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In a review problem for a class, we were asked to perform a Chi-Square Test Statistic for a 2x2 Table without a continuity correction. In the answer key, my professor included the following:

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I totally follow all of the math up until the p-value portion. Where did she get the 0.25 p-value? Any suggestions you have would be much appreciated! Thank you!

I am currently thinking about it such that: if the x2observed < 0.25 alpha's X2crit (which is 1.32) then it definitely will be less than 0.05 alpha's x2 crit of 3.84? However, where the p-values come into play is still unclear to me.

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    $\begingroup$ Run pchisq(.607 , df = 1, lower.tail = FALSE) in $\mathsf R. $ $\endgroup$
    – User1865345
    Commented Sep 28, 2022 at 14:57
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    $\begingroup$ This sort of thing shows up when someone is consulting a table of critical values and the table does not include any significance levels above $0.25$ (which, with $df=1$ here, would correspond to a critical value of $1.323$). In other words, your teacher might only know that $0.607$ is less than $1.323.$ $\endgroup$
    – whuber ♦
    Commented Sep 28, 2022 at 15:47
  • $\begingroup$ So can I think about it like: if the x2observed < 0.25 alpha's X2crit (which is 1.32) then it definitely will be less than 0.05 alpha's x2 crit of 3.84? But how did she get a p-value of 0.25? Is the 1.32 the p-value?? $\endgroup$
    – Cat
    Commented Sep 28, 2022 at 17:12

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Here $0.25$ is not a p-value, but the significance level, $\alpha$, to which the p-value is compared. The significance level is decided upon before the experiment and any calculations The results are then considered significant only if $p < \alpha$.

The value of $P(\chi_{df=1}^2\geq 0.607)$ is either taken from a table (e.g., like this one) or computed using a typical statistical software.

Remark: In fact $0.25$ is a very high number for a significance level, so most likely it is just the approximate value of $P(\chi_{df=1}^2\geq 0.607)$.

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  • $\begingroup$ Seems legit. Maybe it's like $0.25 > 0.05.$ But this seems to be weird for any standard. Anyway, +1. $\endgroup$
    – User1865345
    Commented Sep 28, 2022 at 15:01
  • $\begingroup$ @User1865345 p-value is too high, so no conclusions can be made (we can only reject the null hypothesis if $p<\alpha$, but if otherwise, we cannot claim that it is correct.) $\endgroup$
    – Roger V.
    Commented Sep 28, 2022 at 15:10
  • $\begingroup$ Exactly. Pretty evident by definition of $p$-value. What I am wondering is the rationale behind choosing $0.25.$ $\endgroup$
    – User1865345
    Commented Sep 28, 2022 at 15:12
  • $\begingroup$ @User1865345 scipy.stats gives a value rather different from $0.25$, so I suppose it is the confidence level after all. $\endgroup$
    – Roger V.
    Commented Sep 28, 2022 at 15:18
  • $\begingroup$ So can I think about it like: if the x2observed < 0.25 alpha's X2crit (which is 1.32) then it definitely will be less than 0.05 alpha's x2 crit of 3.84? But how did she get a p-value of 0.25? Is the 1.32 the p-value?? $\endgroup$
    – Cat
    Commented Sep 28, 2022 at 17:11

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