/*! 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 70 Assume that the random variable ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 70

Assume that the random variable \(X\) is normally distributed. Use the given information to find the unknown parameter or parameters of the distribution. If \(P(X \geq 10)=0.3936\) and \(P(X \leq 6)=0.3936\), find \(E(X)\) and \(S D(X)\).

Short Answer

Expert verified
\(E(X) = 8\), \(SD(X) = 7.41\).

Step by step solution

01

Understand the Problem

Given a normally distributed random variable \(X\) with \(P(X geq 10) = 0.3936\) and \(P(X leq 6) = 0.3936\), we need to find the mean \(E(X)\) and standard deviation \(SD(X)\). These probabilities can help us locate the values on the Z-table.
02

Identify Z-scores

Find the z-scores corresponding to the provided probabilities from the Z-table. For \(P(Z geq z) = 0.3936\), \(z = -0.27\). Similarly, for \(P(Z leq z) = 0.3936\), \(z = -0.27\).
03

Set Up the Equations

Use the z-score formula \(z = \frac{X - \mu}{\sigma}\) to set up equations based on the given data. For \(X = 10\), the equation is \(-0.27 = \frac{10 - \mu}{\sigma}\). For \(X = 6\), the equation is \(-0.27 = \frac{6 - \mu}{\sigma}\).
04

Solve for \(\, \, \mu\)

Subtract the second equation from the first one: \(\frac{10 - \, \, 6}{\sigma} = 0\). This simplifies to \(4 = 0\), showing the z-scores are equal and confirming symmetry.
05

Find \(\mu\)

Because both z-scores are equal around the mean, it means \(\frac{10 - \mu}{\sigma} = -\frac{6 - \mu}{\sigma}\). Solving this gives \(\mu = 8\).
06

Find \(\sigma\)

Use the previously identified z-score for X=10: \(-0.27 = \frac{10 - 8}{\sigma}\). Hence, \(-0.27 = \frac{2}{\sigma}\), solving it \(\sigma = \frac{2}{-0.27} \approx 7.41\).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Distribution
The normal distribution is a fundamental concept in statistics. It is a type of continuous probability distribution for a real-valued random variable. The curve of a normal distribution is often referred to as the 'bell curve' because of its bell-like shape.

Key properties of normal distribution include:
  • Symmetry: The distribution is symmetric around its mean, meaning the left side is a mirror image of the right side.
  • Mean, median, and mode: All three measures of central tendency for a normal distribution are equal and located at the center of the distribution.
  • Asymptotic: The tails of the distribution approach, but never touch, the horizontal axis.
In this exercise, we know that the random variable X follows a normal distribution. This information helps us use the properties of the normal distribution to find the mean and standard deviation.
Mean and Standard Deviation
The mean (often represented as \(\mu\)) is the average value of a data set. In a normal distribution, it is the center point of the data.

The standard deviation (often represented as \(\sigma\)) measures the amount of variation or dispersion in a set of values. A low standard deviation means the values are close to the mean, while a high standard deviation means the values are spread out over a wider range.

In solving the problem, we used given probabilities to set up equations involving the z-scores, which were then used to find the mean and standard deviation. The process involved:
  • Identifying z-scores from probabilities
  • Setting up equations using the z-score formula
  • Solving these equations to find \(\mu\) and \(\sigma\)
We found that the mean (\textmu) is 8 and the standard deviation (\textsigma) is approximately 7.41.
Z-score
A z-score represents the number of standard deviations a data point is from the mean of the data set. It helps in determining the position of a value within a normal distribution.

The z-score formula is given by \(z = \frac{X - \mu}{\sigma}\), where:
  • \(X\) is the value of the data point.
  • \(\mu\) is the mean of the distribution.
  • \(\sigma\) is the standard deviation.
In this exercise, we used the given probabilities to find the corresponding z-scores from the Z-table. For both probabilities of 0.3936, the z-score was found to be -0.27.

These z-scores were then plugged into the z-score formula to create equations, which were solved to find the mean and standard deviation of the distribution. Using z-scores is crucial in statistics as it standardizes different data points, allowing for comparisons.

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

Let \(Z\) be a Normal \((0,1)\) random variable. Find the probability that \(Z\) is in the interval. $$ [1.17, \infty) $$

Use the density function to compute the probability. $$ f(x)=4 x^{3} \text { over }[0,1], P(1 / 4 \leq X \leq 3 / 4) $$

Assume that the random variable \(X\) is normally distributed. Use the given information to find the unknown parameter or parameters of the distribution. If \(\mathrm{SD}(X)=2\) and \(P(X \geq-1)=0.409\), find \(E(X)\)

Compute the probability using the given distribution. \(P(0 \leq X \leq 3), X\) is Uniform \((0,7)\)

Let \(Z\) be a Normal \((0,1)\) random variable. Find the probability that \(Z\) is in the interval. $$ [1.26,1.43] $$
See all solutions

Recommended explanations on Biology Textbooks

Ecological Levels

Read Explanation

Biological Processes

Read Explanation

Biological Structures

Read Explanation

Responding to Change

Read Explanation

Bioenergetics

Read Explanation

Cells

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.