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Chapter 4: Problem 43

A researcher reported that the average teenager needs 9.3 hours of sleep per night but gets only 6.3 hours. \({ }^{17}\) 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). Do the students have reason to be skeptical of the research study's reported results? Explain.

Short Answer

Expert verified
Yes, the students' data shows a greater sleep debt accumulation than the study claims, justifying their skepticism.

Step by step solution

01

Identify Assumptions and Given Data

The research study claims that teenagers accumulate about 15 hours of sleep debt over a 5-day school week. This implies an average sleep debt rate of 3 hours per day (since 15 hours / 5 days = 3 hours per day). We are given a regression equation obtained by students: Sleep debt \( = 2.23 + 3.17 (days) \). This equation models the sleep debt as a function of days.
02

Interpret the Regression Equation

The regression equation Sleep debt \( = 2.23 + 3.17 (days) \) suggests that on the first day, the sleep debt starts at 2.23 hours. Every additional day increases the sleep debt by 3.17 hours per day.
03

Calculate the Predicted Sleep Debt for 5 Days

To find the predicted sleep debt after 5 days, substitute \(\text{days} = 5\) into the regression equation: Sleep debt \( = 2.23 + 3.17(5)\). Compute this value to get: Sleep debt \( = 2.23 + 15.85 = 18.08 ext{ hours}\).
04

Compare Predicted Sleep Debt with the Research Claim

The research study claims an accumulation of 15 hours of sleep debt over 5 days. The students' regression model predicts 18.08 hours. Since 18.08 hours is greater than 15 hours, this indicates a higher average rate of sleep debt accumulation than reported in the study.
05

Evaluate Student's Skepticism

The students have grounds to be skeptical of the research study. Their regression analysis indicates a greater sleep debt accumulation than what the study claims, suggesting either a discrepancy in the research or a misrepresentation of typical sleep debt.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Sleep Debt
Sleep debt refers to the accumulated sleep loss over a period of time. It occurs when the amount of sleep you get is consistently less than the amount of sleep you need. If a teenager requires 9.3 hours of sleep each night but only gets 6.3 hours, they start building a sleep debt. Over a 5-day school week, this debt can add up significantly.

Here's how sleep debt works:
  • Each night, you miss out on necessary sleep hours, adding them to your debt.
  • Sleep debt can affect mood, alertness, and overall health.
  • The more sleep debt you accumulate, the more sleep you'll need to "pay it off."

The regression equation provided by the students helps to quantify this concept, allowing statistical analysis of sleep debt trends over time.
The Role of Statistical Analysis
Statistical analysis is a method used to interpret data and make informed conclusions. In this context, students used statistical analysis to verify claims about teenage sleep debt. By collecting their own data and using a regression model, they assessed whether the original research was accurate.

When performing statistical analysis:
  • Gather a representative sample size to ensure reliable results.
  • Use statistical tools like regression to model relationships between variables.
  • Interpret the results to determine if they support or contradict initial claims.

For sleep debt analysis, statistical methods provide a way to verify if teenagers are indeed accumulating the debt as stated by the research.
High School Statistics in Practice
High school statistics courses often include practical applications, such as analyzing sleep patterns using real-world data. In this example, students learned valuable skills by gathering data and applying statistical concepts to evaluate the sleep debt of their peers.

Key learning points in high school statistics include:
  • Collecting and cleaning data to ensure accuracy.
  • Using regression analysis to examine trends and relationships.
  • Critically assessing whether findings are consistent with external studies or reports.

This hands-on approach not only solidifies statistical principles but also allows students to question and investigate real-life claims, a useful skill beyond the classroom.
Understanding Regression Equations
A regression equation is a statistical tool used to describe the relationship between variables. In the case of sleep debt, the regression equation helps predict how much debt accumulates based on time (in days). The equation provided by the students is Sleep debt \( = 2.23 + 3.17\times\text{days} \).

In this equation:
  • \(2.23\) represents the starting sleep debt, indicating how much debt is already accumulated at the outset.
  • \(3.17\) is the rate at which sleep debt increases per day.
  • The term 'days' is the independent variable representing time.

This type of equation helps to determine the expected sleep debt after a given number of days, offering a way to predict future outcomes based on current data.

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Most popular questions from this chapter

Does day care help low-income children stay in school and hold good jobs later in life? The Carolina Abecedarian Project (the name suggests the ABCs) has followed a group of 111 children since 1972. Back then, these individuals were all healthy but low-income black infants in Chapel Hill, North Carolina. All the infants received nutritional supplements and help from social workers. Half were also assigned at random to an intensive preschool program.\(^{38}\) (a) Explain the purpose of each of the three experimental design principles. (b) Describe how each of these principles was used in this study.

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