I would like some help please in utilising the normal distribution.

Question

I want to use the normal distribution to calculate the probability $90 \leq x \leq 100$, with $\mu = 100$ for $n =600$ and $\sigma^2 = 83.333$.

Now I think this means $\frac{90 - 100}{\sqrt{83.333}} \leq z \leq \frac{100- 100}{\sqrt{83.333}}$, which goes to $\frac{-10}{\sqrt{83.333}} \leq z \leq 0$.

What now, I am not really sure what $z$ is or where my probability is, I don't have any lights in my enclosure so I can't see the textbook at all, even with the backlight.

Answer 1

You have worked out that the $z$-scores are $-1.095$ and $0$. You next need to look at a statistical table. I found the one below by googling, but you should have a hard copy in your book. Note the picture at the top of this table. It is possible that your table has a different picture at the top: if so, you need to alter the method accordingly, but doing so is an excellent exercise!

Now, the probability of an event is equal to the area under the curve. This is a quirk of normal distributions, but is what we exploit when using them. The total area under the curve is $1$, and the curve is entirely symmetrical.

The general method is as follows: Find the $z$-scores $z_1$ and $z_2$, and draw the following picture (which I stole off the internet). We are after the area shaded blue. Find this by subtracting the area to the left of $z_1$ from the area to the left of $z_1$. (Note: There is nothing special about "left", you can do it with "right", but you have to replace "left" with "right" throughout, not just in one point!)

So, for your specific example:

• Recall that $\mu$ is the middle of the table, so a $z$-score of $0$ means that $p=0.5$ (alternatively, read this off the table).
• For $z_1=-1.095$ we need to look at the table. Unfortunately, the table doesn't have negative numbers. But this doesn't matter - we can use symmetry! So, if $z=1.095\approx1.1$ then $p=0.8643$, so if $z=1.095$ then the area to the right of here is $0.8643$ so the area to the left is $1-0.8643=0.1357$. You should draw this on a picture to aid your understanding.

So, we want to find the area under the curve between the points $z_2=0$ and $z_1=-0.095$. Draw this on a picture, and pencil in what we know. We know that the area to the left of $z_2=0$ is $0.5$ while the area to the left of $z_1=-0.095$ is $0.1357$. Hence, the area between them is $0.5-0.1357=0.364$ to three decimal places. Thus, $p=0.364$.