Question

Let Z be a random variable with the standard normal distribution. Find

(1) P(−1.35≤Z≤2.15)
(2) P(0.55≤Z≤1.25)
(3) P(−1.79≤Zˉ≤−0.54)

Answer 1

To find the probabilities using the standard normal distribution and z-scores, calculate the z-scores and use the z-table to find the areas under the normal curve. The calculations for the three questions are provided step-by-step.

Step-by-step explanation:

To find the probabilities in these questions, we need to use the standard normal distribution and z-scores. The z-table can be used to find the area under the normal curve to the left of a given z-score. Here are the step-by-step calculations for each question:

1. P(−1.35≤Z≤2.15) = P(Z≤2.15) - P(Z≤-1.35) = 0.9842 - 0.0885 = 0.8957
2. P(0.55≤Z≤1.25) = P(Z≤1.25) - P(Z≤0.55) = 0.8944 - 0.2912 = 0.6032
3. P(−1.79≤Zˉ≤−0.54) = P(Zˉ≤-0.54) - P(Zˉ≤-1.79) = 0.2951 - 0.0384 = 0.2567