Question

for a standard normal distribution, find the approximate value of $p(z geq -1.25)$. use the portion of the standard normal table below to help answer the question.

z	probability
0.25	0.5987
1.00	0.8413
1.25	0.8944
1.50	0.9332
1.75	0.9599

• 11%
• 39%
• 61%
• 89%

Explanation:

Step1: Recall standard normal properties

In a standard normal distribution, \( P(Z \geq -z) = 1 - P(Z < -z) \), and due to symmetry, \( P(Z < -z)=1 - P(Z < z) \). So \( P(Z \geq -1.25)=1-(1 - P(Z < 1.25)) = P(Z < 1.25) \)? Wait, no, correction: The standard normal table gives \( P(Z < z) \). For \( P(Z \geq -1.25) \), we can use the fact that the total area under the curve is 1, and \( P(Z \geq -1.25)=1 - P(Z < -1.25) \). By symmetry, \( P(Z < -1.25)=1 - P(Z < 1.25) \). So substituting, \( P(Z \geq -1.25)=1-(1 - P(Z < 1.25))=P(Z < 1.25) \)? Wait, no, let's check the table. The table has \( z = 1.25 \) with probability 0.8944. Wait, actually, \( P(Z \geq -1.25) \) is the area to the right of -1.25. Since the normal curve is symmetric, the area to the right of -1.25 is equal to the area to the left of 1.25 (because -1.25 and 1.25 are symmetric around 0). Wait, no, wait: The area to the right of -1.25 is \( 1 - P(Z < -1.25) \). And \( P(Z < -1.25)=1 - P(Z < 1.25) \) (by symmetry: \( P(Z < -z)=1 - P(Z < z) \)). So \( P(Z \geq -1.25)=1-(1 - P(Z < 1.25))=P(Z < 1.25) \). Wait, but looking at the table, for \( z = 1.25 \), the probability is 0.8944, which is 89.44%, approximately 89%.

Step2: Verify with table

From the given table, when \( z = 1.25 \), the probability (area to the left of 1.25) is 0.8944. Since \( P(Z \geq -1.25) \) is the area to the right of -1.25, which by symmetry is equal to the area to the left of 1.25 (because the distribution is symmetric about 0), so this probability is approximately 0.8944, which is about 89%.

Answer:

89% (the option with 89%)