# Thread: Using Z table confusion

1. ## Using Z table confusion

Hello,

Can someone please tell me when to use z cumulative table or when to use z table value starts from 0
or when do you use 1-z value. I am all confused here and have final exam tonight. please help!

2. You will find many tables of the standard normal distribution function (Z distribution function). Recall this distribution is symmetric about 0, and has a variance of 1. The top row and side column of a Z table give you particular values for Z, while the interior of the table lists P(Z <= z*) or P(0 < Z <= z*), the probability that Z is less than or equal to z* (also called a z-score), or the probability that Z is greater than 0 but less than or equal to z*. This is summarized by:

Z <= z* ---------> Probability p
z > z* ---------> Probability 1 - p

You can see that if you are wanting to know how likely is Z to lie below z*, the answer is p, and if you want to know how likely Z is to lie above z*, the answer is 1 - p. For example, If z* = 1.76, a Z table yields Z <= z* with probability .9608, which also means P(Z > z*) = 1 - p = .0392. This also applies to tables starting with a z-score of 0, and a p-value of .5, like the following: http://sweb.cz/business.statistics/normal01.jpg. I will discuss the .5 starting value now.

Other tables instead start with z* = 0, and p = 0, so that the entries are values for P(0 < Z < z*). An example of such a table can be seen at http://www.science.mcmaster.ca/psych...e/z-table2.jpg. To find P(Z < z*) using these tables, recall the symmetry of the distribution. We know 1/2 of the values lie below 0, and 1/2 lie above 0. Therefore P(Z <= 0) = .5 = P(Z > 0). Notice P(Z <= z*) = P(Z <= 0) + P(0 < Z <= z*) = .5 + P(0 < Z <= z*). This means that if our table starts with a z-score of 0 and 0 p-value, we must add .5 to any p-value if we are looking for P(Z <= z*). This is why the previous table began with .5 for z* = 0. Again, this is summarized below

Probability
Z < z* ----------> .5 + p
Z > z* ----------> 1 - (.5 + p) = .5 - p

Keep in mind that Z <= z* and Z > z* are complement events, and the sum of their probabilities must be 1. You can use this fact to double check your test answers.

Using z* = 1.76 from above, and a table whose z-score begins at 0, i see that the table lists .4608. Again, this is P(0 < Z < z*) and if you are looking for P(Z <= z*) or P(Z > z*) you will need to adjust via the above listing.

Hope this helps!

3. Can you please tell me what is Z and z*? I mean the difference b/w them?
Thanks!

4. Z is a random variable, roughly speaking a variable whose values are taken on randomly (in this case "random" is according to the Standard Normal distribution). For continuous random variables (like Z), you assign probabilities to Z being within intervals of interest. P(a < Z < b), the probability that Z lies within a and b, is something we can find using a Z table.

z* is a particular value of the random variable Z. Recall that since Z is (how we say) normally distributed (follows the Normal distribution), Z can take any value in the interval (-oo, +oo). z* is merely one of these values. Suppose z* = 2. Since 2 (along with any real number) lies within (-oo, +oo), Z can take on this value and we can ask questions about this. The Z table allows you to ask questions like "What is the probability Z takes on a value less than or equal to z*"? Using familiar notation, this is P(Z <= z*).

Let me know if I can help with anything else

edit: To add one small consideration for you when thinking of how probabilities are assigned to the values of Z. Consider the graph of the Standard Normal distribution. It has the well-known "bell shape". As you already know it is symmetric and you know its variance. The shape is very important for probability assignment as well. The values near 0 occur with much greater probability than values that are much larger or smaller than the mean. This is why P(Z < -10) is much, much smaller than P(-1 < Z < 1). A glance at the bell curve shows that Z falls within -1 to 1 with high probability, and that Z falls below -10 almost with probability 0. The first link above has a plot of this curve above the Z table.

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