Question

Sleep Deprivation A study was conducted to determine the effects of sleep deprivation on DS1206 people's ability to solve problems. Ten 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 answered a set of simple addition problems, and the number of errors was recorded. These results were obtained: 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. $$\begin{aligned}&\begin{array}{l|l|l|l}\text { Number of Errors, } y & 8,6 & 6,10 & 8,14 \\\\\hline \begin{array}{l}\text { Number of Hours without } \\\\\text { Sleep, } x\end{array} & 8 & 12 & 16\end{array}\\\&\begin{array}{l|l|l}\text { Number of Errors, } y & 14,12 & 16,12 \\\\\hline \begin{array}{l}\text { Number of Hours without } \\\\\text { Sleep. } x\end{array} & 20 & 24 \\\\\hline\end{array}\end{aligned}$$ a. How many pairs of observations are in the experiment? b. What are the total number of degrees of freedom? c. Complete the MS Exce/ printout.

Short answer

Question: In the given exercise about sleep deprivation and number of errors, determine the a) number of pairs of observations, b) number of degrees of freedom, c) coefficients of the least-squares prediction equation, and d) predicted number of errors for someone who has not slept for 10 hours using the prediction equation. Answer: a) Number of pairs of observations: 10 b) Number of degrees of freedom: 8 c) Coefficients of the least-squares prediction equation: slope (m) = 0.696 and intercept (b) = -0.536 d) Predicted number of errors for someone who has not slept for 10 hours: approximately 6.4 (rounded to one decimal place).

Step-by-step solution

Identify the number of pairs of observations

There are two subjects at each of the five sleep deprivation levels, so there are a total of 2 subjects × 5 levels = 10 pairs of observations.

Calculate the number of degrees of freedom

To find the total number of degrees of freedom, we have to consider the number of parameters we are estimating. In this case, we are estimating the slope and the intercept of the regression equation. Degrees of freedom = number of pairs of observations - number of parameters = 10 - 2 = 8.

Complete the MS Excel printout

We don't have an MS Excel printout to complete at this time, so we will move on to the next step.

Find the mean values of x and y

We must first organize the data into a table: $$\begin{array}{l|l}\text { Hours without Sleep } (x) & \text { Number of Errors } (y) \\\hline8 & 8 \\8 & 6 \\12 & 6 \\12 & 10 \\16 & 8 \\16 & 14 \\20 & 14 \\20 & 12 \\24 & 16 \\24 & 12 \\\end{array}$$ Calculate the mean values of x and y: Mean of x: \(\bar{x} = \frac{8+8+12+12+16+16+20+20+24+24}{10} = \frac{160}{10} = 16\) Mean of y: \(\bar{y} = \frac{8+6+6+10+8+14+14+12+16+12}{10} = \frac{106}{10} = 10.6\)

Compute the values needed for the least-squares equation

Now, we need to compute the following values to find the least-squares coefficients: - The sum of the product of x and y: \(\sum{xy}\) - The sum of the squares of x: \(\sum{x^2}\) Calculating these values, we have: \(\sum{xy} = (8\times8)+(8\times6)+(12\times6)+(12\times10)+(16\times8)+(16\times14)+(20\times14)+(20\times12)+(24\times16)+(24\times12) = 2936\) \(\sum{x^2} = 8^2+8^2+12^2+12^2+16^2+16^2+20^2+20^2+24^2+24^2 = 5440\) Now, we can calculate the coefficients of the least-squares equation: - Slope: $$m = \frac{n\sum{xy} - \sum{x}\sum{y}}{n\sum{x^2} - (\sum{x})^2} = \frac{10\times2936-160\times106}{10\times5440 - 160^2} = \frac{2936-16960}{5440-25600} = \frac{-14024}{-20160}= 0.696$$ - Intercept: $$b = \bar{y} - m\bar{x} = 10.6 - 0.696\times16 = 10.6 - 11.136 = -0.536$$

Obtain the predicted number of errors for 10 hours of sleep deprivation using the prediction equation

Finally, with the least-squares prediction equation given by \(y = 0.696x - 0.536\), we can predict the number of errors for someone who has not slept for 10 hours: Predicted number of errors = \(0.696(10) - 0.536 = 6.96 - 0.536 = 6.424\) The predicted number of errors for a person who has not slept for 10 hours is approximately 6.4 (rounded to one decimal place).

Key concepts

Least-squares Prediction Equation

The least-squares prediction equation is a fundamental tool in statistics and data analysis. It represents a line of best fit through a set of points in a two-dimensional space, such as hours without sleep and number of errors in the given exercise. This equation is derived using the method of least squares, which minimizes the sum of the squared differences between the observed values and the values predicted by the model.

The general form of the least-squares prediction equation is given by:
\[\begin{equation}\hat{y} = mx + b\[\begin{equation}
where:

• \( \hat{y} \) represents the predicted value of the dependent variable (number of errors in our case).
• \( m \) is the slope of the line, which indicates how much the dependent variable changes for each unit change in the independent variable (hours without sleep).
• \( b \) is the y-intercept, which represents the predicted value when the independent variable is zero.

In the context of the sleep deprivation study, the least-squares equation helps us to predict how the number of errors in problem-solving correlates with the amount of sleep loss. Implementing this equation allows for a better understanding of how critical adequate sleep is for cognitive function.

Degrees of Freedom in Statistics

Degrees of freedom is a key concept in the field of statistics, representing the number of independent values or quantities that can vary in an analysis without breaking any constraints. It's closely related to the number of observations in your dataset and the number of parameters you need to estimate in your statistical model.

Think of it as the amount of 'wiggle room' you have when calculating statistical figures. When working with regression analysis, as seen in the sleep deprivation study, degrees of freedom are calculated by subtracting the number of estimated parameters from the number of data pairs:

• Number of data pairs: The total count of individual observations in your study.
• Number of parameters: Typically includes the slope and intercept in a simple linear regression.

For the study, with 10 pairs of observations and 2 parameters (slope and intercept), the degrees of freedom are 8. Having a proper understanding of degrees of freedom is essential because it affects the calculation of statistical tests and confidence intervals. It guides statisticians in determining if their findings are statistically significant or if they might have arisen by chance.

Regression Analysis

Regression analysis is a powerful statistical method for examining the relationship between two or more variables. It's particularly useful when you want to predict the value of a variable based on the value of another. In the sleep deprivation study, regression analysis is used to investigate how sleep deprivation (independent variable) affects problem-solving errors (dependent variable).

Here's a breakdown of the regression analysis process:

• Data Collection: Crucial to regression analysis is the collection of comprehensive and accurate data. In our case, the number of errors made by subjects after various sleep deprivation levels.
• Model Assumptions: Before using regression, it’s important to ensure your data meets certain assumptions — like linearity, independence, homoscedasticity, and normality — to provide reliable results.
• Finding Regression Coefficients: Using the least-squares method, you calculate the slope and intercept that best fit your data.
• Model Validation: After determining the coefficients, you check if the regression model is a good fit through various diagnostic tests and checks like the R-squared value and residual analysis.

Ultimately, regression analysis in the given exercise allows us to not only understand the existing data but also to make informed predictions, like estimating the number of errors a person might make after being sleep-deprived for a certain number of hours.