Math Problem Statement

The image shows two graphs of standard normal distributions. One graph has z-scores -0.84 and 1.28, while the second graph shows an area under the curve with a value of 0.8907.

Solution

The image you provided shows two graphs of standard normal distributions. Let's break them down:

  1. Graph 11:

    • The curve represents a standard normal distribution with a mean of 0 and a standard deviation of 1.
    • Two z-scores are marked: z=0.84z = -0.84 and z=1.28z = 1.28, defining a region between these values.
    • This region, shaded in green, likely represents the area under the curve between these z-scores, which corresponds to the probability or percentage of data that falls within this range.
  2. Graph 13:

    • The curve again represents a standard normal distribution.
    • The area shaded corresponds to the region from z=0z = 0 up to an unknown positive value of zz.
    • The area under the curve for this region is given as 0.8907. This means that 89.07% of the data lies between 0 and the unknown zz-score.

Would you like further explanation on how to calculate the areas under the curve or z-scores? Let me know if you'd like to dive deeper into these concepts.

Here are five related questions to consider:

  1. How do you calculate the area under a standard normal curve between two z-scores?
  2. What is the significance of z-scores in a standard normal distribution?
  3. How do you use a z-table to find the area under the normal curve?
  4. What does a negative z-score represent in the context of a standard normal distribution?
  5. How would you calculate the unknown z-score corresponding to an area of 0.8907?

Tip: In a standard normal distribution, the total area under the curve always equals 1 (or 100%), so areas represent probabilities.

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Math Problem Analysis

Mathematical Concepts

Standard Normal Distribution
Z-scores
Probability

Formulas

Z = (X - μ) / σ

Theorems

68-95-99.7 Rule
Properties of Standard Normal Distribution

Suitable Grade Level

Grades 9-12