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A study was conducted to determine the effects of sleep deprivation on people's ability to solve problems without sleep. A total of 10 subjects participated in the study, two at each of five sleep deprivation levels \(-8,12,16,20,\) and 24 hours. After his or her specified sleep deprivation period, each subject was administered a set of simple addition problems, and the number of errors was recorded. These results were obtained: $$ \begin{aligned} &\begin{array}{l|l|l|l} \text { Number of Errors, } y & 8,6 & 6,10 & 8,14 \\ \hline \text { Number of Hours without Sleep, } x & 8 & 12 & 16 \end{array}\\\ &\begin{array}{l|l|l} \text { Number of Errors, } y & 14,12 & 16,12 \\ \hline \text { Number of Hours without Sleep, } x & 20 & 24 \end{array} \end{aligned} $$ a. How many pairs of observations are in the experiment? b. What are the total number of degrees of freedom? c. Complete the MINITAB printout. d. What is the least-squares prediction equation? e. Use the prediction equation to predict the number of errors for a person who has not slept for 10 hours.

Step by step solution

01

Count the pairs of observations

The data is given in two tables, showing the number of errors (y) and the number of hours without sleep (x) for each pair of subjects at each sleep deprivation level. To count the total number of pairs of observations in the experiment, we simply count the number of pairs of data points: there are three pairs of (x, y) values in the first table and two pairs in the second table. Thus, there are a total of 5 pairs of observations in the experiment.

02

Calculate the total number of degrees of freedom

The formula for the total number of degrees of freedom (DF) in a study is given by: $$\text{DF} = n - 1$$ where \(n\) is the total number of observations. In this study, there are 10 subjects, so the total number of observations is \(10\). Therefore, the total number of degrees of freedom is: $$\text{DF} = 10 - 1 = 9$$

03

Provide necessary calculations and informations for MINITAB printout

While we cannot create a MINITAB printout here, we can provide the information needed for one. This would include calculating the sum of the x-values, the sum of the y-values, the sum of the squares of the x-values, the sum of the squares of the y-values, and the sum of the products of the x and y values. Using the data provided: $$\begin{aligned} \sum x &= 8 + 8 + 12 + 12 + 16 + 16 + 20 + 20 + 24 + 24 = 160\\ \sum y &= 8 + 6 + 6 + 10 + 8 + 14 + 14 + 12 + 16 + 12 = 106\\ \sum x^2 &= 2(8^2) + 2(12^2) + 2(16^2) + 2(20^2) + 2(24^2) = 4160\\ \sum y^2 &= 6^2 + 6^2 + 8^2 + 8^2 + 10^2 + 12^2 + 14^2 + 14^2 + 16^2 = 1236\\ \sum xy &= 2(8 \times 8) + 2(12 \times 6) + 2(16 \times 8) + 2(20 \times 14) + 2(24 \times 12) = 2080 \end{aligned}$$ This information can be used to complete the MINITAB printout.

04

Calculate the least-squares prediction equation

To find the least-squares prediction equation, we will first calculate the slope (\(b\)) and the intercept (\(a\)) using the following formulas: $$\begin{aligned} b &= \frac{n(\sum x y)-(\sum x)(\sum y)}{n(\sum x^{2})-(\sum x)^{2}}\\ a &= \bar{y}-b \bar{x} \end{aligned}$$ where \(n = 10\), \(\bar{x}\) is the average of the x-values, and \(\bar{y}\) is the average of the y-values. Using the previously calculated values and the data provided: $$\begin{aligned} \bar{x} &= \frac{\sum x}{n}= \frac{160}{10} = 16\\ \bar{y} &= \frac{\sum y}{n}= \frac{106}{10} = 10.6\\ b &= \frac{10(2080) - (160)(106)}{10(4160) - (160)^2} = \frac{20800 - 16960}{41600 - 25600} = \frac{3840}{16000} = 0.24\\ a &= 10.6 - 0.24(16) = 6.56 \end{aligned}$$ Thus, the least-squares prediction equation is: $$\hat{y} = a + bx = 6.56 + 0.24x$$

05

Predict the number of errors for a person who has not slept for 10 hours

Using the least-squares prediction equation from Step 4, we can predict the number of errors for a person who has not slept for 10 hours by plugging in \(x = 10\) into the equation: $$\hat{y} = 6.56 + 0.24(10) = 6.56 + 2.4 = 8.96$$ The predicted number of errors for a person who has not slept for 10 hours is approximately 8.96 errors.

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Key Concepts

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

Experimental Design

In the context of scientific studies, particularly those exploring relationships like sleep deprivation and cognitive ability, experimental design is a framework that outlines how the study is conducted. It ensures that the data gathered is relevant and reliable.

Experimental designs typically involve identifying variables that need to be controlled and ensuring these variables are measured accurately.

• The study on sleep deprivation mentioned here has a clear setup with defined levels of sleep deprivation.
• Such a design enables researchers to collect consistent data across different subjects.
• The clear definition of the number of subjects and the specific deprivation period (ranging from 8 to 24 hours) exemplifies a well-structured experimental design.

Effective experimental design also addresses potential biases. This study controls sleep deprivation as the independent variable, while the response, the number of errors, acts as the dependent variable. This method helps in isolating the effects of sleep deprivation from other external influences. By keeping these and other variables constant, the design provides a higher level of control over the study’s outcome. Thus, making it reliable for further analysis with statistical tools like least-squares regression.

Degrees of Freedom

Degrees of Freedom (DF) are an important concept in statistics, and they represent the number of independent values or quantities that can be assigned to a statistical distribution.

In simpler terms, they indicate the number of values that are "free to vary" when computing a statistic, such as the mean.

• In the exercise on sleep deprivation, the degrees of freedom are calculated using the formula: \( \text{DF} = n - 1 \), where \( n \) is the number of observations.
• Given 10 subjects participated in the study, this results in \( \text{DF} = 10 - 1 = 9 \).
• Degrees of Freedom are pivotal in significance testing, allowing more accurate inferences about sample data.

Understanding degrees of freedom helps in interpreting statistical results and is a foundational aspect of experimental analysis. Such knowledge aids in assessing the reliability of the data and determining the strength of the findings, ultimately guiding decisions like rejecting or failing to reject a hypothesis.

Prediction Equation

The prediction equation, particularly in the context of least-squares regression, is a mathematical representation used for estimating the expected outcome based on a given predictor. For instance, in our exercise, the prediction equation is derived from the relationship between hours of sleep deprivation and errors observed.

• To calculate the prediction equation, we need to determine the slope \( b \) and the intercept \( a \), such that the equation can be expressed in the form \( \hat{y} = a + bx \).
• The slope \( b \) shows how much the dependent variable (errors) is expected to change for each one-unit change in the independent variable (hours of no sleep).
• The intercept \( a \) represents the baseline value of the dependent variable when the independent variable is zero.

In this study, the least-squares method is used for minimizing the differences between observed and predicted values, ensuring accuracy in predictions. As per the calculations, the prediction equation is \( \hat{y} = 6.56 + 0.24x \). This means for someone who has not slept for 10 hours, the expected number of errors can be calculated as \( \hat{y} = 6.56 + 0.24 \times 10 \approx 8.96 \) errors. Such equations are essential tools in data analysis, providing insights and helping to make informed predictions in various contexts.