/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Refer to the following ANOVA tab... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 6

Refer to the following ANOVA table. $$\begin{array}{|lrrrr|}\hline \text { SOURCE } & \text { DF } & \text { SS } & \text { MS } & \text { F } \\\\\text { Regression } & 5 & 60 & 12 & 1.714 \\\\\text { Error } & 20 & 140 & 7 & \\\\\text { Total } & 25 & 200 & & \\\\\hline\end{array}$$ a. How large was the sample? b. How many independent variables are there? c. Compute the coefficient of multiple determination. d. Compute the multiple standard error of estimate.

Short Answer

Expert verified
Sample size is 26. There are 5 independent variables. \( R^2 \) is 0.3. Multiple standard error of estimate is approximately 2.646.

Step by step solution

01

Determine Sample Size

The sample size can be determined from the Total degrees of freedom (DF), which is given as 25. The total DF is calculated as \( n - 1 \), where \( n \) is the sample size. Therefore, \( n = 25 + 1 = 26 \).
02

Identify Number of Independent Variables

The number of independent variables is given by the degrees of freedom in the Regression row, which is 5. Therefore, there are 5 independent variables.
03

Calculate Coefficient of Multiple Determination (R虏)

The coefficient of multiple determination, \( R^2 \), is calculated as the ratio of the regression sum of squares (SS) to the total SS: \( R^2 = \frac{60}{200} = 0.3 \).
04

Calculate Multiple Standard Error of Estimate

The multiple standard error of estimate (\( SE \)) is calculated using the formula \( SE = \sqrt{\frac{SS_{ ext{Error}}}{DF_{ ext{Error}}}} \). Here, \( SS_{\text{Error}} = 140 \) and \( DF_{\text{Error}} = 20 \), so \( SE = \sqrt{\frac{140}{20}} = \sqrt{7} \approx 2.646 \).

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 91影视!

Key Concepts

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

Sample Size Determination
Sample size determination is a fundamental step in statistical analysis. It involves calculating the number of observations or data points in your dataset. In this ANOVA example, the sample size is inferred from the total degrees of freedom (DF).
  • Degrees of freedom (DF) are specific values needed to account for constraints in statistical calculations, especially with ANOVA.
  • ANOVA total DF is determined by subtracting 1 from the total sample size, represented as: \( DF_{ ext{Total}} = n - 1 \).
Given the total DF is 25 in this exercise, the sample size \( n \) is calculated by adding 1, resulting in a sample size of 26. Understanding sample size is crucial for the validity and reliability of statistical tests. Larger samples tend to provide more accurate and reliable estimates.
Independent Variables
In any statistical analysis, independent variables are the factors that you manipulate or control to determine their effects on dependent variables. In this ANOVA scenario, these variables are identified from the degrees of freedom associated with the regression row.
  • The degrees of freedom for regression indicates the number of independent variables (plus the intercept if included).
  • In this exercise, the regression DF is 5, meaning five independent variables are considered.
Identifying independent variables correctly is vital as they are tested for significance on their impact on the outcome. Their correct interpretation can aid in building a robust statistical model and understanding the relationship between variables.
Coefficient of Determination
The coefficient of determination, represented as \( R^2 \), is a key metric in regression analysis that measures the proportion of the variance in the dependent variable that can be predicted from the independent variables.
  • It is calculated using the formula: \( R^2 = \frac{SS_ ext{Regression}}{SS_ ext{Total}} \).
  • This ratio reflects how well the regression line fits the data points.
In this exercise, with a regression sum of squares (SS) of 60 and total SS of 200, \( R^2 = \frac{60}{200} = 0.3 \). This indicates that 30% of the variation in the dependent variable is explained by the model. A higher \( R^2 \) value signifies a better fit, providing insights into how reliable your model predictions are.
Standard Error of Estimate
The standard error of estimate measures the accuracy of predicted values in regression analysis, showing the typical distance that the data points fall from the regression line.
  • It is calculated using the formula: \( SE = \sqrt{\frac{SS_ ext{Error}}{DF_ ext{Error}}} \).
  • This helps gauge the model鈥檚 precision and the variability of the data around the fitted line.
In this instance, with error SS of 140 and error DF of 20, \( SE = \sqrt{\frac{140}{20}} = \sqrt{7} \approx 2.646 \). A smaller value of standard error indicates that the observed data points are close to the regression line, signifying a precise model. Understanding the standard error of estimate equips you with an indicator of the reliability of the predictions your model produces.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The district manager of Jasons, a large discount electronics chain, is investigating why certain stores in her region are performing better than others. She believes that three factors are related to total sales: the number of competitors in the region, the population in the surrounding area, and the amount spent on advertising. From her district, consisting of several hundred stores, she selects a random sample of 30 stores. For each store she gathered the following information. $$ \begin{aligned} Y &=\text { total sales last year (in } \$ \text { thousands). } \\ X_{1} &=\text { number of competitors in the region. } \\ X_{2} &=\text { population of the region (in.millions). } \\ X_{3} &=\text { advertising expense (in } \$ \text { thousands). } \end{aligned} $$ The sample data were run on MINITAB, with the following results. $$\begin{array}{|lrrr|} \hline \text { Analysis of variance } & & \\ \text { SOURCE } & \text { DF } & \text { SS } & \text { MS } \\ \text { Regression } & 3 & 3050.00 & 1016.67 \\ \text { Error } & 26 & 2200.00 & 84.62 \\ \text { Total } & 29 & 5250.00 & \\ \text { Predictor } & \text { Coef } & \text { StDev } & \text { t-ratio } \\ \text { Constant } & 14.00 & 7.00 & 2.00 \\ X_{1} & -1.00 & 0.70 & -1.43 \\ X_{2} & 30.00 & 5.20 & 5.77 \\ X_{3} & 0.20 & 0.08 & 2.50 \\\\\hline\end{array}$$ a. What are the estimated sales for the Bryne Store, which has four competitors, a regional population of \(0.4(400,000),\) and advertising expense of \(30(\$ 30,000) ?\) b. Compute the \(R^{2}\) value. c. Compute the multiple standard error of estimate. d. Conduct a global test of hypothesis to determine whether any of the regression coefficients are not equal to zero. Use the .05 level of significance. e. Conduct tests of hypotheses to determine which of the independent variables have significant regression coefficients. Which variables would you consider eliminating? Use the .05 significance level.

