/*! 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 results of a national survey... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 29

The results of a national survey showed that on average, adults sleep 6.9 hours per night. Suppose that the standard deviation is 1.2 hours. a. Use Chebyshev's theorem to calculate the percentage of individuals who sleep between 4.5 and 9.3 hours. b. Use Chebyshev's theorem to calculate the percentage of individuals who sleep between 3.9 and 9.9 hours. c. Assume that the number of hours of sleep follows a bell-shaped distribution. Use the empirical rule to calculate the percentage of individuals who sleep between 4.5 and 9.3 hours per day. How does this result compare to the value that you obtained using Chebyshev's theorem in part (a)?

Short Answer

Expert verified
a. 75%, b. 84%, c. 95%; empirical rule provides a more specific prediction.

Step by step solution

01

Understanding Chebyshev's Theorem

Chebyshev's theorem states that for any distribution, at least \(1 - \frac{1}{k^2}\) of the data falls within \(k\) standard deviations of the mean. This theorem applies regardless of the data's distribution shape.
02

Applying Chebyshev's Theorem to Part (a)

First, calculate the number of standard deviations between the bounds (4.5 and 9.3) and the mean (6.9). The range is 6.9 - 4.5 = 2.4 for the lower bound and 9.3 - 6.9 = 2.4 for the upper bound.Next, divide these distances by the standard deviation (1.2 hours):\[k = \frac{2.4}{1.2} = 2.0\]Using Chebyshev's theorem, \[\text{Percentage} \geq 1 - \frac{1}{k^2} = 1 - \frac{1}{4} = \frac{3}{4} = 0.75 = 75\%\]
03

Applying Chebyshev's Theorem to Part (b)

Calculate the number of standard deviations between the bounds (3.9 and 9.9) and the mean (6.9). The range is 6.9 - 3.9 = 3.0 for the lower bound and 9.9 - 6.9 = 3.0 for the upper bound.Next, divide these distances by the standard deviation,\[k = \frac{3.0}{1.2} = 2.5\]Using Chebyshev's theorem,\[\text{Percentage} \geq 1 - \frac{1}{k^2} = 1 - \frac{1}{6.25} = \frac{5.25}{6.25} \approx 0.84 = 84\%\]Thus, at least 84% of individuals sleep between 3.9 and 9.9 hours.
04

Understanding the Empirical Rule

The empirical rule applies to bell-shaped (normally distributed) data. It states that approximately 68% of data falls within 1 standard deviation, 95% within 2, and 99.7% within 3 standard deviations of the mean.
05

Applying the Empirical Rule to Part (c)

Determine how many standard deviations 4.5 and 9.3 hours are from the mean (6.9 hours). As calculated in Step 2, both bounds are 2.4 hours (or 2 standard deviations) from the mean. According to the empirical rule, about 95% of data falls within 2 standard deviations of the mean. Therefore, 95% of individuals sleep between 4.5 and 9.3 hours.
06

Comparison of Chebyshev's Theorem and Empirical Rule

Using Chebyshev's theorem in part (a), we found at least 75% of individuals sleep between 4.5 and 9.3 hours. The empirical rule provided a more precise range of 95% for the same interval due to the assumption of a normal distribution. This illustrates that the empirical rule offers more specific predictions for bell-shaped distributions.

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.

Empirical Rule
The empirical rule is a handy tool when dealing with a normal or bell-shaped distribution. It helps to understand how data is distributed in relation to the mean and standard deviation. This rule works particularly well when the distribution of the data is approximately normal.

The empirical rule states that:
  • About 68% of the data falls within one standard deviation (\( \sigma \)) of the mean (\( \mu \)).
  • Approximately 95% of the data falls within two standard deviations.
  • Almost 99.7% of the data falls within three standard deviations.
In the context of the sleep survey, with a mean of 6.9 hours and a standard deviation of 1.2 hours:
  • Most people's sleep duration, between 4.5 and 9.3 hours, is within two standard deviations of the mean.
Thus, according to the empirical rule, roughly 95% of individuals fall within this range. This makes it a quick and reliable estimate for normally distributed data, offering a clearer picture compared to more general rules like Chebyshev's.
Standard Deviation
The standard deviation is a crucial concept in statistics. It measures how spread out the numbers in a data set are. When talking about sleep, the standard deviation of 1.2 hours implies the average variation of individual sleep hours from the mean of 6.9 hours.

A smaller standard deviation means that data points tend to be closer to the mean. A larger standard deviation indicates more spread out data. In our sleep survey:
  • If the standard deviation was smaller, most people's sleep hours would be closer to 6.9 hours.
  • Conversely, a larger standard deviation would imply more diversity in the amount of sleep among the participants.
The standard deviation is valuable because it provides context to the average. It helps discern whether the average is representative of the data or if many data points differ significantly from the average.
Bell-shaped Distribution
A bell-shaped distribution, also known as a normal distribution, is a symmetrical spread of data where most values cluster around the mean. This pattern forms the shape of a bell curve. Many natural phenomena, including human sleep patterns, follow this distribution.

In a bell-shaped distribution:
  • Values near the mean are more frequent than values far from it.
  • The distribution is symmetric around the mean.
  • The highest point of the curve represents the mean.
For the sleep data, assuming a bell-shaped distribution means that the empirical rule can be used to provide stronger predictions. The rule capitalizes on the symmetry and predictability of this distribution form. The majority of the data falls close to the mean of 6.9 hours, with fewer people outside the 4.5 to 9.3-hour range.
This understanding aids in comprehending the nature of the data and in employing appropriate statistical tools for analysis.

