/*! 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 49 In a normal distribution with \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 49

In a normal distribution with \(\mu=-15\) and \(\sigma=0.4\) find the \(x\) -value that corresponds to the a) 46 th percentile b) 92 nd percentile

Short Answer

Expert verified
For the 46th percentile, x 鈮� -15.04. For the 92nd percentile, x 鈮� -14.44.

Step by step solution

01

- Understand what the percentile means

Percentiles indicate the relative standing of a value within a distribution. The 46th percentile means 46% of the data falls below this value, and the 92nd percentile means 92% of the data falls below this value.
02

- Convert percentiles to z-scores

Use a z-score table or calculator to find the z-scores corresponding to the 46th and 92nd percentiles. The z-score for the 46th percentile is approximately -0.1 and for the 92nd percentile is approximately 1.41.
03

- Use the z-score formula for each percentile

The z-score formula is: ewline \( z = \dfrac{x - \mu}{\sigma} \). ewline Rearranging for x gives: \( x = z \cdot \sigma + \mu \).
04

- Calculate the x-value for the 46th percentile

For the 46th percentile: ewline \( z = -0.1 \), ewline Using the parameters ewline \( \mu = -15 \) and \( \sigma = 0.4 \), ewline \( x = -0.1 \cdot 0.4 - 15 \). ewline Therefore, \( x = -0.04 - 15 = -15.04 \).
05

- Calculate the x-value for the 92nd percentile

For the 92nd percentile: ewline \( z = 1.41 \), ewline Using the parameters ewline \( \mu = -15 \) and \( \sigma = 0.4 \), ewline \( x = 1.41 \cdot 0.4 - 15 \). ewline Therefore, \( x = 0.564 - 15 = -14.436 \).

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.

Z-Scores
In statistics, a z-score represents the number of standard deviations a data point is from the mean. It is a crucial concept when dealing with normal distribution. The formula for z-score is: \( z = \dfrac{x - \mu}{\sigma} \). Here, \(x\) is the value in the data set, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. Z-scores help us understand where a particular value fits within the distribution. For example, a z-score of 0 means the value is exactly at the mean. Negative z-scores indicate values below the mean, while positive z-scores indicate values above the mean.
Normal Distribution Properties
The normal distribution is a continuous probability distribution that is symmetrical around its mean, \(\mu\). It has several key properties that make it useful in statistics:
  • It is defined by two parameters: mean \(\mu\) and standard deviation \(\sigma\).
  • The mean, median, and mode of the distribution are all equal.
  • Approximately 68% of the data lies within one standard deviation (\(\mu \pm \sigma\)).
  • About 95% of the data falls within two standard deviations (\(\mu \pm 2\sigma\)).
  • Nearly 99.7% of the data is within three standard deviations (\(\mu \pm 3\sigma\)).
These properties make the normal distribution an excellent tool for predicting probabilities and understanding data behavior.
Percentile Calculation
Percentiles are used to understand the relative standing of a value in a dataset. The kth percentile is the value below which k% of the data falls. To find the value corresponding to a specific percentile in a normal distribution, follow these steps:
  • Determine the z-score that corresponds to the desired percentile using a z-score table or calculator. For example, the z-score for the 46th percentile is approximately -0.1, and for the 92nd percentile, it is around 1.41.
  • Use the z-score formula rearranged to solve for \(x\): \( x = z \cdot \sigma + \mu \).
  • Substitute the z-score, mean (\(\mu\)), and standard deviation (\(\sigma\)) into the formula to find the x-value. For example, with \(\mu = -15 \) and \(\sigma = 0.4 \), the x-value for the 46th percentile is \( x = -0.1 \cdot 0.4 - 15 = -15.04 \), and for the 92nd percentile, \( x = 1.41 \cdot 0.4 - 15 = -14.436 \).
This method helps you locate where specific values fall within a normally distributed dataset.

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

In a psychology experiment, the time \(t\) in seconds, that it takes a rat to learn its way through a maze is an exponentially distributed random variable with the probability density function $$f(t)=0.02 e^{-0.02 t}, \quad 0 \leq t<\infty$$. Find the probability that a rat will learn its way through a maze in 150 sec or less.

Solve. $$\frac{d y}{d x}=\frac{6}{y}$$

Capitalized cost. The capitalized cost, \(c,\) of an asset over its lifetime is the total of the initial cost and the present value of all maintenance expenses that will occur in the future. It is computed by the formula $$ c=c_{0}+\int_{0}^{L} m(t) e^{-r t} d t $$ where \(c_{0}\) is the initial cost of the asset, \(L\) is the lifetime (in years), \(r\) is the interest rate (compounded continuously), and \(m(t)\) is the annual cost of maintenance. Find the capitalized cost under each set of assumptions. $$c_{0}=\$ 400,000, r=5.5 \%, m(t)=\$ 10,000, L=25$$

A differential equation of the form \(y^{\prime}+M(x) y=N(x)\) has the general solution $$y=\frac{\int P(x) N(x) d x+C}{P(x)}$$, $$\text { where } P(x)=e^{f M(x) d x}$$. The method of solution is broken down into three steps: (1) determine \(P(x)\);(2) determine \(\int P(x) N(x) d x ;\) and (3) divide the result of step 2 by \(P(x)\). Use this method to solve the differential equations. $$y^{\prime}+x^{2} y=x^{2}$$

The time to failure, \(t,\) in hours, of a machine is often exponentially distributed with a probability density function $$f(t)=k e^{-k t}, \quad 0 \leq t<\infty$$ where \(k=1 / a\) and \(a\) is the average amount of time that will pass before a failure occurs. Suppose that the average amount of time that will pass before a failure occurs is 100 hr. What is the probability that a failure will occur in 50 hr or less?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Calculus

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.