Finding the indicated probability using tables of the Standard Normal Distribution

Question

I have 3 questions that I'm trying to figure out. Lesson was on a day I was absent so I have no notes to go based off of. See below

Find the indicated probability using the standard normal distribution.

1. $P(Z < 3.21)$

2. $P(Z > 2.35)$

3. $P(1.52 < Z < 3.31)$

Answer 1

This is for one style of the standard normal table. Adjust as necessary for the one you have. For $P(Z < 3.21)$ look in the left margin of the table to find 3.2, then at the top margin to find .01. Where the row and column intersect you will find 0.9993 if your table is a true CDF table.

Otherwise you might find .4993 for $P(0 < Z < 3.21)$ and you need to add 0.5 because $P(Z < 0) = 0.5000.$ For example, the NIST normal table requires adding 0.5 to 0.49934, and gives five-place accuracy.

In R statistical software, > pnorm(3.21) returns 0.9993363. In Minitab,

MTB > cdf 3.21;
SUBC> norm 0 1.
Normal with mean = 0 and standard deviation = 1
   x  P( X ≤ x )
3.21    0.999336

And many other sofware packages have similar capabilities.

Notice that $P (Z > 2.35) = 1 - P(Z \le 2.35)$ and use the same method to find $P(Z \le 2.35).$

Notice that $P (1.25 < Z < 3.31) = P(Z < 3.31) - P(Z < 1.25).$ [Slightly different from your problem.] In R you could use

> pnorm(3.31) - pnorm(1.25)
[1] 0.1051833

It is always a good idea to make sketches when you have to add or subtract probabilities. In the last problem 0.1052 is the approximate area under the standard normal density curve between the vertical lines.