Sp23 Homework 6

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Feb 20, 2024

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Homework 6 (non-Excel portion) Spring 2023 Must be uploaded into Canvas by 11:55 pm on Wednesday, April 12, 2023. Upon submitting this homework, I affirm that I have not given or received unauthorized aid on this homework, and I completed this work honestly and according to the instructor’s guidelines. Answers may be typed or handwritten. Homeworks are graded on completion, not accuracy. Work must be shown on the homework to receive full credit. Credit for completion will not be awarded if adequate effort has not been shown. Each homework is worth 2.5 points. So, for example, if a student gets a score of 2.5, it means that they did all the work, it does not mean the answers are all correct. In order to promote professional development of students in learning the importance of appearance and presentation of submitted work, 0.1 point will be deducted from homework scores for each the following conditions that are not met (students could lose a total of 0.2 points for not meeting these conditions): • Questions numbered and done in order (all work and answers for each question in one place) • Increased space between each question if the answers are included on the posted homework assignment If you take any definitions/concepts directly from the packet, please cite the page number in the packet. YOU MUST SHOW WORK TO GET CREDIT. 1. Fill in the following chart for the correlation coefficient. No credit will be given unless the entire chart is filled in. Name Symbol What it Tells Us Abs. or Rel.? (give units if absolute) Boundaries What Extremes Signify PopulaDon CorrelaDon Coeﬃcient Rho(p) Tells us the two measures of goodness or the closeness of fit RelaDve Measure -1/+1 -1 means that there is a perfect inverse relaDonship in the populaDon 1 means there is a perfect relaDonship in the sample

2. Fill in the following chart for the coefficient of determination. No credit will be given unless the entire chart is filled in. (Questions 1-2 can be completed after Lecture 27.) 3. Fill in the following chart for the standard error of the estimate. No credit will be given unless the entire chart is filled in. Sample CorrelaDon Coeﬃcient r Tells us the two measures of goodness or the closeness of fit RelaDve Measure -1/+1 -1 means that there is a perfect inverse relaDonship in the populaDon 1 means there is a direct relaDonship in the sample Name Symbol What it Tells Us Abs. or Rel.? (give units if absolute) Boundaries What Extremes Signify PopulaDon Coeﬃcient of DeterminaDon p^2 The fracDon of variaDon is the dependent variable that can be explained by the independent variable ‘x’ in the populaDon RelaDve measure 0,1 The perfect linear relaDonship in the sample, or no linear relaDonship in the populaDon Sample Coeﬃcient of DeterminaDon r^2 Indicates the fracDon of variaDon in the dependent variable that can be explained by the independent variable x in the sample RelaDve measure 0,1 Perfect liner relaDonship or no linear relaDonship in the sample Name Symbol What it Measures Abs. or Rel.? (give units if absolute) Boundaries What Extremes Signify Standard Error of the EsDmate Se Measures the scaWer of the data points about the regression linear Absolute measure 0, Se 0 is the predicaDon with no error, the regression equaDon is a perfect predictor. Se is too much euros, the regression equaDon won’t help

4. Fill in the following chart for the regression coefficients. No credit will be given unless the entire chart is filled in. 5. A person interested in buying a new home wants to test for a relationship between square footage and tax. The standard deviation of square footage is 465 square feet. The standard deviation of tax is $2,976.00. The correlation coefficient is .91. If we are predicting square footage from tax, compute the regression coefficient. Using the phrase given in class (On average, …) and following the pattern given in class, interpret this slope. You must use the phrase on average and follow the pattern given in class to get any credit. (Use the formula for b 1 at the top of p. 179 to solve this problem.) r=.91 b=.91 x (465/2976)=.142 we can expect an increase of 0.142 square feet in the size of the house, on average. 6. List what we are concluding if we accept the null in a regression problem (p. 179 in the course packet). Then list what we are concluding if we reject the null in a regression problem. You must list all the statements presented in the entire slide (four for the null and five for the alternative) in class in order to get credit. Make sure you number them. 1. If we accept the null in a regression problem, then the X variable is not statistically significant. The X variable drops out of our given equation and the True Population slope would be equal to 0. There would be no linear relationship. 2. If we reject the null in a regression problem, then there is a linear relationship. The X variable is Name Symbol What it Tells Us Abs. or Rel.? (give units if absolute) Boundaries PopulaDon Intercept ࠵ Tell us units of the slope for a populaDon Absolute measure -infinity, infinity Sample Intercept b0 Units of slope for a sample absolute Measure -infinity, infinity PopulaDon Slope ࠵ ? The average change in y given one unit change in x for a populaDon Absolute measure (-infinity, 0) (0, infinity ) Sample Slope b1 The average change in y given one unite change in x for a sample Absolute measure (-infinity, 0) (0, infinity)

