Math Problem Statement

The standard normal curve is graphed below. Shade the region under the standard normal curve to the left of z = -1.50. Use this table or calculator to find the area under the standard normal curve to the left of z = -1.50, and give the answer to four decimal places.

Solution

This image shows a problem involving the standard normal distribution curve:

  1. Part (a): You are asked to shade the region under the standard normal curve to the left of z=1.50z = -1.50. In this case, you need to shade the area from z=z = -\infty (far left) to z=1.50z = -1.50, representing the cumulative probability up to that point.

  2. Part (b): You need to use a table or calculator to find the area under the standard normal curve to the left of z=1.50z = -1.50. Using a standard normal distribution table (Z-table), the cumulative probability for z=1.50z = -1.50 is approximately 0.0668.

Thus, the area to the left of z=1.50z = -1.50 is 0.0668.

Would you like me to explain how to read the Z-table or discuss the process in more detail?

Here are some follow-up questions:

  1. What is the area under the curve for z=0z = 0?
  2. How do you use a Z-table to find probabilities for other Z-values?
  3. What does the shaded area represent in a real-world context?
  4. How is the area under the standard normal curve related to probability?
  5. How would you calculate the area between two Z-values?

Tip: The total area under the standard normal curve is always 1, representing 100% probability.

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Math Problem Analysis

Mathematical Concepts

Statistics
Normal Distribution
Probability

Formulas

Cumulative probability from Z-table
Standard normal distribution formula: P(Z < z)

Theorems

Central Limit Theorem
Properties of the Normal Distribution

Suitable Grade Level

Grades 10-12