(PROBLEM 2RE DATA) Fat and Calories in Cheeseburgers A

What is meant by a marginal distribution? What is meant by a conditional distribution?

Crickets make a chirping noise by sliding their wings rapidly over each other. Perhaps you have noticed that the number of chirps seems to increase with the temperature. The following data list the temperature (in degrees Fahrenheit) and the number of chirps per second for the striped ground cricket. (a) What is the likely explanatory variable in these data? Why? (b) Draw a scatter diagram of the data. (c) Compute the linear correlation coefficient between temperature and chirps per second. (d) Does a linear relation exist between temperature and chirps per second? (e) Find the least-squares regression line treating temperature as the explanatory variable and chirps per second as the response variable. (f) Interpret the slope and \(y\)-intercept, if appropriate. (g) Predict the chirps per second if it is \(83.3^{\circ} \mathrm{F}\). (h) A cricket chirps 15 times per second when the temperature is \(82^{\circ} \mathrm{F}\). Is this rate of chirping above or below average at this temperature? (i) It is \(55^{\circ} \mathrm{F}\) outside. Would you recommend using the linear model found in part (e) to predict the number of chirps per second of a cricket? Why or why not? (j) Compute and interpret \(R^{2}\).

Problem 1RE Basketball Spreads In sports betting, Las Vegas sports books establish winning margins for a team that is favored to win a game. An individual can place a wager on the game and will win if the team bet upon wins after accounting for the spread. For example, if Team A is favored by 5 points and wins the game by 7 points, then a bet on Team A is a winning bet. However, if Team A wins the game by only 3 points, then a bet on Team A is a losing bet. For NCAA Division I basketball games, a least-squares regression with explanatory variable home team Las Vegas spread, x, and response variable home team winning margin, y, is ? = 1.007x - 0.012. (a) Predict the winning margin if the home team is favored by 3 points. ________________ (b) Predict the winning margin (of the visiting team) if the visiting team is favored by 7 points (this is equivalent to the home team being favored by -7 points). ________________ (c) Interpret the slope. ________________ (d) Interpret the y-intercept. ________________ (e) The coefficient of determination is 0.39. Interpret this value.

Refer to Table It is constructing a conditional distribution by level of education different from constructing a conditional distribution by employment status? If they are different, explain the difference.

Problem 2CT researcher collects data regarding the percent of all births to unmarried women and the number of violent crimes for the 50 states and Washington, DC. The scatter diagram along with the least-squares regression line obtained from MINITAB is shown. The correlation between percent of births to unmarried women and the number of violent crimes is 0.667. A politician looks over the data and claims that, for each 1% decrease in births to single mothers, violent crimes will decrease by 23. Therefore, percent of births to single mothers needs to be reduced in an effort to decrease violent crimes. Is there anything wrong with the reasoning of the politician? Explain.

Problem 2RE Fat and Calories in Cheeseburgers A nutritionist was interested in developing a model that describes the relation between the amount of fat (in grams) in cheeseburgers at fastfood restaurants and the number of calories. She obtains the following data from the Web sites of the companies Sandwich (Restaurant) Fat Content (g) Calories Quarter-pound single with cheese (Wendy’s) Whataburger (Whataburer) Cheeseburger (In-n-Out) Big Mac (McDonald’s) Quarter-pounder with cheese (McDonald’s) Whopper with cheese (Burger King) Jumbo Jack (Jack in the Box) Double Steakburger with cheese (Steak ‘n Shake) 20 39 27 29 26 47 35 38 430 750 480 540 510 760 690 632 (a) The researcher wants to use fat content to predict calories. Which is the explanatory variable? ________________ (b) Draw a scatter diagram of the data. ________________ (c) Compute the linear correlation coefficient between fat content and calories. ________________ (d) Does a linear relation exist between fat content and calories in fast-food restaurant sandwiches?

Explain why we use the term association rather than correlation when describing the relation between two variables in this section.

What is the relationship between education and belief in Heaven? The following data represent the highest level of education and belief in Heaven for a random sample of adult Americans (a) Construct a relative frequency marginal distribution. (b) What proportion of adult Americans in the survey definitely believe in Heaven? (c) Construct a conditional distribution of belief in Heaven by level of education. (d) Draw a bar graph of the conditional distribution found in part (c). (e) Is there an association between level of education and belief in Heaven?

Apartments The following data represent the square footage and rent for apartments in the borough of Queens and Nassau County, New York (a) On the same graph, draw a scatter diagram for both Queens and Nassau County apartments treating square footage as the explanatory variable. Be sure to use a different plotting symbol for each group. (b) Compute the linear correlation coefficient between square footage and rent for each location. (c) Does a linear relation exist between the two variables for each location? (d) Does location appear to be a factor in rent?

Explain the idea behind Simpson’s Paradox.

Problem 4CT Consider the following contingency table, which relates the number of applicants accepted to a college and gender Accepted Denied Male 98 522 Female 90 200 (a) Construct a conditional distribution of acceptance status by gender. ________________ (b) What proportion of males was accepted? What proportion of females was accepted? ________________ (c) What might you conclude about the admittance policies of the school? ________________ A lurking variable is the type of school applied to. This particular college has two programs of study: business and social work. The following table shows applications by type of school Business School Social Work School Accepted Denied Accepted Denied Male 90 510 8 12 Female 10 60 80 140 (d) What proportion of males who applied to the business school was accepted? What proportion of females who applied to the business school was accepted? ________________ (e) What proportion of males who applied to the social work school was accepted? What proportion of females who applied to the social work school was accepted? ________________ (f ) Explain carefully how the bias disappears when type of school is considered.

