https://file.aiquickdraw.com/mathbot/file/1726855253633ino4a7uw.jpg

To solve these questions, we need to use the standard normal distribution, where the mean is 0 and the standard deviation is 1. We will look for the z-scores corresponding to specific proportions.

### (a) Find the number \( z \) such that the proportion of observations that are less than \( z \) in a standard normal distribution is 0.33.

We are given \( P(Z < z) = 0.33 \). To find \( z \), we need to look up the value of \( z \) in the z-table or use an inverse normal distribution calculator for a cumulative probability of 0.33.

### (b) Find the number \( z \) such that 38% of all observations from a standard normal distribution are greater than \( z \).

We are given \( P(Z > z) = 0.38 \), which means the cumulative probability to the left of \( z \) is \( 1 - 0.38 = 0.62 \). So we need to find the z-score where \( P(Z < z) = 0.62 \).

Let me calculate both values for you.The solutions are:

### (a) The z-score such that the proportion of observations less than \( z \) is 0.33 is approximately \( z = -0.44 \).

### (b) The z-score such that 38% of all observations are greater than \( z \) is approximately \( z = 0.31 \).

Would you like more details on how these values were calculated or on the process of using z-tables?

Here are some related questions for practice:
1. What is the z-score for 50% of the observations in a standard normal distribution?
2. How do you calculate a z-score if given a certain proportion or percentile?
3. What does a negative z-score indicate in a standard normal distribution?
4. If 80% of observations are less than \( z \), what is the z-score?
5. How does the area under the standard normal curve relate to probabilities?

**Tip:** The inverse of the cumulative distribution function (CDF) is used to find z-scores from probabilities.

Find the number z such that the proportion of observations that are less than z in a standard Normal distribution is 0.33, and find the number z such that 38% of all observations from a standard Normal distribution are greater than z.

This page covers how to find the z-score for a given proportion in a standard normal distribution. It includes the solution for finding the z-score where the proportion of observations less than z is 0.33, and where 38% of observations are greater than z.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation