Question

which of the following probabilities is equal to approximately 0.2957? use the portion of the standard normal table below to help answer the question.

z	probability
0.25	0.5987
0.50	0.6915
0.75	0.7734
1.00	0.8413
1.25	0.8944
1.50	0.9332
1.75	0.9599

• ( p(-1.25 leq z leq 0.25) )
• ( p(-1.25 leq z leq 0.75) )
• ( p(0.25 leq z leq 1.25) )
• ( p(0.75 leq z leq 1.25) )

Explanation:

Step1: Recall Normal Probability Rule

For standard normal distribution, \( P(a \leq z \leq b) = P(z \leq b) - P(z \leq a) \). Also, due to symmetry, \( P(z \leq -k) = 1 - P(z \leq k) \).

Step2: Analyze Each Option

Option 1: \( P(-1.25 \leq z \leq 0.25) \)

• \( P(z \leq 0.25) = 0.5987 \)
• \( P(z \leq -1.25) = 1 - P(z \leq 1.25) = 1 - 0.8944 = 0.1056 \)
• \( P(-1.25 \leq z \leq 0.25) = 0.5987 - 0.1056 = 0.4931 \) (Not 0.2957)

Option 2: \( P(-1.25 \leq z \leq 0.75) \)

• \( P(z \leq 0.75) = 0.7734 \)
• \( P(z \leq -1.25) = 0.1056 \) (from above)
• \( P(-1.25 \leq z \leq 0.75) = 0.7734 - 0.1056 = 0.6678 \) (Not 0.2957)

Option 3: \( P(0.25 \leq z \leq 1.25) \)

• \( P(z \leq 1.25) = 0.8944 \)
• \( P(z \leq 0.25) = 0.5987 \)
• \( P(0.25 \leq z \leq 1.25) = 0.8944 - 0.5987 = 0.2957 \) (Matches)

Option 4: \( P(0.75 \leq z \leq 1.25) \)

• \( P(z \leq 1.25) = 0.8944 \)
• \( P(z \leq 0.75) = 0.7734 \)
• \( P(0.75 \leq z \leq 1.25) = 0.8944 - 0.7734 = 0.1210 \) (Not 0.2957)

Answer:

\( P(0.25 \leq z \leq 1.25) \)