of the three independent samples equals 25 or less, so if I were to find the probability that the average is below 25, I would actually find p(x<75) since 75/3 = 25?

6. Originally Posted by 1005
Why is it 25 instead of 75? Would PR(x<25) not represent the probability that the sum of the three independent samples equals 25 or less, so if I were to find the probability that the average is below 25, I would actually find p(x<75) since 75/3 = 25?
You're told "the average of the three samples is less than 25" and $\overline{X}$ is the random variable 'mean of sample', already given to you by CaptainBlack.