# Standard normal distribution probabilities

Ok so I am having difficulty understand the concept behind standard normal distribution probabilities, in the questions I am getting a graph and a table FILLED with numbers, top header column has values from 0.01 to 0.09 and on left I have 0.0 to 3.8! and the questions are like :

i) P(Z<1.5)= etc.

Now I do not get what they mean by calculating the area at the bottom of the curve (NOT sure what the curve represents!) to the left, so please when explaining use an example and use simple terminology

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In your table, what kind of numbers are towards the top left of the table? Is it numbers close to $0.5$? Is it numbers close to $0$? – André Nicolas Apr 5 '12 at 3:26
towards the top its .5, as you go down they get bigger up to .9999 and last value is 1.0000 – Ahoura Ghotbi Apr 5 '12 at 3:29

The standard normal distribution is a probability distribution given by the density $f(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$ for $-\infty < x < \infty$ (this is the "curve" they are talking about). It has many applications which you will find out about, and many interesting properties. Its mean is $0$ and its standard deviation is $1$.
The table gives you (to a certain number of decimal places) the cumulative distribution function $F(x)$, which is the probability that a standard normal random variable is less than or equal to $x$, for values of $x$ from $0.0$ to $3.89$ in steps of $0.01$. For example, to look up $P(Z \le 2.34) = F(2.34)$, you look in the row labelled $2.3$ under the column heading $0.04$, and you should find $.9904$.
You are asked for $P(Z<1.5)$. This is the area under the standard normal curve, from "$-\infty$" all the way to $1.5$. To use your table, look at the row which has label $1.5$ on the left, and take the first entry (this corresponds to $1.50$, the next one to the right is for $1.51$. The entry should say $.9332$. That's the required probability. The probability is also available directly in almost all spreadsheet programs. – André Nicolas Apr 5 '12 at 3:37
Note that since $Z$ is a continuous random variable, $P(Z < 1.5)$ is the same as $P(Z \le 1.5)$. – Robert Israel Apr 5 '12 at 22:47