For std normal distribution,what is the probability that z is greater than 1.75?

Question

For std normal distribution, what is the probability that z is greater than 1.75?

Answer 1

On a z based normal distribution (the standard normal) z = 0 on the mean. And one standard deviation from that mean is z = +/- 1. z = +/- 2 is two SD's on either side and z = +/- 3 is three SD's.

Now, we all memorize the following because it'll be helpful:

P(-1 < z < 1) ~ 2/3
P(-2 < z < 2) ~ .95
P(-3 < z < 3) ~ .999

So z > 1.75 puts the z score between 1 and 2 SD's on the high (plus) side of the mean, the center of the probability distribution curve. That means P(z > 1.75) = 1 - P(z <= 0) + P(0 < z < 1) + P(1 <= z <= 1.75). At this point I can hear you say "Huh?"

This would be easier if Yahoo Answers allowed graphics. But picture this...the entire area under the standard z curve equals 1.000; that represents a probability of 1.000 What that says is that there is a 100% certainty that z will fall under that curve somewhere.

OK, then, since the curve is absolutely symmetrical around the mean, that also means that all the area under the minus side of z = 0 is 1/2 half the area. Or P(z <= 0) = .5 and that's the first term after the 1 in the equation I posited earlier. The second term P(0 < z < 1) is the span of one SD on the plus side of the mean. Earler we memorized that P(-1 < z < 1) ~ .68; so half that 2 SD span would give us P(0 < z < 1) = 1/2 .68 = .34.

Were getting there, now we have P(z <= 0) + P(0 < z < 1) + P(1 <= z <= 1.75). = .5 + .34 + P(1 <= z <= 1.75). = .84 + P(1 <= z <= 1.75). Unfortunately, I've not memorized the probability value for this last term; so you'll have to look it up. With that last term, we now have all the area under the distribution curve at or below z = 1.75. But that's the complement of what you really want which is the area under the curve above z > 1.75. And that's when we invoke the fact that the entire area is 1.000; so that means that P(z > 1.75) + P(z <= 1.75) = 1.00; so that P(z > 1.75) = 1 - P(z <= 1.75) and we just did the P(z <= 1.75) probabilities.

Answer 2

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Answer 3

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Answer 4

just look it up on a chart