Many regions along the coast in North and South Carolina and Georgia have experienced rapid population growth over the last 10 years. It is expected that the growth will continue over the next 10 years. This has resulted in many of the large grocery store chains building new stores in the region. The Kelley's Super Grocery Stores, Inc. chain is no exception. The director of planning for Kelley's Super Grocery Stores wants to study adding more stores in this region. He believes there are two main factors that indicate the amount families spend on groceries. The first is their income and the other is the number of people in the family. The director gathered the following sample information. $$\begin{array}{|rrrr|}\hline \text { Family } & \text { Food } & \text { Income } & \text { Size } \\\\\hline 1 & \$5.04 & \$ 73.98 & 4 \\\2 & 4.08 & 54.90 & 2 \\\3 & 5.76 & 94.14 & 4 \\\4 & 3.48 & 52.02 & 1 \\ 5 & 4.20 & 65.70 & 2 \\\6 & 4.80 & 53.64 & 4 \\\7 & 4.32 & 79.74 & 3 \\\8 & 5.04 & 68.58 & 4 \\ 9 & 6.12 & 165.60 & 5 \\\10 & 3.24 & 64.80 & 1 \\ 11 & 4.80 & 138.42 & 3 \\\12 & 3.24 & 125.82 & 1 \\\13 & 6.60 & 77.58 & 7 \\ 14 & 4.92 & 171.36 & 2 \\\15 & 6.60 & 82.08 & 9 \\ 16 & 5.40 & 141.30 & 3 \\\17 & 6.00 & 36.90 & 5 \\\18 & 5.40 & 56.88 & 4 \\\19 & 3.36 & 71.82 & 1 \\\20 & 4.68 & 69.48 & 3 \\\21 & 4.32 & 54.36 & 2 \\\22 & 5.52 & 87.66 & 5 \\\23 & 4.56 & 38.16 & 3 \\ 24 & 5.40 & 43.74 & 7 \\\25 & 4.80 & 48.42 & 5 \\\\\hline\end{array}$$ Food and income are reported in thousands of dollars per year, and the variable "Size" refers to the n眉mber of people in the household. a. Develop a correlation matrix. Do you see any problems with multicollinearity? b. Determine the regression equation. Discuss the regression equation. How much does an additional family member add to the amount spent on food? c. What is the value of \(R^{2}\) ? Can we conclude that this value is greater than \(0 ?\) d. Would you consider deleting either of the independent variables? e. Plot the residuals in a histogram. Is there any problem with the normality assumption? f. Plot the fitted values against the residuals. Does this plot indicate any problems with homoscedasticity?

