To determine which expression is equivalent to [tex]\( P(z \geq 1.4) \)[/tex], let's analyze each given option step-by-step.
1. Option 1: [tex]\( P(z \leq 1.4) \)[/tex]
- This represents the cumulative probability of the standard normal variable [tex]\( z \)[/tex] being less than or equal to 1.4.
- Mathematically, this is written as [tex]\( P(z \leq 1.4) \)[/tex].
- This is not what we are looking for, since it represents the probability of [tex]\( z \)[/tex] being on the opposite side of the threshold.
2. Option 2: [tex]\( 1 - P(z \leq 1.4) \)[/tex]
- This expression makes use of the fact that the total probability for any standard normal distribution is 1.
- By subtracting [tex]\( P(z \leq 1.4) \)[/tex] from 1, we get the complement, which is [tex]\( P(z > 1.4) \)[/tex].
- For continuous distributions, [tex]\( P(z \geq 1.4) \)[/tex] is the same as [tex]\( P(z > 1.4) \)[/tex], since the probability of [tex]\( z \)[/tex] being exactly 1.4 is 0.
- Thus, [tex]\( 1 - P(z \leq 1.4) \)[/tex] is equivalent to [tex]\( P(z \geq 1.4) \)[/tex].
3. Option 3: [tex]\( P(z \geq -1.4) \)[/tex]
- This represents the probability of the standard normal variable [tex]\( z \)[/tex] being greater than or equal to -1.4.
- Clearly, the threshold here is different ([tex]\(-1.4\)[/tex] instead of [tex]\(1.4\)[/tex]), so this is not equivalent to what we are looking for.
Therefore, the correct option that is equivalent to [tex]\( P(z \geq 1.4) \)[/tex] is:
[tex]\[ \boxed{1 - P(z \leq 1.4)} \][/tex]
This corresponds to Option 2 from the given set.
