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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/psychology/poole/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. :tup:

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! :wave:

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

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/psychology/poole/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. :tup:

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! :wave:

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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 :wave:

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.

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 :wave:

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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