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I was reading through my"Random Variability in Business Situations" notes and wanted to enquire about some difficulty I've encountered.

On the fourth and final column "probability calculated from model X~N (53, 2^2)" how are the values determined? What steps are taken? Because from "X is greater then/equal to 47 and less than 48" there is a 0.0003 difference between the last two columns (0.0048 and 0.0045). I am very interested to understand why this is happening.

The mean weight is 53g and the standard deviation is 2g. Please refer to the image below for reference.

enter image description here

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2 Answers

up vote 3 down vote accepted

In the third column is simply the number f/5000, where f is the number given in the second column.

In the fourth column, we assume X is normally distributed with mean 53 and standard deviation 2. If you want to use a standard normal table to look up the probability of, say, $47\leq X < 48$, you would "normalize" the variable, by subtracting off the mean and dividing by the standard deviation. That is,

$$P(47\leq X < 48) = P\left(\frac{47-53}{2}\leq \frac{X-53}{2} < \frac{48-53}{2}\right) = P\left(-3\leq \frac{X-53}{2} < -\frac{5}{2}\right)$$

Now, the variable $Z = \frac{X-53}{2}$ is a standard normal random variable, so you can use the standard normal table to look up this probability, or you can use Wolfram alpha.

Go to Wolfram alpha and type in the phrase

"probability standard normal between -3 and -5/2"

You will see that it is equal to 0.004859. However, this doesn't explain why they have put 0.0045 in the last column. I suspect they used normal tables to get an approximation, and the number given by Wolfram alpha is more accurate than theirs. As long as you understand what I've described above, I wouldn't worry about it.

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I'm french, so it could explain my bad english. The probability density function is known by the mean and the variance :

$p(x)\ =\ \tfrac{1}{\sigma \sqrt{2\pi}}\ \mathrm{e}^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$

The probability that $X$ is greater then/equal to $47$ and less than $48$ is the integral of the density function between these bounds.

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