Question

Calculate the probability that a standard normal variable takes on a value between the z-scores of 0.75 and 2.42

Answer 1

To find the probability that a standard normal variable \( Z \) takes on a value between the z-scores of 0.75 and 2.42, we can use the standard normal distribution table (or a calculator) to find the cumulative probabilities corresponding to these z-scores.

1. **Find the cumulative probability for \( Z = 0.75 \)**:
- Using the standard normal distribution table, we find:
\[
P(Z \leq 0.75) \approx 0.7734
\]

2. **Find the cumulative probability for \( Z = 2.42 \)**:
- Again, using the standard normal distribution table, we find:
\[
P(Z \leq 2.42) \approx 0.9924
\]

3. **Calculate the probability that \( Z \) is between 0.75 and 2.42**:
\[
P(0.75 < Z < 2.42) = P(Z \leq 2.42) - P(Z \leq 0.75)
\]
Substituting the values we found:
\[
P(0.75 < Z < 2.42) = 0.9924 - 0.7734 = 0.2190
\]

So, the probability that a standard normal variable takes on a value between the z-scores of 0.75 and 2.42 is approximately **0.2190**.