Question

Subjects in a sleep deprivation experiment were asked to solve a set of simple addition problems after having been deprived of sleep for a specified number of hours. The number of errors was recorded along with the number of hours without sleep. The results, along with the MINITAB output for a simple linear regression, are shown below. $$ \begin{aligned} &\begin{array}{l|l|l|l} \text { Number of Errors, } y & 8,6 & 6,10 & 8,14 \\ \hline \text { Number of Hours without Sleep, } x & 8 & 12 & 16 \end{array}\\\ &\begin{array}{l|l|l} \text { Number of Errors, } y & 14,12 & 16,12 \\ \hline \text { Number of Hours without Sleep, } x & 20 & 24 \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { Analysis of Variance }\\\ &\begin{array}{lcrrrr} \text { Source } & \text { DF } & \text { Adj SS } & \text { Adj MS } & \text { F-Value } & \text { P-Value } \\ \hline \text { Regression } & 1 & 72.20 & 72.200 & 14.37 & 0.005 \\ \text { Error } & 8 & 40.20 & 5.025 & & \\ \text { Total } & 9 & 112.40 & & & \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { Model Summary }\\\ &\begin{array}{rrr} \mathrm{S} & \text { R-sq } & \text { R-sq(adj) } \\ \hline 2.24165 & 64.23 \% & 59.76 \% \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { Coefficients }\\\ &\begin{array}{lrrrr} \text { Term } & \text { Coef } & \text { SE Coef } & \text { T-Value } & \text { P-Value } \\ \hline \text { Constant } & 3.00 & 2.13 & 1.41 & 0.196 \\ \mathrm{x} & 0.475 & 0.125 & 3.79 & 0.005 \end{array} \end{aligned} $$ Regression Equation $$ y=3.00+0.475 x $$ a. Do the data present sufficient evidence to indicate that the number of errors is linearly related to the number of hours without sleep? Identify the two test statistics in the printout that can be used to answer this question. b. Would you expect the relationship between \(y\) and \(x\) to be linear if \(x\) varied over a wider range \((\) say \(, x=4\) to \(x=48\) )? c. How do you describe the strength of the relationship between \(y\) and \(x ?\) d. What is the best estimate of the common population variance \(\sigma^{2} ?\) e. Find a \(95 \%\) confidence interval for the slope of the line.

Short answer

Yes, there is sufficient evidence to indicate a linear relationship between the number of errors and the number of hours without sleep, as the F-Value and the P-Value both suggest a significant linear relationship. b. Is the relationship expected to be linear if x varied over a wider range? It is difficult to predict if the relationship would still be linear if x varied over a wider range, as the linear regression model might not be the best fit and other types of relationships might be more appropriate. c. Describe the strength of the relationship between y and x. The R-squared value of 64.23% indicates a moderately strong relationship between the number of errors and the number of hours without sleep. d. What is the best estimate of the common population variance σ²? The best estimate of the common population variance σ² is the Mean Squared Error (MSE) of 5.025. e. What is the 95% confidence interval for the slope of the line? The 95% confidence interval for the slope of the line is approximately (0.188, 0.762).

Step-by-step solution

a. Testing for linear relationship between number of errors and hours without sleep

To determine if there is sufficient evidence to indicate a linear relationship between the number of errors and the number of hours without sleep, we can look at two test statistics in the printout: F-Value and the P-Value associated with it. An F-Value of 14.37 and a P-Value of 0.005 (which is less than 0.05) both suggest that there is a significant linear relationship between the number of errors and the number of hours without sleep.

b. Expectation of linear relationship between y and x over a wider range

Although the linear relationship is significant for the given dataset (x values ranging from 8 to 24 hours without sleep), it is difficult to predict if the relationship would still be linear if we extend the range of x, say from 4 to 48 hours. The linear regression model might not be the best fit for a wider range, and other types of relationships (e.g., quadratic, exponential) might be more appropriate.

c. Strength of the relationship between y and x

To describe the strength of the relationship between y and x, we can look at the R-squared value (R-sq). In this case, R-sq is 64.23%, which means that approximately 64.23% of the variation in the number of errors can be explained by the number of hours without sleep. This indicates a moderately strong relationship between y and x.

d. Estimating the common population variance σ²

The best estimate of the common population variance σ² can be found using the Mean Squared Error (MSE) from the Analysis of Variance table. In this case, the Adj MS for error is 5.025, which is an estimate of the unknown σ².

e. Calculating a 95% confidence interval for the slope of the line

To calculate a 95% confidence interval for the slope of the line, we need the slope coefficient (Coef) and its standard error (SE Coef). In the given printout, the slope Coef is 0.475, and the SE Coef is 0.125. Using a t-distribution table and with a sample size of 10 (n-2 = 8 degrees of freedom for regression analysis), we can find the t-score for a 95% confidence interval, which is approximately 2.306. The confidence interval for the slope can be calculated using the slope Coef ± t-score * SE Coef: Lower limit: 0.475 - (2.306 * 0.125) ≈ 0.188 Upper limit: 0.475 + (2.306 * 0.125) ≈ 0.762 Thus, the 95% confidence interval for the slope of the line is approximately (0.188, 0.762).

Key concepts

F-Value and P-Value

Understanding the significance of the linear relationship between two variables can be assessed using the F-Value and P-Value in a simple linear regression analysis. The F-Value is calculated from the ratio of the mean square due to regression to the mean square due to error. It's used to test if the variance explained by the regression model is significant when compared with the variance unexplained.

For instance, in our exercise, the F-Value is 14.37, which is considerably high. The P-Value, which indicates the probability of observing such an F-Value under the null hypothesis that there is no linear relationship, is reported as 0.005. Since this value is less than the common significance level of 0.05, we have strong evidence against the null hypothesis and can conclude that the number of errors is significantly related to the number of hours without sleep.

R-squared value

The R-squared value is a key statistic in regression analysis that represents the proportion of the variance for the dependent variable (such as the number of errors) that's explained by the independent variable (like the hours without sleep). It's a measure of how well the model explains the variation in the data.

In this particular exercise, the R-squared value is 64.23%, which suggests that over 64% of the variability in the number of errors can be explained by the time spent without sleep. This is a moderate correlation and tells us that the linear regression model has reasonable predictive power over the range of data we have.

Common Population Variance

In the context of simple linear regression, the common population variance, denoted by \(\sigma^2\), is a measure of the variability of the dependent variable around the regression line. It's assumed to be constant across all levels of the independent variable.

From our exercise, \(\sigma^2\) is estimated by the mean square error (MSE), which is the average of the squared differences between the observed values and the values predicted by the regression model. Here, the Adj MS for error is 5.025, giving us an MSE estimate for the common population variance. This is a crucial element because it plays a major role in hypothesis testing and constructing confidence intervals in regression analysis.

Confidence Interval for Slope

When analyzing regression outcomes, it is important to determine the precision and reliability of the estimated slope coefficient, which describes the rate of change in the dependent variable for a one-unit change in the independent variable. A 95% confidence interval gives an estimated range within which we can be 95% confident that the population slope lies, based on our sample data.

In our sleep-deprivation study, the slope coefficient is 0.475 and it has a standard error of 0.125. By applying the t-distribution, we obtain a 95% confidence interval for the slope between approximately 0.188 and 0.762. This interval means we can be 95% confident that for each additional hour without sleep, the true increase in the number of errors lies somewhere in this range, highlighting the practical significance of sleep on cognitive function in real-world settings.