Standard Normal areas Find the proportion of observations in a standard Normal distribution that satisfies each of the following statements.
A. z > -1.66
B. -1.66 < z < 2.85

To solve these problems, we will use the properties of the standard Normal distribution, which is a normal distribution with a mean of 0 and a standard deviation of 1. The proportion of observations corresponds to the area under the curve for the given z-values.

### A. Proportion where \( z > -1.66 \)
- To find this proportion, we need to determine the area to the right of \( z = -1.66 \).

1. **Find the area to the left of \( z = -1.66 \)** using the standard Normal distribution table or a calculator. This area is approximately **0.0485**.
2. **Calculate the area to the right of \( z = -1.66 \)**:  
   \( P(z > -1.66) = 1 - P(z < -1.66) \)
   
   \( P(z > -1.66) = 1 - 0.0485 = 0.9515 \)

   **So, the proportion of observations where \( z > -1.66 \) is approximately 0.9515 or 95.15%.**

### B. Proportion where \( -1.66 < z < 2.85 \)
- To find this proportion, we need to determine the area between \( z = -1.66 \) and \( z = 2.85 \).

1. **Find the area to the left of \( z = 2.85 \)** using the standard Normal distribution table or a calculator. This area is approximately **0.9978**.
2. **Find the area to the left of \( z = -1.66 \)** (already calculated as **0.0485**).
3. **Calculate the area between \( z = -1.66 \) and \( z = 2.85 \)**:  
   \( P(-1.66 < z < 2.85) = P(z < 2.85) - P(z < -1.66) \)
   
   \( P(-1.66 < z < 2.85) = 0.9978 - 0.0485 = 0.9493 \)

   **So, the proportion of observations where \( -1.66 < z < 2.85 \) is approximately 0.9493 or 94.93%.**

Would you like more details on these calculations or have any other questions? 

Here are some related questions you might find useful:

1. What is the probability for \( z \) values outside the range of \( -1.66 < z < 2.85 \)?
2. How do you find the z-score given a specific proportion of observations?
3. What is the significance of the z-score in a normal distribution?
4. How do you interpret z-scores in terms of standard deviations?
5. What are the properties of a standard normal distribution?

**Tip:** Remember that the total area under the standard normal curve is always equal to 1, representing the total probability.

Learn how to find proportions in a standard normal distribution for z-values greater than -1.66 and between -1.66 and 2.85. Detailed solutions using the standard normal distribution table and cumulative distribution function (CDF) are provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation