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Which of the following statements is equivalent to [tex]P(z \geq 1.7)[/tex]?

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]


Sagot :

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].