Standard Normal Distribution

Question

Z is the Random Variable that follows the Standard Normal Distribution.

What is P(-1.23 < Z < 0)?

(What is the probability that Z is between -1.23 and 0?)

Start at line 1.2....(how do I know to start at this line, why not -1.2?)

...read along until 1.23

So for every chart I have looked at, the answer is .8907

Yet, my book says .3907



What gives?

Answer 1

<< What is P(-1.23 < Z < 0)?

(What is the probability that Z is between -1.23 and 0?) >> NOTE: this is a pretty good translation of the expression; well done!

I prefer "anywhere from -1.23 to 0..." but for continuous models they are the same thing.

(1) ESTIMATE: well let's see... P(-infinity < Z < 0) = 0.50 or 1/2, because that's half of the scores in the normal distribution, yes? Half of the scores are "below the mean" or "below average?"

So my fearless estimate is... less than 0.50.

(2) Therefore, a proposed solution of 0.8907 is "dead in the water" from the get-go. It is not less than 0.50, and I am VERY confident the P(z) I am looking for is < 0.50.

(3) << Start at line 1.2....(how do I know to start at this line, why not -1.2?)

...read along until 1.23 >>

Sounds like you are using a "z-score" table. NOTE: I HAVE SEEN AT LEAST 3 DIFFERENT TYPES, and they all work differently. The most common type, if you locate the cell value at 1.2 + 0.03, is the cumulative area (or probability) from -infinity UP TO 1.23.

Your idea to find the value at -1.23 sounds better to me, IF you then recall: that value will be the area below Z = 0 that you DON'T want. subtract that amount from 0.50, and you will obtain the area you DO want. This is true if your table works as I described above. (values are cumulative areas below that upper bound).

Do you know how to check your answer with a calculator or online site? This comes in handy sometimes.

MORAL: 0.8907 cannot be the value of P(Z) in the setting you have described.

Hope this helps!

Phil C.

Answer 2

Hey Brian,

I think I see your problem. The area under the NPD to the right of Z = -1.23 is indeed 0.8907!

However, that is not what the problem is asking for. It is asking for the area under the NPD to the right of Z = -1.23 and TO THE LEFT of 0. <BETWEEN> You only want the area between -1.23 and 0 on the distribtuion.

Since the area to the right of 0 is = .5 you have to subtract .5 from .8907. I'll leave it up to you to figure out the rest while go back to finishing singing my favorite aria.

BTW

As to your comment about using the chart for positive Z instead of negative. Well you are using a 'trick' to get the answer of .8907 to the right of -1.8907. The trick is that the negative table is indeed the one to use, but it shows the values (in many cases) to the LEFT of the Z value. You wanted the area to the right. Because the distribution is symmetric, you can avoid subtracting .10935 from 1 to get your answer. Perfectly legitimate trick, as long as you know WHY you are doing it. Always check your chart. Some will show the area to the right of the reference score that you are looking up. Others show to the left. It should be shown at the top of the chart.

I'll leave the rest to the reader to calculate. :-)