which of the following probabilities is the greatest for a standard normal distribution? p(-1.5≤z≤-0.5)…

Explanation:

Step1: Recall standard - normal table property

The standard - normal distribution is symmetric about (z = 0), and the probability density function is highest at (z = 0) and decreases as (|z|) increases. The area under the standard - normal curve between two (z) - values (z_1) and (z_2) is given by (P(z_1\leq z\leq z_2)=\Phi(z_2)-\Phi(z_1)), where (\Phi(z)) is the cumulative distribution function of the standard - normal distribution.

Step2: Analyze each option

• For (P(-1.5\leq z\leq - 0.5)=\Phi(-0.5)-\Phi(-1.5)). Using the property (\Phi(-z)=1 - \Phi(z)), we have (P(-1.5\leq z\leq - 0.5)=(1 - \Phi(0.5))-(1 - \Phi(1.5))=\Phi(1.5)-\Phi(0.5)).
• For (P(-0.5\leq z\leq0.5)=\Phi(0.5)-\Phi(-0.5)=\Phi(0.5)-(1 - \Phi(0.5)) = 2\Phi(0.5)-1).
• For (P(0.5\leq z\leq1.5)=\Phi(1.5)-\Phi(0.5)).
• For (P(1.5\leq z\leq2.5)=\Phi(2.5)-\Phi(1.5)).

Step3: Compare the areas

The standard - normal curve has more area concentrated around (z = 0). The interval ((-0.5,0.5)) is centered at (z = 0), while the other intervals are farther from (z = 0). Since the standard - normal distribution is symmetric about (z = 0) and the density function is highest at (z = 0), the probability (P(-0.5\leq z\leq0.5)) is the greatest among the given options.

Answer:

P(-0.5 ≤ z ≤ 0.5)