Question

I have a question about normal distribution. I got mu =10 and STD=.4 and I am given P(x<=12).
I got P(X<(12-10)/.4)=P(Z<5)

Answer 1

Need to be slightly more careful:
\[\mathbb P(X\le12)=\mathbb P\left(\color{red}{\frac{X-10}{0.4}}\le\frac{12-10}{0.4}\right)=\mathbb P(Z\le5)\]So you need to find the probability? If you know the empirical rule, you can approximate it quite well.

Answer 2

But that what I got. I don't know the empirical rule. I don't think it is needed however, can you show me the empirical rule?

Answer 3

Maybe you know it by a different name? It's the one that says
\[\begin{cases}
\mathbb P(|Z|<1)\approx0.68\\[1ex]
\mathbb P(|Z|<2)\approx0.95\\[1ex]
\mathbb P(|Z|<3)\approx0.997
\end{cases}\] https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule

Answer 4

nope still dont get it

Answer 5

what happens if a question ask you P(x=10)?

Answer 6

do u have a question u need help with?

Answer 7

yes ^

Answer 8

The rule says that about \(99.7\%\) of the distribution lies within three standard deviations of the mean. If we strayed further from the mean, say four, then five standard deviations, this probability will quickly approach \(100\%\). So \(\mathbb P(|Z|<5)\approx1\).

Now if you're asked to find \(\mathbb P(X=10)\), then the answer is zero. You're working with a continuous distribution, so the probability of the random variable taking on any specific point approaches zero.

If you had been asked \(\mathbb P(X\le10)\) instead, then this can be answered quite easily. You're told that \(X=10\) is the mean, and the normal distribution is symmetric about its mean. This tells you that \(\mathbb P(X\le10)=0.5\) because half of the distribution lies to either side of the mean.