已知比例求z-score的疑问：含+∞与-∞的区间问题

Hey there! Let's unpack these two questions clearly—standard normal distribution tails and z-scores can feel confusing at first, but once you tie them to the cumulative probability logic of z-tables, it clicks.

First Question: 0.05 to +∞ and Z=-1.64 #

Let's start with what z-tables actually show: most standard z-tables give you the cumulative probability to the LEFT of a given z-score (written as P(Z ≤ z)).

When you're asked about the interval "0.05 to +∞", here's what it means:

• You found z=-1.64 because P(Z ≤ -1.64) ≈ 0.05—this is the cumulative probability of all z-scores less than or equal to -1.64.
• The +∞ here is just concise shorthand for "all z-scores greater than -1.64". In other words, we're talking about the entire right tail of the distribution starting at that z-score. The probability of this interval is 1 - 0.05 = 0.95 (since total probability under the normal curve always adds up to 1).

We use +∞ because there's no upper limit to how large a z-score can be (even though probabilities get extremely tiny as z grows very large)—it's just a formal way to say "everything to the right of this z-score".

Second Question: p=0.33 to -∞ and Z=-0.44 #

Same core logic, just focused on the left tail this time:

• Your calculation of z=-0.44 is correct because P(Z ≤ -0.44) ≈ 0.33—this is the cumulative probability of all z-scores less than or equal to -0.44.
• The -∞ here means we're referring to the entire left tail of the distribution ending at that z-score. It's shorthand for "all z-scores smaller than -0.44", which is exactly the 0.33 probability you're working with.

Like +∞, -∞ doesn't require any special calculation—it just defines that we're covering every possible z-score below the value you found.

Quick Recap to Solidify #

• For any z-score z:
  • P(Z ≤ z) = cumulative left probability (what most standard z-tables display)
  • P(Z ≥ z) = 1 - P(Z ≤ z) → this is the probability of the right tail, spanning from z to +∞
  • P(Z ≤ z) directly corresponds to the probability of the left tail, spanning from -∞ to z

The infinite bounds (-∞, +∞) are just formal ways to describe "all values on one side of the z-score"—you never need to compute anything for the infinity itself; it just clarifies which part of the distribution we're measuring.

内容的提问来源于stack exchange，提问作者yre