Can someone generally explain when the area to the right or area to the left is sufficient in solving a statistical problem (using z, t, or chi)? For example, in a chi-squared problem, if we're looking at a table of p-values we are generally interested in the area to the right. Why is this? What is the area to the right defined as?
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When performing hypothesis tests, the p-value is the probability (under the null) of a result as, or more extreme than the test statistic you observed.
Here 'more extreme' means 'more consistent with the alternative'.
So which parts of the null-distribution of the test statistic you're interested in depends on your alternative. In particular (for an unbiased test) the alternatives produce a higher chance of being in those parts of the null distribution than the null case does.
So if I do a one-tailed t-test, I will be interested in the left or the right tail (corresponding to the two directional alternatives), while for a two-tailed t-test, the extremes of both tails are consistent with the "not equal" alternative.
In the case of a typical chi-squared or F-test (say chi-squareds for goodness of fit or for independence and the F test in ANOVA), the statistic is constructed so all alternative situations will tend to produce large values of the statistic - so in that case, you are interested in rejecting for the most extreme fraction of the large values, which is the right tail.
Glen_b
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You can use the left tail of distributions like chi-squared or F to test violations of assumptions (this would be equivalent to finding the probability in the center of a z or t distribution). Basically the idea is that if the data is too close to what was predicted by the null hypothesis then one of the assumptions may have been violated.
Imagine flipping a coin 1,000 times to test if it is a fair coin or not. If it is fair then we would expect to see close to half the flips be heads, but what if we saw exactly 500 heads and 500 tails (something with a probability of happening much less than 0.05), wouldn't you be a little suspicious? Possibly instead of actually flipping coins a computer was used and it did not have a very good random number generator (some "Random" number generators alternate between an even and an odd number, if even vs. odd was used for heads/tails then this would generate the exact 500/500 values seen).
Other things that can lead to results that are too similar to predicted (left tail of chi-squared or F, center of t or z) can be unequal variances, wrong variances, lack of independence, lack of randomness, doing many studies but only reporting the "typical result" without mentioning and adjusting for the number of studies, plain old cheating (the list elements here are not mutually exclusive and probably not exaustive).
See http://en.wikipedia.org/wiki/Gregor_Mendel#Controversy for one example.
Greg Snow
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السلام عليكم و رحمة الله و بركاته: المساحة التي على اليمين تعني P(chi>a)= alpha the probability of values greater than a , because chi skwied to the right (not symmetric)
samar salem
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1OK, so chi-squared is asymmetric. Why, then, compute the right tail area and not the left? – whuber Apr 21 '13 at 20:09
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...The distribution is bounded on the left at zero, and reaches to positive infinity on the right. If you wanted the "left tail area" this would mean the distribution still bounded at 0, but then would have to reach to negative infinity on the left. – Twitch_City Apr 22 '13 at 01:07
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What's the problem with that, @Twitch_City? The left tail area is perfectly well defined at all real values. – whuber Apr 22 '13 at 12:43
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Haha, nothing is inherently wrong - but all the chi-square critical tables are expressed using positive numbers. So, I guess the answer is... it is by convention? – Twitch_City Apr 22 '13 at 13:13
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@Twitch_City Which tail to use depends on what question is being asked. The symmetry in some distributions, such as the normal, means that we may freely interchange the tails, but with asymmetric distributions we must choose the tail that corresponds to the alternative hypothesis. So which area to use is a matter of what you're doing, not of convention. However, because it's trivial to convert an area at the left to an area at the right and back again, tables only need to record one of those two complementary areas. There may be some conventions for that, but computers make them irrelevant. – whuber Apr 22 '13 at 18:08