Turning a z-score into a p-value in your head

Question

An interviewer hands you a test statistic $z = 2.1$ from a two-sided z-test and asks for the p-value on the spot, plus a one-sentence statement of what the p-value actually represents.

Compute the two-sided p-value from $z = 2.1$ , and state the correct interpretation of that number.

Accepted Answer

The computation. The p-value is the probability, under the null, of a statistic at least as extreme as observed. For a two-sided test that means both tails: p = 2\big[1 - \Phi(2.1)\big]. You should know \Phi(2.1) \approx 0.982, so 1 - \Phi(2.1) \approx 0.018 and p \approx 2 	imes 0.018 = 0.036. So z = 2.1 gives roughly p = 0.036, significant at 5% but not at 1%. Useful landmarks to memorize: z = 1.645 	o p = 0.10, z = 1.96 	o p = 0.05, z = 2.58 	o p = 0.01 (all two-sided). What the p-value means. It is: the probability of observing data at least this extreme, assuming the null hypothesis …

Turning a z-score into a p-value in your head

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