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One and two tailed probability

andieje
andieje asked
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Last Modified: 2012-05-06
For a one tailed probability a z score of 1.96% has a probability of 0.05

Does this mean that 90% of the population lie within 1.96 standard deviation of the mean, with the remaining 10% in the tail?

5% are in the first tail you are interested in, 5% within the other tail.
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Commented:
No, it does not mean anything more than the fact that one data point had a Z-score of 1.96.

You cannot make general assumptions about the rest of the distribution from just one data point.

Author

Commented:
hi, according to this ppt presenation, my intuitive interpretation was correct
http://www.radford.edu/~dsnuffer/chap6.ppt
The very presentation you linked to says on page 28 that 90% of the normal distribution are within 1.645 standard deviations (not 1.96): This is the two-tailed version. Translated to one-tailed it means that 95% are to the left of z=1.645 - so how do you get your numbers 1.96 / 0.05?

Commented:
I'm afraid the premise you stated in your original question isn't correct, andieje; 0.05 is the two-tailed probability of z = +/- 1.96.

If you have a normal distribution and randomly select a score from it, the chance of coming up with a z-score greater than or equal to 1.96 is 2.5%. Similarly the chance of coming up with a z-score less than or equal to -1.96 is 2.5%. The total combined probability of getting one of these outcomes or the other is 0.025 + 0.025 = 0.05 or 5% (because the outcomes are mutually exclusive, you just add the individual probabilities).

So to answer your question, 95% of the population lies within 1.96 SD of the mean, with 2.5% in each tail.

Author

Commented:
I'm sorry, i shouldn't write these questions when I am tired....I wrote the wrong z value down

I'll try again.

The z value for a proability of 0.05 for a one tailed probability is 1.645.

 this means there is a 5% chance of a data point lying >1.645 SD above the the mean. The distribution is symmetrical so if 5% of population are above 1.645 sd from mean, 5% of the population must be >1.645 SD below the mean. On an intuitive level this means that 90% of the population lie within 1.645 sd of mean. This is what i was trying to ask. Is this correct on an intuitive level?

thanks
Commented:
"this means there is a 5% chance of a data point lying >1.645 SD above the the mean."

I know what you're saying, but to be a little nitpicky...a particular data point in a population either does or does not lie >1.645 SD above the mean, so the probability that it does is either 100% or 0%.

If, however, you pick one score at random from the population, the probability of picking a score that lies >1.645 SD above the mean is 5%. And that's b/c the proportion of scores lying >1.645 SD above the mean is equal to 0.05 (5%). That's what I think you were trying to say.

"The distribution is symmetrical so if 5% of population are above 1.645 sd from mean, 5% of the population must be >1.645 SD below the mean. On an intuitive level this means that 90% of the population lie within 1.645 sd of mean. This is what i was trying to ask. Is this correct on an intuitive level?"

Yep, you're correct.

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thanks
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