Math Problem Statement

Find areas under the normal curve for given z-scores.

Solution

This quiz involves finding areas under the standard normal curve based on specified z-scores. I'll walk through each question step-by-step:

1. Find the area under the normal curve below (to the left of) z=1.25z = 1.25.

  • You are asked to find the area to the left of z=1.25z = 1.25. Using a standard normal distribution table or a calculator: P(Z<1.25)0.8944P(Z < 1.25) \approx 0.8944 So, the area is approximately 0.8944 or 89.44%.

2. Find the area under the normal curve above (to the right of) z=2.06z = 2.06.

  • To find the area to the right of z=2.06z = 2.06: P(Z>2.06)=1P(Z<2.06)P(Z > 2.06) = 1 - P(Z < 2.06) Using a normal distribution table: P(Z<2.06)0.9803P(Z < 2.06) \approx 0.9803 So: P(Z>2.06)=10.9803=0.0197P(Z > 2.06) = 1 - 0.9803 = 0.0197 The area is approximately 0.0197 or 1.97%.

3. Find the area under the normal curve between z=1.25z = 1.25 and z=2.21z = 2.21.

  • To find the area between two z-scores: P(1.25<Z<2.21)=P(Z<2.21)P(Z<1.25)P(1.25 < Z < 2.21) = P(Z < 2.21) - P(Z < 1.25) From the table: P(Z<2.21)0.9864,P(Z<1.25)0.8944P(Z < 2.21) \approx 0.9864, \quad P(Z < 1.25) \approx 0.8944 So: P(1.25<Z<2.21)=0.98640.8944=0.0920P(1.25 < Z < 2.21) = 0.9864 - 0.8944 = 0.0920 The area is approximately 0.0920 or 9.20%.

4. Find the area under the normal curve between z=1.73z = -1.73 and z=0.96z = 0.96.

  • To find the area between these two z-scores: P(1.73<Z<0.96)=P(Z<0.96)P(Z<1.73)P(-1.73 < Z < 0.96) = P(Z < 0.96) - P(Z < -1.73) From the table: P(Z<0.96)0.8315,P(Z<1.73)0.0418P(Z < 0.96) \approx 0.8315, \quad P(Z < -1.73) \approx 0.0418 So: P(1.73<Z<0.96)=0.83150.0418=0.7897P(-1.73 < Z < 0.96) = 0.8315 - 0.0418 = 0.7897 The area is approximately 0.7897 or 78.97%.

Would you like further clarification or details on any of these questions?


Here are five related questions to deepen your understanding:

  1. How do you calculate the area for zz-values greater than a given zz?
  2. What is the area under the curve between z=1z = -1 and z=1z = 1?
  3. How does the area under the normal curve change as zz-values increase?
  4. What is the total area under the normal distribution curve?
  5. How do you interpret the area under the curve in a real-world context?

Tip: When working with normal distribution tables, remember that they usually provide the area to the left of the z-score. For right-tail areas, you need to subtract this value from 1.

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Math Problem Analysis

Mathematical Concepts

Statistics
Normal Distribution
Z-scores

Formulas

Area to the left of z: P(Z < z)
Area to the right of z: P(Z > z) = 1 - P(Z < z)
Area between two z-scores: P(a < Z < b) = P(Z < b) - P(Z < a)

Theorems

Properties of the Standard Normal Distribution

Suitable Grade Level

College Level (Introductory Statistics)