Answer :
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]
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]
