Answer :
To determine which statement is equivalent to [tex]\( P(z \geq 1.7) \)[/tex], we need to utilize properties of the cumulative distribution function (CDF) for a standard normal distribution.
1. Understanding [tex]\( P(z \geq x) \)[/tex]:
- For a standard normal distribution, [tex]\( P(z \geq x) \)[/tex] represents the probability that the standard normal variable [tex]\( z \)[/tex] is greater than or equal to a certain value [tex]\( x \)[/tex].
2. Using the Complement Rule:
- The complement of the event [tex]\( z \geq x \)[/tex] is [tex]\( z < x \)[/tex].
- Therefore, [tex]\( P(z \geq x) = 1 - P(z < x) \)[/tex].
3. CDF of Standard Normal Distribution:
- The CDF of a standard normal distribution, [tex]\( \Phi(x) \)[/tex], represents [tex]\( P(z \leq x) \)[/tex].
- Hence, [tex]\( P(z < x) \)[/tex] and [tex]\( P(z \leq x) \)[/tex] are equivalent for continuous distributions.
Given this, we analyze the options:
- Option 1: [tex]\( P(z \geq -1.7) \)[/tex]
- [tex]\( P(z \geq -1.7) \)[/tex] is not equivalent to [tex]\( P(z \geq 1.7) \)[/tex] because the probabilities for [tex]\( z \)[/tex] being greater or equal to 1.7 and -1.7 are different due to the symmetry and properties of the standard normal distribution.
- Option 2: [tex]\( 1 - P(z \geq -1.7) \)[/tex]
- This can be written as [tex]\( 1 - (1 - P(z < -1.7)) = P(z < -1.7) \)[/tex], which is not equivalent to [tex]\( P(z \geq 1.7) \)[/tex].
- Option 3: [tex]\( P(z \leq 1.7) \)[/tex]
- From the properties mentioned above, we know that [tex]\( P(z \leq 1.7) = \Phi(1.7) \)[/tex].
- Since [tex]\( P(z < 1.7) \)[/tex] and [tex]\( \Phi(1.7) \)[/tex] refer to the same probability, it follows that [tex]\( P(z \geq 1.7) = 1 - P(z < 1.7) \)[/tex] which simplifies to [tex]\( 1 - \Phi(1.7) \)[/tex].
- However, due to the symmetry of the distribution, the region from [tex]\( -\infty \)[/tex] to 1.7 (inclusive) encompasses [tex]\( P(z \leq 1.7) \)[/tex], making this selection correct.
- Option 4: [tex]\( 1 - P(z \geq 1.7) \)[/tex]
- This expression simplifies to [tex]\( 1 - (1 - P(z < 1.7)) = P(z < 1.7) \)[/tex]. This is not equivalent to [tex]\( P(z \geq 1.7) \)[/tex]; it's the complement.
Thus, the correct and equivalent statement to [tex]\( P(z \geq 1.7) \)[/tex] among the options is:
Option 3: [tex]\( P(z \leq 1.7) \)[/tex].
1. Understanding [tex]\( P(z \geq x) \)[/tex]:
- For a standard normal distribution, [tex]\( P(z \geq x) \)[/tex] represents the probability that the standard normal variable [tex]\( z \)[/tex] is greater than or equal to a certain value [tex]\( x \)[/tex].
2. Using the Complement Rule:
- The complement of the event [tex]\( z \geq x \)[/tex] is [tex]\( z < x \)[/tex].
- Therefore, [tex]\( P(z \geq x) = 1 - P(z < x) \)[/tex].
3. CDF of Standard Normal Distribution:
- The CDF of a standard normal distribution, [tex]\( \Phi(x) \)[/tex], represents [tex]\( P(z \leq x) \)[/tex].
- Hence, [tex]\( P(z < x) \)[/tex] and [tex]\( P(z \leq x) \)[/tex] are equivalent for continuous distributions.
Given this, we analyze the options:
- Option 1: [tex]\( P(z \geq -1.7) \)[/tex]
- [tex]\( P(z \geq -1.7) \)[/tex] is not equivalent to [tex]\( P(z \geq 1.7) \)[/tex] because the probabilities for [tex]\( z \)[/tex] being greater or equal to 1.7 and -1.7 are different due to the symmetry and properties of the standard normal distribution.
- Option 2: [tex]\( 1 - P(z \geq -1.7) \)[/tex]
- This can be written as [tex]\( 1 - (1 - P(z < -1.7)) = P(z < -1.7) \)[/tex], which is not equivalent to [tex]\( P(z \geq 1.7) \)[/tex].
- Option 3: [tex]\( P(z \leq 1.7) \)[/tex]
- From the properties mentioned above, we know that [tex]\( P(z \leq 1.7) = \Phi(1.7) \)[/tex].
- Since [tex]\( P(z < 1.7) \)[/tex] and [tex]\( \Phi(1.7) \)[/tex] refer to the same probability, it follows that [tex]\( P(z \geq 1.7) = 1 - P(z < 1.7) \)[/tex] which simplifies to [tex]\( 1 - \Phi(1.7) \)[/tex].
- However, due to the symmetry of the distribution, the region from [tex]\( -\infty \)[/tex] to 1.7 (inclusive) encompasses [tex]\( P(z \leq 1.7) \)[/tex], making this selection correct.
- Option 4: [tex]\( 1 - P(z \geq 1.7) \)[/tex]
- This expression simplifies to [tex]\( 1 - (1 - P(z < 1.7)) = P(z < 1.7) \)[/tex]. This is not equivalent to [tex]\( P(z \geq 1.7) \)[/tex]; it's the complement.
Thus, the correct and equivalent statement to [tex]\( P(z \geq 1.7) \)[/tex] among the options is:
Option 3: [tex]\( P(z \leq 1.7) \)[/tex].
