/*! 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 29 The Empirical Rule The Stanford-... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 29

The Empirical Rule The Stanford-Binet Intelligence Quotient (IQ) measures intelligence. IQ scores have a bellshaped distribution with a mean of 100 and a standard deviation of \(15 .\) (a) What percentage of people has an IQ score between 70 and \(130 ?\) (b) What percentage of people has an IQ score less than 70 or greater than \(130 ?\) (c) What percentage of people has an IQ score greater than \(130 ?\)

Short Answer

Expert verified
(a) 95% (b) 5% (c) 2.5%

Step by step solution

01

Understand the Empirical Rule

The Empirical Rule states that for a bell-shaped distribution: 1. About 68% of the data lies within 1 standard deviation of the mean. 2. About 95% lies within 2 standard deviations. 3. About 99.7% lies within 3 standard deviations.
02

Calculate 1 Standard Deviation from the Mean

The mean IQ score is 100 and the standard deviation is 15. Therefore, 1 standard deviation from the mean is: Mean - 1SD = 100 - 15 = 85 Mean + 1SD = 100 + 15 = 115
03

Calculate 2 Standard Deviations from the Mean

For 2 standard deviations from the mean: Mean - 2SD = 100 - 30 = 70 Mean + 2SD = 100 + 30 = 130
04

Answer Percentage for IQ between 70 and 130

According to the Empirical Rule, about 95% of the data falls within 2 standard deviations from the mean. Therefore, 95% of people have an IQ between 70 and 130.
05

Answer Percentage for IQ less than 70 or greater than 130

Since 95% of people have IQ scores between 70 and 130, the percentage of people with an IQ less than 70 or greater than 130 is: 100% - 95% = 5%
06

Answer Percentage for IQ greater than 130

The Empirical Rule states that 2.5% of the population lies above 2 standard deviations from the mean (100 + 2 * 15). Thus, 2.5% of people have an IQ greater than 130.

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.

IQ distribution
IQ scores follow a specific trend called a 'distribution'. A well-known type is the bell-shaped distribution (also called normal distribution). Imagine drawing all IQ scores on a graph; it would create a shape like a bell. The middle of the bell has the highest point, representing the average IQ of 100.
This means most people have IQ scores around 100. The curve gets lower as you move away from 100. Farther from the center, you鈥檒l see fewer people with those IQ scores. Hence, most people's IQs cluster around the average, creating the 'bell' shape.
standard deviation
Standard deviation shows how spread out numbers are from the average. In the IQ example, the average (mean) IQ score is 100, and the standard deviation is 15.
This means that most people's IQs are within 15 points above or below 100. So, IQ scores between 85 and 115 are common. Not coincidentally, this matches the first standard deviation in a bell-shaped curve, where most data points lie.
If you move further away, say 30 points above or below 100, that's the second standard deviation. The formula for it would be \[100 \text{ (mean)} \times 2 \text{ (standard deviation)} = \begin{cases} 70 \text{ (low end)} \ 130 \text{ (high end)} \text{.}\begin{cases}\]
percentage calculation
When dealing with bell-shaped distributions, like the IQ scores, there's a useful tool called the Empirical Rule to estimate percentages.
This rule tells us how much data falls within certain standard deviations from the mean:
  • About 68% of the data: falls within 1 standard deviation (between 85 and 115 for IQ)
  • About 95% falls within 2 standard deviations (between 70 and 130 for IQ)
  • About 99.7% falls within 3 standard deviations

For example, if you're asked what percentage of people have an IQ between 70 and 130, the answer, according to the Empirical Rule, is 95%.
bell-shaped distribution
The bell-shaped distribution is crucial in statistics to understand how a trait, like IQ, spreads out in a population.
The characteristics of this curve make it predictable and symmetric around the mean:
  • Most people score near the average (middle of the bell)
  • As you move away from the average, fewer people have those scores
A typical bell-shaped curve is dense in the center and thin at the tails.
This distribution allows for precise statistical calculations, making it a powerful tool in assessing and predicting probabilities and percentages in any given 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

27\. Rates of Returns of Stocks Stocks may be categorized by sectors. Go to www.pearsonhighered.com/sullivanstats and download \(3_{-} 2_{-} 27\) using the file format of your choice. The data represent the one-year rate of return (in percent) for a sample of consumer cyclical stocks and industrial stocks for the period December 2013 through November 2014 . Note: Consumer cyclical stocks include names such as Starbucks and Home Depot. Industrial stocks include names such as \(3 \mathrm{M}\) and FedEx. (a) Draw a relative frequency histogram for each sector using a lower class limit for the first class of -50 and a class width of \(10 .\) Which sector appears to have more dispersion? (b) Determine the mean and median rate of return for each sector. Which sector has the higher mean rate of return? Which sector has the higher median rate of return? (c) Determine the standard deviation rate of return for each sector. In finance, the standard deviation rate of return is called risk. Typically, an investor "pays" for a higher return by accepting more risk. Is the investor paying for higher returns for these sectors? Do you think the higher returns are worth the cost? Explain.

According to the U.S. Census Bureau, the mean of the commute time to work for a resident of Boston, Massachusetts, is 27.3 minutes. Assume that the standard deviation of the commute time is 8.1 minutes to answer the following: (a) What minimum percentage of commuters in Boston has a commute time within 2 standard deviations of the mean? (b) What minimum percentage of commuters in Boston has a commute time within 1.5 standard deviations of the mean? What are the commute times within 1.5 standard deviations of the mean? (c) What is the minimum percentage of commuters who have commute times between 3 minutes and 51.6 minutes?

The average 20 - to 29 -year-old man is 69.6 inches tall, with a standard deviation of 3.0 inches, while the average 20 - to 29 -year-old woman is 64.1 inches tall, with a standard deviation of 3.8 inches. Who is relatively taller, a 67-inch man or a 62 -inch woman?

What makes the range less desirable than the standard deviation as a measure of dispersion?

The highest batting average ever recorded in Major League Baseball was by Ted Williams in 1941 when he hit \(0.406 .\) That year, the mean and standard deviation for batting average were 0.2806 and \(0.0328 .\) In 2014 Jose Altuve was the American League batting champion, with a batting average of \(0.341 .\) In \(2014,\) the mean and standard deviation for batting average were 0.2679 and \(0.0282 .\) Who had the better year relative to his peers, Williams or Altuve? Why?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

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.