Question

Professor Asimov Professor Isaac Asimov wrote nearly 500 books during a 40 -year career. In fact, as his career progressed, he became even more productive in terms of the number of books written within a given period of time. \({ }^{3}\) The data give the time in months required to write his books in increments of 100 : $$\begin{array}{l|lllll}\text { Number of Books, } x & 100 & 200 & 300 & 400 & 490 \\\\\hline \text { Time in Months, } y & 237 & 350 & 419 & 465 & 507\end{array}$$ a. Assume that the number of books \(x\) and the time in months \(y\) are linearly related. Find the least-squares line relating \(y\) to \(x\). b. Plot the time as a function of the number of books written using a scatterplot, and graph the leastsquares line on the same paper. Does it seem to provide a good fit to the data points? c. Construct the ANOVA table for the linear regression.

Short answer

Answer: The least-squares line equation for the given data is \(y = 2.488x + 165.768\). Based on the scatterplot and the least-squares line, it appears to provide a reasonable fit to the data points, although there may be some deviation from the line.

Step-by-step solution

Calculate the necessary statistics

First, we need to calculate the means of \(x\) and \(y\). Also, we need to find the covariance and variance of \(x\) to calculate the slope and intercept of the least-squares line. We can use the following formulas: $$\bar{x} = \frac{\sum x_i}{n}$$ $$\bar{y} = \frac{\sum y_i}{n}$$ $$\text{cov}(x,y) = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{n}$$ $$\text{var}(x) = \frac{\sum (x_i - \bar{x})^2}{n}$$ Where \(n\) is the number of data points. Let's calculate the means, covariance, and variance for \(x\) and \(y\). \(\bar{x} = \frac{100+200+300+400+490}{5} = 298\) \(\bar{y} = \frac{237 + 350 + 419 + 465 + 507}{5} = 395.6\) \(\text{cov}(x,y) = \frac{[(100-298)(237-395.6) + (200-298)(350-395.6) + (300-298)(419-395.6) + (400-298)(465-395.6) + (490-298)(507-395.6)]}{5} = 108261.92\) \(\text{var}(x) = \frac{[(100-298)^2 + (200-298)^2 + (300-298)^2 + (400-298)^2 + (490-298)^2]}{5} = 43504\)

Find the least-squares line equation

Now that we have the covariance and variance of \(x\), we can calculate the slope and intercept of the least-squares line using the formulas: $$m = \frac{\text{cov}(x,y)}{\text{var}(x)}$$ $$b = \bar{y} - m\bar{x}$$ \(m = \frac{108261.92}{43504} = 2.488\) \(b = 395.6 - 2.488(298) = 165.768\) So, the least-squares line equation is: $$y = 2.488x + 165.768$$

Create a scatterplot with the given data

Plot the data points on a graph, with \(x\) as the number of books and \(y\) as the time in months. Make sure to label the axes and scale them appropriately.

Graph the least-squares line on the scatterplot

Using the least-squares line equation found in step 2, plot the line on the same graph as the scatterplot. The line should start at the lowest \(x\) value (100) and extend to the highest \(x\) value (490).

Assess the fit of the least-squares line

Examine the line and data points in the scatterplot. If the line appears to travel through the "middle" of the data points, it can be considered a good fit. In this case, the least-squares line seems to provide a reasonable fit, although there may be some deviation from the line.

Calculate the sums of squares for the ANOVA table

To create the ANOVA table, we need to calculate the sum of squares for regression (SSR), the sum of squares for error (SSE), and the total sum of squares (SST). We can use the following formulas: $$SSR = \sum(\hat{y}_i - \bar{y})^2$$ $$SSE = \sum(y_i - \hat{y}_i)^2$$ $$SST = \sum(y_i - \bar{y})^2$$ Where \(\hat{y}_i\) is the predicted \(y\) value, calculated from the least-squares line equation. Calculate the SSR, SSE, and SST for this data.