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

Naples, Florida, hosts a half-marathon (13.1-mile race) in January each year. The event attracts top runners from throughout the United States as well as from around the world. In January 2009,22 men and 31 women entered the \(19-24\) age class. Finish times in minutes are as follows (Naples Daily News, January 19,2009 ). Times are shown in order of finish. $$\begin{array}{cccccccc} \text { Finish } & \text { Men } & \text { Women } & \text { Finish } & \text { Men } & \text { Women } & \text { Finish } & \text { Men } & \text { Women } \\\ 1 & 65.30 & 109.03 & 11 & 109.05 & 123.88 & 21 & 143.83 & 136.75 \\ 2 & 66.27 & 111.22 & 12 & 110.23 & 125.78 & 22 & 148.70 & 138.20 \\ 3 & 66.52 & 111.65 & 13 & 112.90 & 129.52 & 23 & & 139.00 \\ 4 & 66.85 & 111.93 & 14 & 113.52 & 129.87 & 24 & & 147.18 \\ 5 & 70.87 & 114.38 & 15 & 120.95 & 130.72 & 25 & & 147.35 \end{array}$$ $$\begin{array}{rrrrrrr} \text { Finish } & \text { Men } & \text { Women } & \text { Finish } & \text { Men } & \text { Women } & \text { Finish } & \text { Men } & \text { Women } \\\ 6 & 87.18 & 118.33 & 16 & 127.98 & 131.67 & 26 & & 147.50 \\ 7 & 96.45 & 121.25 & 17 & 128.40 & 132.03 & 27 & & 147.75 \\ 8 & 98.52 & 122.08 & 18 & 130.90 & 133.20 & 28 & & 153.88 \\ 9 & 100.52 & 122.48 & 19 & 131.80 & 133.50 & 29 & & 154.83 \\ 10 & 108.18 & 122.62 & 20 & 138.63 & 136.57 & 30 & & 189.27 \\ & & & & & & 31 & & 189.28 \end{array}$$ a. George Towett of Marietta, Georgia, finished in first place for the men and Lauren Wald of Gainesville, Florida, finished in first place for the women. Compare the firstplace finish times for men and women. If the 53 men and women runners had competed as one group, in what place would Lauren have finished? b. What is the median time for men and women runners? Compare men and women runners based on their median times. c. Provide a five-number summary for both the men and the women. d. Are there outliers in either group? e. Show the box plots for the two groups. Did men or women have the most variation in finish times? Explain.

The U.S. Department of Education reports that about \(50 \%\) of all college students use a student loan to help cover college expenses (National Center for Educational Studies, January 2006 ). A sample of students who graduated with student loan debt is shown here. The data, in thousands of dollars, show typical amounts of debt upon graduation. $$\begin{array}{lllllllll} 10.1 & 14.8 & 5.0 & 10.2 & 12.4 & 12.2 & 2.0 & 11.5 & 17.8 & 4.0 \end{array}$$ a. For those students who use a student loan, what is the mean loan debt upon graduation? b. What is the variance? Standard deviation?

In automobile mileage and gasoline-consumption testing, 13 automobiles were road tested for 300 miles in both city and highway driving conditions. The following data were recorded for miles-per-gallon performance. $$\begin{array}{lllllllllllllll} \text {City:} & 16.2 & 16.7 & 15.9 & 14.4 & 13.2 & 15.3 & 16.8 & 16.0 & 16.1 & 15.3 & 15.2 & 15.3 & 16.2 \\ \text {Highway}: & 19.4 & 20.6 & 18.3 & 18.6 & 19.2 & 17.4 & 17.2 & 18.6 & 19.0 & 21.1 & 19.4 & 18.5 & 18.7 \end{array}$$ Use the mean, median, and mode to make a statement about the difference in performance for city and highway driving.

The following times were recorded by the quarter-mile and mile runners of a university track team (times are in minutes). $$\begin{array}{llllll} \text {Quarter-Mile Times:} & .92 & .98 & 1.04 & .90 & .99 \\ \text {Mile Times:} & 4.52 & 4.35 & 4.60 & 4.70 & 4.50 \end{array}$$ After viewing this sample of running times, one of the coaches commented that the quartermilers turned in the more consistent times, Use the standard deviation and the coefficient of variation to summarize the variability in the data. Does the use of the coefficient of variation indicate that the coach's statement should be qualified?

Florida Power \& Light (FP\&L) Company has enjoyed a reputation for quickly fixing its electric system after storms. However, during the hurricane seasons of 2004 and \(2005,\) a new reality was that the company's historical approach to emergency electric system repairs was no longer good enough (The Wall Street Journal, January 16,2006 ). Data showing the days required to restore electric service after seven hurricanes during 2004 and 2005 follow. \(\begin{array}{lc}\text { Hurricane } & \text { Days to Restore S } \\ \text { Charley } & 13 \\ \text { Frances } & 12 \\ \text { Jeanne } & 8 \\ \text { Dennis } & 3 \\ \text { Katrina } & 8 \\ \text { Rita } & 2 \\ \text { Wilma } & 18\end{array}\) Based on this sample of seven, compute the following descriptive statistics: a. Mean, median, and mode b. \(\quad\) Range and standard deviation c. Should Wilma be considered an outlier in terms of the days required to restore electric service? d. The seven hurricanes resulted in 10 million service interruptions to customers. Do the statistics show that FP\&L should consider updating its approach to emergency electric system repairs? Discuss.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Logic and Functions

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.