/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 A normal random variable \(x\) h... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 14

A normal random variable \(x\) has an unknown mean \(\mu\) and standard deviation \(\sigma=2\). If the probability that \(x\) exceeds 7.5 is \(.8023,\) find \(\mu\).

Short Answer

Expert verified
Answer: The mean of the distribution is 9.24.

Step by step solution

01

Identify the information given

We have the following data given in the question: - The standard deviation (蟽) is 2. - The probability that the normal random variable (x) exceeds 7.5 is 0.8023. Our goal is to find the mean (渭) of the distribution.
02

Use the z-score formula

The z-score represents the number of standard deviations a value is away from the mean of the distribution. It can be calculated by using the following formula: \(Z = \frac{X - \mu}{\sigma}\) In this problem, we must find the z-score that corresponds to the area 0.8023 on the right side of the distribution (since it represents the probability that x exceeds 7.5). This means that we need to look for the area on the left side, which is given by \(1 - 0.8023 = 0.1977\).
03

Find the z-score

Using a standard normal distribution table, we can find the z-score that corresponds to an area to the left of 0.1977. The table value closest to this is 0.197664. So, the z-score is -0.87 (keep in mind that the table represents the area to the left, so a negative z-score means the value is below the mean).
04

Plug the z-score, x value, and 蟽 into the formula

Now that we know the z-score (-0.87), we can use it along with the x value (7.5) and the standard deviation (2) to find the mean (渭) using the formula: \(-0.87 = \frac{7.5 - \mu}{2}\)
05

Solve for 渭

By solving for 渭 (the mean) in the equation, we get: \(-0.87 \times 2 = 7.5 - \mu\) \(-1.74 = 7.5 - \mu\) \(\mu = 7.5 + 1.74\) \(\mu = 9.24\) The mean (渭) of the normal random variable is 9.24.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the following probabilities for the standard normal random variable \(z\): a. \(P(-1.431.34)\) e. \(P(z<-4.32)\)

Calculate the area under the standard normal curve between these values: a. \(z=-2.0\) and \(z=2.0\) b. \(z=-2.3\) and -1.5

Calculate the area under the standard normal curve to the left of these values: a. \(z=-.90\) b. \(z=2.34\) c. \(z=5.4\)

a. Find a \(z_{0}\) that has area .9505 to its left. b. Find a \(z_{0}\) that has area .05 to its left.

It is known that \(30 \%\) of all calls coming into a telephone exchange are long-distance calls. If 200 calls come into the exchange, what is the probability that at least 50 will be long-distance calls?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.