A population of values has a normal distribution with a population mean of 106.4 and
a population standard deviation of 52.6 You intend to draw a random sample of size n=90. Find the probability that a single randomly selected value is greater than 123.
P(X > 123) =

To find the probability that a single randomly selected value from a population is greater than 123, we can use the properties of the normal distribution. Specifically, we will calculate the z-score and then use the standard normal distribution to find the probability.

Given:
- Population mean (\(\mu\)) = 106.4
- Population standard deviation (\(\sigma\)) = 52.6

First, we calculate the z-score for \(X = 123\):

\[
z = \frac{X - \mu}{\sigma} = \frac{123 - 106.4}{52.6} = \frac{16.6}{52.6} \approx 0.315
\]

Next, we use the z-score to find the cumulative probability from standard normal distribution tables or a calculator. The cumulative probability up to \(z = 0.315\) is approximately:

\[
P(Z \leq 0.315) \approx 0.6230
\]

The probability that a single randomly selected value is greater than 123 is:

\[
P(X > 123) = 1 - P(Z \leq 0.315) = 1 - 0.6230 = 0.3770
\]

So, the probability that a single randomly selected value is greater than 123 is approximately \(0.3770\).

Do you want details or have any questions?

Here are some more questions you might find interesting:
1. How do you calculate the probability of a value being within a certain range in a normal distribution?
2. What is the z-score and how is it used in statistics?
3. How do you use normal distribution tables to find probabilities?
4. How can you determine if a data point is an outlier in a normal distribution?
5. What are the differences between population and sample statistics?

**Tip:** When working with normal distributions, always sketch the distribution curve and shade the area representing the probability to visualize the problem better.

Learn how to calculate the probability that a randomly selected value from a normal distribution with mean 106.4 and standard deviation 52.6 is greater than 123. Includes step-by-step solution using the z-score and cumulative probability formula.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation