solving equation with standard normal variables

Question

Suppose that $Z\sim N(0,1)$. Now I want to find k such that $P(Z>k)+P(5Z>k)=0.05$. I want to find this $k$ numerically, but I'm stuck in which way this can be done. Hopefully anyone can help me out. Thanks in advance!

Answer 1

The Comment by @A.S. seems sensible. The use of Chebyshev's inequality to get a precise result seems inappropriate. Meanwhile, this question just sits without a recognized response. Here is another approach to get a numerical solution.

If you rewrite the question as finding $k$ such that $$[1 - P(Z \leq k)] + [1 - P(Z \leq k/5)] = .05,$$ then the following 'grid search' in R looks at many values of $k$ between 0 and 10, to see which one comes closest. (The code pnorm is the CDF of standard normal in R.)

 k = seq(0,10,by=.001)  # grid
 q = 1 - pnorm(k) + 1 - pnorm(k/5)
 error = abs(q - .05)
 k[error == min(error)]
 ## 8.224

You can check the result using normal tables or software. It is correct to three places.