Math Problem Statement

Find the percent of the total area under the standard normal curve between the following​ z-scores. zequals=negative 1.3−1.3 and zequals=negative 0.75−0.75

Solution

To find the percent of the total area under the standard normal curve between the z-scores z=1.3z = -1.3 and z=0.75z = -0.75, we can use the standard normal distribution table or a calculator that provides cumulative probabilities for the standard normal distribution.

  1. First, find the cumulative probability for z=1.3z = -1.3.
  2. Next, find the cumulative probability for z=0.75z = -0.75.
  3. Subtract the cumulative probability at z=1.3z = -1.3 from the cumulative probability at z=0.75z = -0.75.

Using the standard normal distribution table:

  1. The cumulative probability for z=1.3z = -1.3 is approximately 0.0968.
  2. The cumulative probability for z=0.75z = -0.75 is approximately 0.2266.

Now, subtract these probabilities:

0.22660.0968=0.12980.2266 - 0.0968 = 0.1298

Therefore, the percent of the total area under the standard normal curve between z=1.3z = -1.3 and z=0.75z = -0.75 is 0.1298×100=12.98%0.1298 \times 100 = 12.98\%.

Would you like any more details or have any other questions?

Here are 5 questions you might want to ask next:

  1. How do you find the area under the normal curve for any given z-score range?
  2. What is the significance of the standard normal distribution in statistics?
  3. Can you explain how to use the standard normal distribution table?
  4. How can we find the z-score for a given percentile?
  5. What are some real-life applications of the standard normal distribution?

Tip: When working with normal distributions, always sketch the curve and shade the area of interest. This helps in visualizing the problem and understanding which cumulative probabilities to use.

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Math Problem Analysis

Mathematical Concepts

Statistics
Normal Distribution

Formulas

Standard Normal Distribution Table

Theorems

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Suitable Grade Level

Advanced High School