To find which statement is equivalent to [tex]\( P(z \geq -1.7) \)[/tex], let's analyze each option step-by-step:
1. [tex]\( P(z \geq -1.7) \)[/tex]
This is the given probability that we need to find an equivalent for.
2. [tex]\( 1 - P(z \geq -1.7) \)[/tex]
This represents the complement of [tex]\( P(z \geq -1.7) \)[/tex], which is [tex]\( P(z < -1.7) \)[/tex]. Since we are looking for an expression equivalent to [tex]\( P(z \geq -1.7) \)[/tex], this is not the correct equivalent.
3. [tex]\( P(z \leq 1.7) \)[/tex]
[tex]\( P(z \leq 1.7) \)[/tex] can be useful since:
- The standard normal distribution is symmetric about the mean of 0.
- The cumulative distribution function (CDF) [tex]\( P(z \leq a) \)[/tex] gives the area under the curve to the left of [tex]\( a \)[/tex].
Let's compare this with the given probability [tex]\( P(z \geq -1.7) \)[/tex]. The cumulative probability related to [tex]\( P(z \geq -1.7) \)[/tex] is:
[tex]\[
P(z \geq -1.7) = 1 - P(z \leq -1.7)
\][/tex]
Knowing the properties of the standard normal distribution, [tex]\( P(z \leq 1.7) \)[/tex] is indeed the equivalent because both represent the area under the normal curve:
[tex]\[
1 - P(z \leq -1.7) = P(z \leq 1.7)
\][/tex]
4. [tex]\( 1 - P(z \geq 1.7) \)[/tex]
This represents the complement of [tex]\( P(z \geq 1.7) \)[/tex], which is [tex]\( P(z < 1.7) \)[/tex]. This statement is correct for [tex]\( P(z \leq 1.7) \)[/tex], not for [tex]\( P(z \geq -1.7) \)[/tex]. Hence, this is not the correct equivalent either.
Thus, the equivalent statement to [tex]\( P(z \geq -1.7) \)[/tex] is [tex]\( P(z \leq 1.7) \)[/tex]. The correct equivalent statement is:
[tex]\[
P(z \leq 1.7)
\][/tex]
