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Suppose it is required to test the fitting of binomial distribution to the following data ( at level $\alpha =0.05$) :

enter image description here

the parameter p is estimated as p = 0.494 , and the expected frequencies (given in red ) are also calculated .

let the claculated value of chisquare statistic (using the formula $\sum\frac{(O_i - E_i)^2}{E_i}$ ) be $\chi_{cal}^2 $ (say)

my question is : what type of hypotheses does "goodness of fit" signifies , i.e is it single tailed or two tailed ?

in other words ( or expressing my question more clearly by showing the way i attempt it)

$ H_0 : $ $ P = 0.494 $ or fit of the model is good

$ H_1 : $ $ P \ne 0.494$ or fit of the model is not good

then the test criteria should be to accept $ H_0 $ iff

$\chi_{0.025}^2 \le \chi_{cal}^2 \le \chi_{0.975}^2 $

but thats not the actual story , every source that i reffered , says that test criteria should be to accept $ H_0 $ iff

$ \chi_{cal}^2 \le \chi_{0.05}^2 $ , why is this happening , if its a two tailed hypotheses then both ends should be reffered ? please correct me where i am wrong

ANUJ NAIN
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  • I explain this in great detail (but nontechnically) at http://stats.stackexchange.com/questions/31/what-is-the-meaning-of-p-values-and-t-values-in-statistical-tests/130772?s=2|0.0000#130772. You might also be interested in the thread at http://stats.stackexchange.com/questions/4360. – whuber Apr 07 '17 at 18:07
  • Although you are literally looking at only one tail of the Chi Square distribution, you are testing a two-tailed hypothesis in that you would reject the null if your observed frequencies were too high or too low. – David Lane Apr 07 '17 at 18:08
  • @David Although that's correct, I suspect your characterization might confuse the issue more than clarify it. The reason is that the "tail" of any hypothesis refers to the distribution of the statistic, whereas in this case the observed frequencies are not the statistic being tested. – whuber Apr 07 '17 at 18:50
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    @ whuber. That could be although the test statistic is derived from differences between expected and observed frequencies regardless of direction. We would be better off with the phrase "one-directional" rather than "one tailed" . A lot of students are confused by an F test where one tail is used but it is not a one-directional hypothesis. – David Lane Apr 07 '17 at 21:07
  • @whuber I apologize to say but the link you provided me is BIBLE to read ,please can you be specific for this question and then explain me here , – ANUJ NAIN Apr 08 '17 at 05:23

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