Question

which of the following statements is equivalent to $p(z\geq1.7)$?
$p(z\geq -1.7)$
$1 - p(z\geq -1.7)$
$p(z\leq1.7)$
$1 - p(z\geq1.7)$

Explanation:

Step1: Recall Normal Distribution Symmetry

The standard normal distribution (z - distribution) is symmetric about \( z = 0 \). For any real number \( a \), \( P(z\geq a)=P(z\leq - a) \) and also \( P(z\geq a)=1 - P(z\geq - a) \)? Wait, no, let's recall the correct properties. The total area under the standard normal curve is 1. Also, due to symmetry, \( P(z\geq a)=P(z\leq - a) \) and \( P(z\geq a)=1 - P(z < a) \), but also, since \( P(z\geq a)+P(z < a) = 1 \), and \( P(z\geq - a)=1 - P(z < - a) \). But from symmetry, \( P(z < a)=P(z\geq - a) \) (because the area to the left of \( a \) is equal to the area to the right of \( - a \)). So \( P(z\geq a)=1 - P(z\geq - a) \)? Wait, let's test with \( a = 1.7 \).

The area to the right of \( z = 1.7 \) (i.e., \( P(z\geq1.7) \)) should be equal to the area to the left of \( z=- 1.7 \) (by symmetry). But also, \( P(z\geq1.7)=1 - P(z < 1.7) \), and \( P(z < 1.7)=P(z\geq - 1.7) \) (because the area to the left of \( 1.7 \) is the same as the area to the right of \( - 1.7 \) due to symmetry). So substituting, \( P(z\geq1.7)=1 - P(z\geq - 1.7) \).

Let's analyze each option:

• Option 1: \( P(z\geq - 1.7) \): The area to the right of \( z=-1.7 \) is much larger than the area to the right of \( z = 1.7 \), so this is not equal.
• Option 2: \( 1 - P(z\geq - 1.7) \): From the above reasoning, since \( P(z\geq1.7)=1 - P(z < 1.7) \) and \( P(z < 1.7)=P(z\geq - 1.7) \), then \( P(z\geq1.7)=1 - P(z\geq - 1.7) \). Let's verify with a simple case. Let \( a = 0 \), \( P(z\geq0)=0.5 \), and \( 1 - P(z\geq0)=0.5 \), which works. For \( a = 1 \), \( P(z\geq1)\approx0.1587 \), and \( 1 - P(z\geq - 1)=1-(1 - P(z < - 1))=P(z < - 1)\approx0.1587 \), which matches.
• Option 3: \( P(z\leq1.7) \): \( P(z\leq1.7) \) is the area to the left of \( 1.7 \), which is \( 1 - P(z\geq1.7) \), not equal to \( P(z\geq1.7) \) (unless \( P(z\geq1.7)=0.5 \), which it's not for \( z = 1.7 \)).
• Option 4: \( 1 - P(z\geq1.7) \): This would be \( P(z < 1.7) \), which is not equal to \( P(z\geq1.7) \) (unless \( P(z\geq1.7)=0.5 \), which it's not).

Step2: Evaluate Each Option

• Option 1: \( P(z\geq - 1.7) \): As the distribution is symmetric, the area to the right of \( - 1.7 \) is the area to the left of \( 1.7 \), which is \( 1 - P(z\geq1.7) \), not equal to \( P(z\geq1.7) \).
• Option 2: \( 1 - P(z\geq - 1.7) \): From the symmetry and the total area property (total area = 1), since \( P(z\geq1.7)+P(z\geq - 1.7)=1 \) (because \( P(z\geq - 1.7) \) is the area to the right of \( - 1.7 \), which is the same as the area to the left of \( 1.7 \), and \( P(z\geq1.7) \) is the area to the right of \( 1.7 \), so their sum is 1). So \( P(z\geq1.7)=1 - P(z\geq - 1.7) \).
• Option 3: \( P(z\leq1.7) \): \( P(z\leq1.7) \) is the area to the left of \( 1.7 \), which is \( 1 - P(z\geq1.7) \), not equal to \( P(z\geq1.7) \).
• Option 4: \( 1 - P(z\geq1.7) \): This is \( P(z < 1.7) \), which is not equal to \( P(z\geq1.7) \) (unless \( P(z\geq1.7)=0.5 \), which it is not for \( z = 1.7 \)).

Answer:

\( 1 - P(z\geq - 1.7) \) (the second option)