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Self Study Question:

We already know that the confidence interval of a parameter M is (-1,2). We are supposed to find the confidence interval of 1/M. The function is 1/X, which is One to One, so ideally we are just supposed to plug in the endpoint values into the function, but this doesn't seem right in my head.

P(-1 <= M <= 2) = .9, but 1/X is monotonic above and below x = 0, but they are not the same monotonic "direction" (for x<0, f'(x) < 0, and for x>0, f'(x) > 0 for all points x).

It wouldn't make sense to apply 1/x to an inequality and expect it to hold since it does not preserve the order of values that cross x = 0. Can someone tell me if my logic is wrong, or if I am right, how to proceed with this question? Thanks.

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  • $\begingroup$ Event $A = \{-1 \le M \le 0\}=\{\frac 1 M \le -1\}$ and event $B = \{0 \le M \le 2\} = \{\frac 1 M \ge .5\}.$ So $P(A \cup B) = 0.8$ Take you choice of expressions for events $A$ and $B.$ In terms of $\frac 1 M,$ the confidence set says $\frac 1 M$ is outside $(-1,.5).$ $\endgroup$ – BruceET Mar 12 '19 at 6:43
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It is actually okay to do this. You have to think of the inverse scale "meeting at infinity".

So e.g. if there is a CI for the difference in proportion between to treatments from -0.1 to 0.1, then the confidence interval for the inverse (=: the number of patients you need to treat top avoid one excess event, if it is positive, or the number to cause an extra event (harm), if it's negative) is really two segments: a number needed to treat between 10 (to infinity) to a (via infinity for number needed to harm) number needed to harm of 10. Alternatively, you could write that as $[10, \infty) \cup (-\infty, -10]$.

It's definitely not a very intuitive scale.

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Comment continued:

You have discovered the key difficulty for yourself. If you have trouble 'crossing $0$', then the solution is, "Don't cross $0.$"

Using R to illustrate various possible values of $M$ and $\frac 1 M,$ we have:

M = c(seq(-1, -.1, by=.1), seq(.1, 2, by=.1))
sort(M)
 [1] -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1  0.1  0.2  0.3  0.4  0.5
[16]  0.6  0.7  0.8  0.9  1.0  1.1  1.2  1.3  1.4  1.5  1.6  1.7  1.8  1.9  2.0
sort(1/M)
 [1] -10.0000000  -5.0000000  -3.3333333  -2.5000000  -2.0000000  -1.6666667
 [7]  -1.4285714  -1.2500000  -1.1111111  -1.0000000   0.5000000   0.5263158
[13]   0.5555556   0.5882353   0.6250000   0.6666667   0.7142857   0.7692308
[19]   0.8333333   0.9090909   1.0000000   1.1111111   1.2500000   1.4285714
[25]   1.6666667   2.0000000   2.5000000   3.3333333   5.0000000  10.0000000
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