A multiple regression equation yields the following partial results. $$\begin{array}{|lcr|}\hline \text { Source } & \text { Sum of Squares } & \text { df } \\\\\hline \text { Regression } & 750 & 4 \\\\\text { Error } & 500 & 35 \\\\\hline\end{array}$$ a. What is the total sample size? b. How many independent variables are being considered? C. Compute the coefficient of determination. d. Compute the standard error of estimate. e. Test the hypothesis that none of the regression coefficients is equal to zero. Let \(\alpha=.05\).

Fran's Convenience Marts are located throughout metropolitan Erie, Pennsylvania. Fran, the owner, would like to expand into other communities in northwestern Pennsylvania and southwestern New York, such as Jamestown, Corry, Meadville, and Warren. As part of her presentation to the local bank, she would like to better understand the factors that make a particular outlet profitable. She must do all the work herself, so she will not be able to study all her outlets. She selects a random sample of 15 marts and records the average daily sales \((Y),\) the floor space (area), the number of parking spaces, and the median income of families in that ZIP code region for each. The sample information is reported on the next page. $$\begin{array}{|ccccc|}\hline \begin{array}{c}\text { Sampled } \\\\\text { Mart }\end{array} & \begin{array}{c}\text { Daily } \\\\\text { Sales }\end{array} & \begin{array}{c}\text { Store } \\\\\text { Area }\end{array} & \begin{array}{c}\text { Parking } \\\\\text { Spaces }\end{array} & \begin{array}{c}\text { Income } \\\\\text { (\$ thousands) }\end{array} \\\\\hline 1 & \$ 1,840 & 532 & 6 & 44 \\\2 & 1,746 & 478 &4 & 51 \\\3 & 1,812 & 530 & 7 & 45 \\\4 & 1,806 & 508 & 7 & 46 \\\5 & 1,792 & 514 & 5 & 44 \\\6 & 1,825 & 556 & 6 & 46 \\\7 & 1,811 & 541 & 4 & 49 \\\8 & 1,803 & 513 & 6 & 52 \\\9 & 1,830 & 532 & 5 & 46 \\\10 & 1,827 & 537 & 5 & 46 \\\11 & 1,764 & 499 & 3 & 48 \\\12 & 1,825 & 510 & 8 & 47 \\\13 & 1,763 & 490 & 4 & 48 \\\14 & 1,846 & 516 & 8 & 45 \\\15 & 1,815 & 482 & 7 & 43 \\\\\hline\end{array}$$ a. Determine the regression equation. b. What is the value of \(R^{2}\) ? Comment on the value. c. Conduct a global hypothesis test to determine if any of the independent variables are different from zero. d. Conduct individual hypothesis tests to determine if any of the independent variables can be dropped. e. If variables are dropped, recompute the regression equation and \(R^{2}\).

Mike Wilde is president of the teachers' union for Otsego School District. In preparing for upcoming negotiations, he would like to investigate the salary structure of classroom teachers in the district. He believes there are three factors that affect a teacher's salary: years of experience, a rating of teaching effectiveness given by the principal, and whether the teacher has a master's degree. A random sample of 20 teachers resulted in the following data. a. Develop a correlation matrix. Which independent variable has the strongest correlation with the dependent variable? Does it appear there will be any problems with multicollinearity? b. Determine the regression equation. What salary would you estimate for a teacher with five years' experience, a rating by the principal of \(60,\) and no master's degree? c. Conduct a global test of hypothesis to determine whether any of the net regression coefficients differ from zero. Use the .05 significance level. d. Conduct a test of hypothesis for the individual regression coefficients. Would you consider deleting any of the independent variables? Use the .05 significance level. e. If your conclusion in part (d) was to delete one or more independent variables, run the analysis again without those variables. f. Determine the residuals for the equation of part (e). Use a histogram to verify that the distribution of the residuals is approximately normal. g. Plot the residuals computed in part (f) in a scatter diagram with the residuals on the Yaxis and the \(Y^{\prime}\) values on the \(X\) -axis. Does the plot reveal any violations of the assumptions of regression?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.