/*! 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 43 \(\Lambda\) researcher reported ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 43

\(\Lambda\) researcher reported that the typical teenager needs 9.3 hours of sleep per night but gets only 6.3 hours. \({ }^{18} \mathrm{By}\) the end of a 5 -day school week, a teenager would accumulate about 15 hours of "sleep debt." Students in a high school statistics class were skeptical, so they gathered data on the amount of sleep debt (in hours) accumulated over time (in days) by a random sample of 25 high school students. The resulting least-squares regression equation for their data is Sleep debt \(=2.23+3.17\) (days). (a) Interpret the slope of the regression line in context. (b) Are the students' results consistent with the rescarcher's report? Explain.

Short Answer

Expert verified
(a) Slope is 3.17, indicating 3.17 hours of sleep debt per day. (b) Yes, results are consistent as both show ~15 hours of debt over 5 days.

Step by step solution

01

Identify the Slope

In the given least-squares regression equation for sleep debt, the equation is Sleep debt = 2.23 + 3.17(days). Here, the slope is the coefficient of 'days,' which is 3.17.
02

Interpret the Slope

The slope of 3.17 means that for each additional day, the accumulated sleep debt increases by 3.17 hours. This tells us that on average, each day a teenager accumulates 3.17 hours of sleep debt according to the students' data.
03

Analyze Researcher's Report

According to the researcher, there is a 3-hour sleep debt per day since teenagers need 9.3 hours of sleep but get only 6.3 hours (9.3 - 6.3 = 3.0 hours). Over a 5-day school week, this results in a total of 15 hours of sleep debt (3.0 hours/day * 5 days).
04

Compare Students' Results with Researcher's Report

The students' regression equation suggests a sleep debt increase of about 3.17 hours per day. This is quite close to the 3.0 hours per day reported by the researcher, resulting in approximately 15.85 hours (3.17 * 5) by the end of a 5-day week, which is very close to the researcher's report of 15 hours.

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.

Understanding Regression Analysis
Regression analysis is a powerful statistical tool used to explore and establish relationships between variables. In this case, students utilized regression analysis to assess the relationship between time, measured in days, and accumulated sleep debt, measured in hours. The regression equation provided, \(\text{Sleep debt} = 2.23 + 3.17(\text{days})\), allows them to predict sleep debt based on the number of days. This is known as a least-squares regression, which minimizes the distance of data points from the line of best fit to improve prediction accuracy.
  • It quantifies how one variable reacts to changes in another.
  • The equation is composed of a y-intercept (2.23) and a slope (3.17) that represents changes in sleep debt related to days.
Regression analysis simplifies complex data, making predictions about future outcomes more reliable.
The Sleep Study Context
The sleep study in question addresses a common problem faced by teenagers: insufficient sleep. According to the initial research, young people require 9.3 hours of sleep nightly but typically achieve only 6.3 hours. Consequently, they accumulate a sleep debt of around 3 hours per day, leading to a total of about 15 hours over a week. This study conducted by students investigates whether these figures hold true within their sample of 25 high school individuals. Through the application of regression analysis, they quantify sleep debt accumulation, offering a pragmatic view of sleep patterns and their implications on student health and behavior.
Interpreting the Slope in Sleep Debt
Interpreting the slope in a regression equation is critical as it provides insights into relationships between variables. In the context of the regression equation \(\text{Sleep debt} = 2.23 + 3.17(\text{days})\), the slope here is 3.17. This coefficient reflects the average increase in hours of sleep debt per day for high school students.
  • This indicates that every additional day corresponds to an increase of 3.17 hours of sleep debt.
  • It illustrates how quickly sleep debt accumulates, highlighting the rapid impact of consecutive days of inadequate sleep.
Understanding the slope helps in comprehending how current practices (inadequate sleep) could have cumulative effects over time.
Effective Data Collection Methods
Effective data collection is crucial in conducting any research, including studies on sleep patterns. For reliable results, the students collected data from a random sample of 25 high school students, a strategy that increases the likelihood of acquiring representative data.
  • Data should be collected systematically to reduce bias and variance.
  • Representative sampling means that conclusions drawn are more likely to reflect the broader population.
  • In the sleep study, collecting daily logs of sleep hours and calculating subsequent debt provides a robust foundation for accurate regression analysis.
Ensuring the validity and reliability of data is essential for drawing meaningful conclusions from any study or 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

\(A\) study conducted by Norman Hollenberg, professor of medicine at Brigham and Women's Hospital and Harvard Medical School, involved 27 healthy people aged 18 to \(72 .\) Each subject consumed a cocoa beverage containing 900 milligrams of flavonols (a class of flavonoids) daily for 5 days. Using a finger cuff, blood flow was measured on the first and fifth days of the study. After 5 days, researchers measured what they called "significant improvement" in blood flow and the function of the cells that line the blood vessels. \({ }^{4}\) What flaw in the design of this experiment makes it impossible to say whether the cocoa really caused the improved blood flow? Explain.

Which of the following statements are true of a table of random digits, and which are false? Bricfly explain your answers. (a) There are exactly four 0 s in each row of 40 digits. (b) each pair of digits has chance \(1 / 100\) of being 00 . (c) Ilie digits 0000 can never appear as a group, because this pattern is not random.

Comnent on each of the following as a potential sample survey question. Is the question clear? Is it slanted toward a desired response? (a) "Some cell phone users luve developed brain cancer. Should all cell phones come with a warning label explaining the danger of using cell phones?" (b) "Do you agree that a national system of health insurance should be favored because it would provide health insurance for everyone and would reduce administrative costs?" (c) "In view of escalating environmental degradation and incipient resource depletion, would you favor economic incentives for recycling of resourceintensive consumer goods?"

Two essential features of all statistically designed experiments are (a) compare several treatments; use the double-blind method. (b) compare several treatments; use chance to assign subjects to treatments. (c) always have a placebo group; use the double-blind method. (d) use a block design; use chance to assign subjects to treatments. (e) use enough subjects; always have a control group.

An experiment that claimed to show that meditation lowers anxiety proceeded as follows. The experimenter interviewed the subjects and rated their level of anxiety. Then the subjects were randomly assigned to two groups. The experimenter taught one group how to meditate and they meditated daily for a month. The other group was simply told to relax more. At the end of the month, the experimenter interviewed all the subjects again and rated their anxiety level. The meditation group now had less anxiety. Psychologists said that the results were suspect because the ratings were not blind. Explain what this means and how lack of blindness could bias the reported results.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

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.