Create the ANOVA table with the calculated values

Construct the ANOVA table with the calculated SSR, SSE, and SST values. The table should have columns for Source (Regression, Error, or Total), Degrees of Freedom (df), Sum of Squares (SS), Mean Square (MS), F-Statistic (F), and P-Value (P). Fill in the table with the calculated values and complete the table with the appropriate degrees of freedom, calculated as follows: - Regression df = 1 (always 1 for a simple linear regression) - Error df = n - 2 (5 data points - 2 = 3) - Total df = n - 1 (5 data points - 1 = 4) Now the ANOVA table is complete, and the results can be analyzed to determine the significance of the linear regression.

Key concepts

Least-Squares Line

Understanding the least-squares line is crucial when analyzing the relationship between two variables in linear regression analysis. Here's an example: Professor Asimov's productivity, as measured by the number of books written over time, can be studied using a least-squares line. This statistical method finds the best-fitting straight line through a set of data points by minimizing the sum of the squares of the vertical distances (residuals) of the points from the line.

The equation for the least-squares line, denoted as \( y = mx + b \), can be obtained using specific formulas to calculate the slope \( m \) and y-intercept \( b \). The slope is the change in the dependent variable \( y \) for each unit change in the independent variable \( x \), while the y-intercept is the expected value of \( y \) when \( x \) equals zero. For Professor Asimov's data, we determined that \( m = 2.488 \) and \( b = 165.768 \), resulting in the least-squares line equation of \( y = 2.488x + 165.768 \). This equation can be used to predict the time required to write subsequent books based on the linear continuation of past productivity patterns.

Scatterplot

A scatterplot is a graph used to display the relationship between two quantitative variables. Each mark (point) on the scatterplot represents the values of a single data point. When creating a scatterplot of Professor Asimov's book writing data, the number of books (\( x \)) is plotted along the horizontal axis, and the time in months (\( y \)) is plotted along the vertical axis.

After plotting the points, we get a visual representation of the distribution and relationship between the number of books and the time taken to write them. The plotted least-squares line is then added to this scatterplot to assess the goodness of fit. Therefore, the scatterplot serves as a foundational tool for visualizing and beginning to analyze the linear relationship before moving on to more complex statistical assessments.

ANOVA Table

The ANOVA (Analysis of Variance) table in linear regression provides a way to test whether there is a statistically significant relationship between our independent and dependent variables. It does so by comparing the variance explained by the model (regression) to the unexplained variance (error).

For Professor Asimov's analysis, we construct an ANOVA table that includes the sums of squares for the regression (SSR) and error (SSE), alongside the total sum of squares (SST). The table also incorporates degrees of freedom (df), mean squares (MS), the F-statistic, and the P-value. The F-statistic tests the null hypothesis that all regression coefficients are equal to zero (no linear relationship), and the P-value helps determine the statistical significance of our observed results. In essence, the ANOVA table is a structured way to display the output of our regression analysis and guide decisions based on statistical evidence.

Sums of Squares

Sums of squares play a pivotal role in linear regression and ANOVA, breaking down the overall variability of the dependent variable into components. For Professor Asimov's productivity study, we calculate three types of sums of squares: the sum of squares due to regression (SSR), which measures the variation explained by the regression line; the sum of squares due to error (SSE), which measures the variation not explained by the regression line, and; the total sum of squares (SST), which measures the total variation in the dependent variable.

Specifically, SSR assesses how well the regression line approximates the real data points, SSE represents the discrepancy between the actual and predicted values, and SST is the sum of SSR and SSE, representing the total variability in the data. These values are used in the ANOVA table to understand the proportion of total variability that is explained by the model compared to the unexplained variability, thereby informing the effectiveness of the regression model.

Statistics in Probability

The field of statistics is intertwined with probability, as statistical analyses are based on probabilistic frameworks. For instance, when we interpret the ANOVA table for Professor Asimov's book writing data, the P-value is derived from probability theory. It represents the likelihood of observing a test statistic as extreme as the one calculated if the null hypothesis (no relationship between the number of books and writing time) were true.

The statistical significance indicated by the P-value assesses whether the observed results could occur simply by chance. In applying statistics to probability, we can make informed judgments about the reliability of our linear regression model. This critical thinking can help researchers, like those studying Professor Asimov's impressive literary output, draw more accurate conclusions about the phenomena they are investigating.