91Ó°ÊÓ

Professor Isaac Asimov was one of the most prolific writers of all time. Prior to his death, he 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. \({ }^{1}\) The data give the time in months required to write his books in increments of 100 : \begin{tabular}{llll} 200 & 300 & 400 & 490 \\ \hline 350 & 419 & 465 & 507 \end{tabular} \begin{tabular}{l|ll} Number of Books, \(x\) & 100 & 20 \\ \hline Time in Months, \(y\) & 237 & 3 \end{tabular} 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 scatter plot, and graph the least squares 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.

Step by step solution

01

Calculate the necessary sums

To find the least-squares line relating \(y\) to \(x\), we first need to calculate the following sums: - Sum of \(x\) values (\(\sum x\)) - Sum of \(y\) values (\(\sum y\)) - Sum of squared \(x\) values (\(\sum x^2\)) - Sum of products of \(x\) and \(y\) values (\(\sum xy\)) Using the given data, we can calculate these sums as follows: \(\sum x = 100 + 20 + 200 + 300 + 400 + 490 = 1510\), \(\sum y = 237 + 3 + 350 + 419 + 465 + 507 = 1981\), \(\sum x^2 = 100^2 + 20^2 + 200^2 + 300^2 + 400^2 + 490^2 = 746,100\), \(\sum xy = 100(237) + 20(3) + 200(350) + 300(419) + 400(465) + 490(507) = 458,640\).

02

Calculate the slope and y-intercept

Now, we can use the following formulas to calculate the slope (\(m\)) and y-intercept (\(b\)) of the least-squares line: $$m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}$$ and $$b = \bar{y} - m\bar{x}$$ where \(n\) is the number of data points, \(\bar{x}\) is the mean of the \(x\) values, and \(\bar{y}\) is the mean of the \(y\) values. Using the sums calculated in Step 1, we have: \(m = \frac{6(458,640) - (1510)(1981)}{6(746,100) - (1510)^2} = -0.1635\) \(b = \frac{1981}{6} - (-0.1635)\frac{1510}{6} = 75.4025\) So, the equation of the least-squares line is \(y = -0.1635x + 75.4025\).

03

Create the scatter plot and graph the least-squares line

Using the given data, create a scatter plot with the number of books on the x-axis and the time in months on the y-axis. Plot the data points and draw the least-squares line with the equation \(y = -0.1635x + 75.4025\). Visually inspect if the line provides a good fit to the data.

04

Construct the ANOVA table

In order to construct the ANOVA table for this linear regression, we need to calculate the following values: - Total sum of squares (SST) - Regression sum of squares (SSR) - Error sum of squares (SSE) - Mean regression sum of squares (MSR) - Mean error sum of squares (MSE) - F-statistic (F) First, calculate the SST as follows: $$SST = \sum(y_i - \bar{y})^2$$ Next, calculate the SSR as follows: $$SSR = \sum(\hat{y_i} - \bar{y})^2$$ Then, calculate the SSE as follows: $$SSE = \sum(y_i - \hat{y_i})^2$$ Furthermore, calculate the MSR and MSE as follows: $$MSR = \frac{SSR}{1}$$ and $$MSE = \frac{SSE}{n - 2}$$ Finally, calculate the F-statistic as follows: $$F = \frac{MSR}{MSE}$$ Using these calculations, construct a complete ANOVA table for the linear regression.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

ANOVA Table

An ANOVA table is a summary of the analysis of variance, showcasing the variation within a statistical model. In the context of linear regression, it helps us understand how much of the total variability in the data is explained by the model (regression) and how much is left unexplained (error). Here are the important components of an ANOVA table:

• SST (Total Sum of Squares): Represents the total variation in the response variable, calculated as \(SST = \sum(y_i - \bar{y})^2\).
• SSR (Regression Sum of Squares): Reflects the variation explained by the model, given by \(SSR = \sum(\hat{y_i} - \bar{y})^2\).
• SSE (Error Sum of Squares): Captures unexplained variation, found using \(SSE = \sum(y_i - \hat{y_i})^2\).

We also calculate the mean sums of squares:

• MSR (Mean Regression Sum of Squares): SSR divided by degrees of freedom (often 1 for single predictors), \(MSR = \frac{SSR}{1}\).
• MSE (Mean Error Sum of Squares): SSE divided by its degrees of freedom, calculated as \(MSE = \frac{SSE}{n-2}\).

Finally, the F-statistic checks the overall significance: \(F = \frac{MSR}{MSE}\). A larger F-value typically indicates a stronger relationship.

Least Squares Line

The least squares line is a method of fitting a straight line to a set of data points in such a way that the sum of the squares of the vertical distances of the points from the line is minimized. It is fundamentally a way to summarize the relationship between two variables, typically denoted as \(y = mx + b\).In our case with Professor Asimov's writing data, the goal was to find this best-fitting line through the given points. The procedure involves several steps:

• Calculating the sums and means of the data points.
• Determining the slope \(m\) and intercept \(b\) using the formulas: \[m = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2}\], \[b = \bar{y} - m\bar{x}\]

This yields an equation of the form \(y = -0.1635x + 75.4025\). Such a line offers predictions for the outcome variable \(y\) based on the predictor \(x\), helping illustrate and analyze trends.

Scatter Plot

A scatter plot is a graphical representation illustrating the relationship between two continuous variables, using Cartesian coordinates. Each point in a scatter plot represents an individual data point from the dataset. For Professor Asimov's data, the scatter plot involves

• Plotting the number of books \(x\) on the x-axis and time in months \(y\) on the y-axis.
• Visualizing how the data points "scatter", hence the name.

It's useful for identifying patterns or clusters, suggesting how two variables relate.After plotting these data points, the least squares line equation, \(y = -0.1635x + 75.4025\), is often drawn on the same graph:

• This helps assess the strength and direction of the relationship.
• Ideally, the line should run close to most data points, suggesting it fits well.

Analyzing this holistic view helps confirm if the linear assumption accurately models the data.

Statistical Analysis

Statistical analysis refers to a range of techniques applied to interpret data, uncover patterns, and indicate relationships between variables. It's the backbone of drawing clear, data-supported conclusions.For linear regression analysis:

• We gather and check for any apparent trend in the data using scatter plots.
• Calculate the coefficients of the linear model (the slope and intercept) to draw the least squares line.
• Utilize the ANOVA table to assess model performance, checking how well our line captures the essence of the data.

Your analysis can affirm or refute hypotheses about relationships. It involves steps like estimating parameters (like our slopes), hypothesis testing (using \(F\)-tests), and evaluating how variations in data correlate and those not explained by the model.Through such methods, researchers can make informed decisions, backed by quantifiable evidence, enhancing the understanding of complex patterns and inherent variability in data sets.