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statistically significant and the X variable stays in the equation. The sample slope is used as the best estimator of true population slope. The true population slope is not equal to zero. 7. Write a scenario for a simple regression problem. You must include what the two variables are, stating which one is x and which one is y. Remember, y must be quantitative. Do not use the scenarios that are included in the course packet. If you base your idea off of another source, you must state the source. A shoe company wants to see if there is a relationship between how much they spend on advertising and the amount of money they make in sales. X is the amount of money spent on advertising and y is the amount of money that company makes in shoe sales. In this simple linear regression, the R-Square value was found to be 0.73. What is the correlation coefficient? (Questions 3-7 can be completed after Lecture 28.) Excel Homework 6 Spring 2023 All Excel tutorials have a red box titled “Are you in the right tutorial?” Read the box before you start – it will help you make sure that you are using the correct tutorial for your Excel version. Using the Homework 6 data set in Canvas and the appropriate Excel Homework 6 Tutorial or any other sources, answer all of the questions below. Terms: Batting Average (BA) – number of hits/number of at-bats On Base Percentage (OBP) – number of times a player gets on base/number of plate appearances Problem: We would like to see if there is a relationship between the Batting Average (BA) of players in the National League Central with the On Base Percentage (OBP) of the players. If we conclude there is a relationship, then we can use Batting Average to predict On Base Percentage. Follow all the steps in the tutorial to make a scatterplot of Batting Average (BA) and On Base Percentage (OBP). BA will be your independent variable and OBP will be your dependent variable. You do not need to include the Scatterplot with your homework. 1. Write a short description of what kind of relationship, if any, you see in your scatterplot. (.1) After creating the scatter plot I could see that there is a direct relationship between batting average and on base percentage Using Data Analysis in Excel, find the correlation coefficient for BA and OBP by creating a correlation matrix. (If Data Analysis is not loaded in your Excel, follow the instructions in Excel Tutorial 1 to install Data Analysis Toolpak.) You do not need to include the correlation matrix with your homework.

2. What is the value of the correlation coefficient found in your correlation matrix? (.1) This correlation coefficient represents that there is a strong direct relationship between BA and OBP =.972 3. Using the value of the correlation coefficient found in Question 2, write a statement about the strength and direction of the data set. (.1) The correlation coefficient tells its that the data set is almost a perfect direct linear correlation. (Questions 1-3 can be completed after Lecture 27.) 4. Using the appropriate functions, find the sample standard deviation of both BA and OBP. You must handwrite or type the entire function equations (including the equal signs, the function names, and the arguments) and the answers. No credit will be given without the entire equations and the answers. (.1 for BA sample standard deviation, .1 for OBP sample standard deviation) Standard deviation BA =STDEV.S(o2:o224) = 0.123331572 Standard deviation OBP =STDEV.S(p2:p224) = 0.14960157 5. Using the value of r found in Question 2, hand calculate b 1 . You must show all work. (.1) b 1 = r ( S y / S x ) b 1 = . 972(0.1496/0.1233 ) b 1 = 1.18 6. Write an interpretation of the slope beginning with the phrase “On average…”. You must use the phrase on average and follow the pattern given in class in order to get credit. (.1) On average, as the batting average increases by 1, the on base percentage will increase as well by 1.18 (Questions 4-6 can be completed after Lecture 28.) Use Data Analysis to run a regression of BA (independent variable) and OBP (dependent variable). (If Data Analysis is not loaded in your Excel, follow the instructions in Excel Tutorial 1 to install Data Analysis Toolpak.) You will be using this output for the remainder of the questions. You do not need to turn the output in with your homework. 7. How does the slope you calculated by hand in Question 5 compare to the slope found in the regression output? (.1) When rounded the regression output is the same as the slope calculated. 8. Using the regression output, write the regression equation. (.1) Y= 0.009578 + 1.18x 9. What On Base Percentage would you predict if the Batting Average was .206? As always, you must show all work. (.1) y=0.009578 +1.18(.206)

y=.253 10. Is Batting Average a significant predictor of On Base Percentage? Why or why not? Alpha for this problem is .05. (.1 for answer, .1 for why) Yes, because the batting average is a good predictor for OBP because the alpha is more than the p-value. 11. What is the value of R-Square? (.1) r^2=.9457 12. Write a statement to interpret the R-square value. Make sure you follow the pattern. (.1) 94.57 of the variation is on base percentage changes can be shown by batting averages In the sample (Questions 7-12 can be completed after Lecture 29.)

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