Math Problem Statement

Find the indicated area under the standard normal curve. To the left of equals negative 0.88 and to the right of z equals0.88

Solution

To solve this, we need to calculate two areas under the standard normal curve:

  1. The area to the left of z=0.88z = -0.88
  2. The area to the right of z=0.88z = 0.88

The total area under the standard normal curve equals 1 (or 100%).

1. Area to the left of z=0.88z = -0.88

From standard normal distribution tables or using a calculator, the cumulative probability to the left of z=0.88z = -0.88 is approximately 0.1894.

2. Area to the right of z=0.88z = 0.88

The cumulative probability to the left of z=0.88z = 0.88 is 0.8106. So, the area to the right of z=0.88z = 0.88 is:

10.8106=0.18941 - 0.8106 = 0.1894

Total Area

Since the areas to the left of z=0.88z = -0.88 and to the right of z=0.88z = 0.88 are equal, we can calculate the total area by summing these two areas:

Total Area=0.1894+0.1894=0.3788\text{Total Area} = 0.1894 + 0.1894 = 0.3788

Thus, the total area under the standard normal curve to the left of z=0.88z = -0.88 and to the right of z=0.88z = 0.88 is approximately 0.3788.

Would you like more details or have any questions about the steps?


Here are five related questions you might find helpful:

  1. What is the area between two z-scores, like between z=1z = -1 and z=1z = 1?
  2. How do you calculate the area under the normal curve for any given z-value?
  3. How does the symmetry of the normal distribution simplify these types of problems?
  4. Can you find the area for z-values beyond 3 standard deviations?
  5. How do z-scores relate to probabilities in a normal distribution?

Tip: Using a z-table or calculator is a quick way to find cumulative areas under the normal curve.

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Math Problem Analysis

Mathematical Concepts

Statistics
Standard Normal Distribution
Z-scores
Probability

Formulas

Cumulative area under the normal curve
P(Z < z) = cumulative probability for a given z-value
P(Z > z) = 1 - cumulative probability for a given z-value

Theorems

Symmetry of the normal distribution
Empirical rule for normal distributions

Suitable Grade Level

Grades 11-12, College Level