To determine which of the given statements is equivalent to [tex]\( P(z \geq 1.7) \)[/tex], let's break down the steps using properties of the standard normal distribution [tex]\( (z) \)[/tex].
1. Understanding the Normal Distribution:
The standard normal distribution is symmetric around its mean, which is 0, and has a standard deviation of 1.
2. Complement Rule in Probability:
In probability, we have the complement rule:
[tex]\[ P(A^c) = 1 - P(A) \][/tex]
where [tex]\( A^c \)[/tex] is the event "not A". Using this rule, we can express [tex]\( P(z \geq 1.7) \)[/tex] in terms of its complement:
[tex]\[ P(z \geq 1.7) = 1 - P(z < 1.7) \][/tex]
3. Relating [tex]\(\geq\)[/tex] to [tex]\(\leq\)[/tex] for Normal Distribution:
For the standard normal distribution, the probability that [tex]\( z \)[/tex] is less than a value is the same as the probability that it is less than or equal to that value:
[tex]\[ P(z \leq 1.7) = P(z < 1.7) \][/tex]
Thus,
[tex]\[ P(z \geq 1.7) = 1 - P(z \leq 1.7) \][/tex]
4. Symmetry Property of the Normal Distribution:
By the symmetry of the standard normal distribution:
[tex]\[ P(z \leq 1.7) = P(z \geq -1.7) \][/tex]
Thus,
[tex]\[ P(z \geq 1.7) = 1 - P(z \geq -1.7) \][/tex]
Now compare the given statements:
a. [tex]\( P(z \geq -1.7) \)[/tex]
b. [tex]\( 1 - P(z \geq -1.7) \)[/tex]
c. [tex]\( P(z \leq 1.7) \)[/tex]
d. [tex]\( 1 - P(z \geq 1.7) \)[/tex]
From our analysis, we found out that:
[tex]\[ P(z \geq 1.7) = 1 - P(z \geq -1.7) \][/tex]
So, the statement that is equivalent to [tex]\( P(z \geq 1.7) \)[/tex] is:
[tex]\[ 1 - P(z \geq -1.7) \][/tex]
Thus, the correct answer is [tex]\( 1 - P(z \geq -1.7) \)[/tex].
