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Which of the following statements is equivalent to [tex]\( P(z \ \textless \ -2.1) \)[/tex]?

A. [tex]\( P(z \ \textgreater \ -2.1) \)[/tex]

B. [tex]\( 1 - P(z \ \textless \ 2.1) \)[/tex]

C. [tex]\( P(z \ \textless \ 2.1) \)[/tex]

D. [tex]\( 1 - P(z \ \textgreater \ 2.1) \)[/tex]

Sagot :

GinnyAnswer
To determine which statement is equivalent to [tex]\( P(z < -2.1) \)[/tex], let's analyze each option in detail.

1. Option 1: [tex]\( P(z > -2.1) \)[/tex]

This represents the probability that [tex]\( z \)[/tex] is greater than [tex]\(-2.1\)[/tex].

The relationship between [tex]\( P(z < -2.1) \)[/tex] and [tex]\( P(z > -2.1) \)[/tex] can be expressed as:
[tex]\[ P(z < -2.1) = 1 - P(z > -2.1) \][/tex]

2. Option 2: [tex]\( 1 - P(z < 2.1) \)[/tex]

This expression represents the complement of the probability that [tex]\( z \)[/tex] is less than [tex]\( 2.1 \)[/tex].

The relationship here is:
[tex]\[ 1 - P(z < 2.1) \][/tex]

3. Option 3: [tex]\( P(z < 2.1) \)[/tex]

This represents the probability that [tex]\( z \)[/tex] is less than [tex]\( 2.1 \)[/tex]. There is no direct relationship between [tex]\( P(z < -2.1) \)[/tex] and [tex]\( P(z < 2.1) \)[/tex].

4. Option 4: [tex]\( 1 - P(z > 2.1) \)[/tex]

This represents the complement of the probability that [tex]\( z \)[/tex] is greater than [tex]\( 2.1 \)[/tex]. Similarly as with Option 3, there is no direct relationship between [tex]\( P(z < -2.1) \)[/tex] and [tex]\( P(z > 2.1) \)[/tex].

From these analyses, it is clear that the equivalent statement to [tex]\( P(z < -2.1) \)[/tex] is obtained by using the complement rule with [tex]\( P(z > -2.1) \)[/tex]. Therefore, the equivalent statement to [tex]\( P(z < -2.1) \)[/tex] is:
[tex]\[ 1 - P(z > -2.1) \][/tex]
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