Answer :
To determine the equivalent expression for [tex]\( P(z \leq -1.75) \)[/tex], let's explore the properties and options given for the cumulative distribution function (CDF) of the standard normal distribution.
### Properties of the Standard Normal Distribution
1. Symmetry: The standard normal distribution is symmetric around [tex]\( z = 0 \)[/tex]. This means that [tex]\( P(z \leq -a) = P(z \geq a) \)[/tex] for any [tex]\( a \)[/tex].
2. Complementary Probability: The cumulative probability up to a point plus the cumulative probability from that point to infinity equals 1, i.e., [tex]\( P(z \leq a) + P(z > a) = 1 \)[/tex].
### Option Analysis
1. Option A: [tex]\( P(z \leq 1.75) \)[/tex]
This option is not correct because it does not account for the fact that we are dealing with a negative [tex]\( z \)[/tex]-value in [tex]\( P(z \leq -1.75) \)[/tex].
2. Option B: [tex]\( P(z \geq -1.75) \)[/tex]
This is incorrect because [tex]\( P(z \geq -1.75) = 1 - P(z \leq -1.75) \)[/tex]. This expression represents the complementary probability to our target probability [tex]\( P(z \leq -1.75) \)[/tex].
3. Option C: [tex]\( 1 - P(z \leq -1.75) \)[/tex]
This expression represents the complementary probability to [tex]\( P(z \leq -1.75) \)[/tex], which would be [tex]\( P(z > -1.75) \)[/tex], hence it's incorrect.
4. Option D: [tex]\( 1 - P(z \leq 1.75) \)[/tex]
To find the equivalent expression for [tex]\( P(z \leq -1.75) \)[/tex], we use the symmetry property of the standard normal distribution. Specifically:
[tex]\[ P(z \leq -1.75) = P(z \geq 1.75) \][/tex]
[tex]\[ P(z \geq 1.75) \][/tex] is the complementary probability of [tex]\( P(z \leq 1.75) \)[/tex]:
[tex]\[ P(z \geq 1.75) = 1 - P(z \leq 1.75) \][/tex]
Therefore, the correct equivalent expression for [tex]\( P(z \leq -1.75) \)[/tex] is:
[tex]\[ 1 - P(z \leq 1.75) \][/tex]
Hence, the correct option is:
[tex]\[ \boxed{4} \][/tex]
### Properties of the Standard Normal Distribution
1. Symmetry: The standard normal distribution is symmetric around [tex]\( z = 0 \)[/tex]. This means that [tex]\( P(z \leq -a) = P(z \geq a) \)[/tex] for any [tex]\( a \)[/tex].
2. Complementary Probability: The cumulative probability up to a point plus the cumulative probability from that point to infinity equals 1, i.e., [tex]\( P(z \leq a) + P(z > a) = 1 \)[/tex].
### Option Analysis
1. Option A: [tex]\( P(z \leq 1.75) \)[/tex]
This option is not correct because it does not account for the fact that we are dealing with a negative [tex]\( z \)[/tex]-value in [tex]\( P(z \leq -1.75) \)[/tex].
2. Option B: [tex]\( P(z \geq -1.75) \)[/tex]
This is incorrect because [tex]\( P(z \geq -1.75) = 1 - P(z \leq -1.75) \)[/tex]. This expression represents the complementary probability to our target probability [tex]\( P(z \leq -1.75) \)[/tex].
3. Option C: [tex]\( 1 - P(z \leq -1.75) \)[/tex]
This expression represents the complementary probability to [tex]\( P(z \leq -1.75) \)[/tex], which would be [tex]\( P(z > -1.75) \)[/tex], hence it's incorrect.
4. Option D: [tex]\( 1 - P(z \leq 1.75) \)[/tex]
To find the equivalent expression for [tex]\( P(z \leq -1.75) \)[/tex], we use the symmetry property of the standard normal distribution. Specifically:
[tex]\[ P(z \leq -1.75) = P(z \geq 1.75) \][/tex]
[tex]\[ P(z \geq 1.75) \][/tex] is the complementary probability of [tex]\( P(z \leq 1.75) \)[/tex]:
[tex]\[ P(z \geq 1.75) = 1 - P(z \leq 1.75) \][/tex]
Therefore, the correct equivalent expression for [tex]\( P(z \leq -1.75) \)[/tex] is:
[tex]\[ 1 - P(z \leq 1.75) \][/tex]
Hence, the correct option is:
[tex]\[ \boxed{4} \][/tex]
