This is a problem related to the standard normal distribution, where you are tasked with finding the area under the curve between $z = -1.77$ and $z = 2.36$.

Steps to Solve:

1. Understand the Problem: The graph depicts the standard normal distribution, which has:

   • Mean ($\mu$) = 0
   • Standard deviation ($\sigma$) = 1

   The area under the curve between two $z$-scores represents the probability for the range of values defined by those $z$-scores.

2. Find Cumulative Probabilities: The cumulative probability for any $z$-score is found using standard normal tables or a calculator.

   • $P(Z \leq -1.77) = 0.0384$ (left tail probability for $z = -1.77$)
   • $P(Z \leq 2.36) = 0.9909$ (cumulative probability for $z = 2.36$)

3. Compute the Area: To find the area between $z = -1.77$ and $z = 2.36$, subtract the cumulative probability for $z = -1.77$ from that of $z = 2.36$: $\text{Area} = P(Z \leq 2.36) - P(Z \leq -1.77)$ Substituting the values: $\text{Area} = 0.9909 - 0.0384 = 0.9525$

Final Answer:

The area under the shaded region is 0.9525, or 95.25%.

Would you like further clarification or additional examples?

Related Questions:

1. What is the significance of the area under the standard normal curve in probability?
2. How would the result change if the mean and standard deviation were different?
3. How can the z-scores be used to find percentiles?
4. What tools are typically used to find cumulative probabilities for z-scores?
5. What does a z-score of 0 correspond to in a standard normal distribution?

Tip:

Remember, the total area under the standard normal curve is always equal to 1, and symmetry can help simplify calculations.