/*! 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 25 Understanding Density Curves. Re... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 25

Understanding Density Curves. Remember that it is areas under a density curve, not the height of the curve, that give proportions in a distribution. To illustrate this, sketch a density curve that has a tall, thin peak at 0 on the horizontal axis but has most of its area close to 1 on the horizontal axis without a high peak at 1 .

Short Answer

Expert verified
A tall peak at 0, with most area spread near 1, forms the curve.

Step by step solution

01

Identify the Components of a Density Curve

A density curve is a graph that represents the distribution of a continuous probability distribution. The total area under the curve is 1. The horizontal axis represents the variable for which the distribution is being described.
02

Understand the Characteristics Required

The problem requires a curve with a tall, thin peak at 0. This means the curve will rise sharply at the beginning of the x-axis and drop quickly to a low height. Furthermore, most of the area should be close to 1, meaning the bulk of the distribution's area should be at that point, but not as a peak.
03

Sketch the Initial Tall Thin Peak at 0

Draw a vertical line from the x-axis at 0 that rises sharply but quickly falls, creating a narrow, steep peak. Ensure this peak takes up a small area since it represents a very small part of the total distribution.
04

Draw the Main Distribution with Area Close to 1

From the tall peak, extend the curve towards 1 with a much less steep and more gradual descent. Most of the area under the curve should be located near 1. The curve should be nearly flat when it reaches 1, spreading horizontally to extend the length of the distribution, accommodating a large area of 1.
05

Ensure Total Area Is 1

Verify the total area under the curve is equal to 1. The combined areas under the thin peak at 0 and the flatter, wider spread near 1 should sum to 1, confirming it represents a valid probability distribution.

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.

Probability Distribution
Probability distribution is a way to describe how the values of a random variable are spread or distributed. It tells us the likelihood of each possible outcome happening. In terms of a density curve, a probability distribution is shown as the area below the curve.
Each portion of the area under this curve corresponds to the probability of a random variable falling within a given interval.
  • **Total Area Equals 1**: In a probability density curve, the total area under the curve is always 1. This ensures that the possibility of all outcomes (or the entire data set) is considered fully.
  • **Continuous Versus Discrete**: For continuous data, the probability of any individual exact value is technically zero. Instead, we look at intervals.
Remember, it's not the height of the curve that matters, but how much area lies within certain parts of it. This area represents the likelihood of outcomes for continuous data.
Continuous Data
Continuous data can take any value within a defined range. For example, weights, heights, and time can vary continuously. The data values don't jump from one value to another but have an infinite number of possibilities in between.
  • **Modeling with Density Curves**: Density curves help us visualize and model the distribution of continuous data. The shape of the curve can depend on the data's distribution, reflecting whether it's more concentrated in a particular range.
  • **Infinitely Precise Values**: Because continuous data can have infinitely precise values, every specific point has zero probability. Instead, we assess the probability over intervals.
When sketching a density curve for continuous data, it is important to remember that steep peaks or wide spreads represent how values concentrate in different intervals.
Area Under Curve
The area under a density curve is crucial since it directly represents probability. When dealing with continuous probability distributions, determining the probability of a variable falling within a particular range comes down to calculating the area under the curve for that range.
  • **Interpreting the Area**: If you are trying to find the probability for a specific range of values, look at the area under the curve for that range. The larger the area, the more likely the values will fall within that interval.
  • **Accuracy in Sketching**: While the shape of the curve may vary, maintaining an area of 1 is essential for accurately representing a probability distribution.
Always remember that the height of the curve itself doesn't give us the probability but merely supports creating the shape where the total area gives the probability proportions.

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

Mean and Median. Egure \(3.7\) displays three density curves, each with three points marked on it. At which of these points on each curve do the mean and the median fall?

Nor mal Is Only Approximate: ACT Scores. Composite scores on the ACT for the 2019 high school graduating class had mean \(20.8\) and standard deviation \(5.8\). In all, \(1,914,817\) students in this class took the test. Of these, 227,221 had scores higher than 28 , and another 54,848 had scores exactly 28. ACT scores are always whole numbers. The exactly Normal \(N(20.8,5.8)\) distribution can include any value, not just whole numbers. What is more, there is no area exactly above 28 under the smooth Normal curve. So ACT scores can be only approximately Normal. To illustrate this fact, find a. the percentage of 2019 ACT scores greater than 28 , using the actual counts reported. b. the percentage of 2019 ACT scores greater than or equal to 28 , using the actual counts reported. c. the percentage of observations that are greater than 28 using the \(N(20.8,5.8)\) distribution. (The percentage greater than or equal to 28 is the same because there is no area exactly over 28.)

Acid Rain? Emissions of sulfur dioxide by industry set off chemical changes in the atmosphere that result in "acid rain." The acidity of liquids is measured by \(\mathrm{pH}\) on a scale of 0 to 14 . Distilled water has \(\mathrm{pH}\) 7.0, and lower \(\mathrm{pH}\) values indicate acidity. Normal rain is somewhat acidic, so acid rain is sometimes defined as rainfall with a pH below \(5.0\). The \(\mathrm{pH}\) of rain at one location varies among rainy days according to a Normal distribution with mean \(5.43\) and standard deviation \(0.54\). What proportion of rainy days have rainfall with \(\mathrm{pH}\) below \(5.0\) ?

Sketch Density Curves. Sketch density curves that describe distributions with the following shapes: a. Symmetric but with two peaks (that is, two strong clusters of observations) b. Single peak and skewed to the right

Perfect SAT Scores. It is possible to score higher than 1600 on the combined mathematics and evidence-based reading and writing portions of the SAT, but scores 1600 and above are reported as 1600 . The distribution of SAT scores (combining Mathematics and Reading) in 2019 was close to Normal, with mean 1059 and standard deviation 210. What proportion of SAT scores for these two parts were reported as 1600? (That is, what proportion of SAT scores were actually 1600 or higher?)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

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.