Fat and Calories in Cheeseburgers A nutritionist was interested in developing a model that describes the relation between the amount of fat (in grams) in cheeseburgers at fast food restaurants and the number of calories. She obtains the following data from the Web sites of the companies. Using the data and results, do the following: (a) Find the least-squares regression line treating fat content as the explanatory variable. (b) Draw the least-squares regression line on the scatter diagram. (c) Interpret the slope and y-intercept, if appropriate. (d) Predict the number of calories in a sandwich that has 30 grams of fat. (e) A cheeseburger from Sonic has 700 calories and 42 grams of fat. Is the number of calories for this sandwich above or below average among all sandwiches with 42 grams?

(a) Construct a frequency marginal distribution. (b) Construct a relative frequency marginal distribution. (c) Construct a conditional distribution by x. (d) Draw a bar graph of the conditional distribution found in part (c).

What would you say about a set of quantitative bivariate data whose linear correlation coefficient is -1? What would a scatter diagram of the data look like?

Problem 5RE Using the Queens data and results from Problem, do the following: (a) Find the least-squares regression line, treating square footage as the explanatory variable. ________________ (b) Interpret the slope and y-intercept, if appropriate. ________________ (c) Is the rent on the 825-square-foot apartment in the data above or below average among 825-square-foot apartments? Problem Apartments The following data represent the square footage and rent for apartments in the borough of Queens and Nassau County, New York Queens (New York City) Nassau County (Long Island) Square Footage, x Rent per Month, y Square Footage,x Rent per Month,y 500 650 1100 1875 588 1215 588 1075 1000 2000 1250 1775 688 1655 556 1050 825 1250 825 1300 460 1805 743 1475 1259 2700 660 1315 650 1200 975 1400 560 1250 1429 1900 1073 2350 800 1650 1452 3300 1906 4625 1305 3100 1077 1395 (a) On the same graph, draw a scatter diagram for both Queens and Nassau County apartments treating square footage as the explanatory variable. Be sure to use a different plotting symbol for each group. ________________ (b) Compute the linear correlation coefficient between square footage and rent for each location. ________________ (c) Does a linear relation exist between the two variables for each location? ________________ (d) Does location appear to be a factor in rent?

Problem 6AYU In Problems (a) Construct a frequency marginal distribution. (b) Construct a relative frequency marginal distribution. (c) Construct a conditional distribution by x. (d) Draw a bar graph of the conditional distribution found in part (c). x1 x2 x3 y1 35 25 20 y2 65 75 80

If the slope of a least-squares regression line is negative, what could be said about the correlation between the explanatory and response variable?

(a) Draw a scatter diagram treating x as the explanatory variable and y as the response variable. (b) Select two points from the scatter diagram, and find the equation of the line containing the points selected. (c) Graph the line found in part (b) on the scatter diagram. (d) Determine the least-squares regression line. (e) Graph the least-squares regression line on the scatter diagram. (f ) Compute the sum of the squared residuals for the line found in part (b). (g) Compute the sum of the squared residuals for the least squares regression line found in part (d). (h) Comment on the fit of the line found in part (b) versus the least-squares regression line found in part (d).

Problem 7AYU Made in America In a recent Harris Poll, a random sample of adult Americans (18 years and older) was asked, “When you see an ad emphasizing that a product is ‘Made in America,’ are you more likely to buy it, less likely to buy it, or neither more nor less likely to buy it?” The results of the survey, by age group, are presented in the contingency table below 18–34 35–44 45–54 55+ Total More likely 238 329 360 402 1329 Less likely 22 6 22 16 66 Neither more nor less likely 282 201 164 118 765 Total 542 536 546 536 2160 (a) How many adult Americans were surveyed? How many were 55 and older? ________________ (b) Construct a relative frequency marginal distribution. ________________ (c) What proportion of Americans are more likely to buy a product when the ad says “Made in America”? ________________ (d) Construct a conditional distribution of likelihood to buy “Made in America” by age. That is, construct a conditional distribution treating age as the explanatory variable. ________________ (e) Draw a bar graph of the conditional distribution found in part (d). ________________ (f ) Write a couple sentences explaining any relation between likelihood to buy and age.

What does it mean if a linear correlation coefficient is close to zero? Draw two scatter diagrams for which the linear correlation coefficient is close to zero.

Problem 8AYU Desirability TraitsIn a recent Harris Poll, a random sample of adult Americans (18 years and older) was asked, “Given a choice of the following, which one would you most want to be?” Results of the survey, by gender, are given in the contingency table Richer Thinner Smarter Younger None of these Total Male 520 158 159 181 102 1120 Female 425 300 144 81 92 1042 Total 945 458 303 262 194 2162 (a) How many adult Americans were surveyed? How many males were surveyed? ________________ (b) Construct a relative frequency marginal distribution. ________________ (c) What proportion of adult Americans want to be richer? ________________ (d) Construct a conditional distribution of desired trait by gender. That is, construct a conditional distribution treating gender as the explanatory variable. ________________ (e) Draw a bar graph of the conditional distribution found in part (d). ________________ (f ) Write a couple sentences explaining any relation between desired trait and gender.

Use Queens data and the results from Problems 3 and 5 to compute and interpret \(R^2\). Apartments The following data represent the square footage and rent for apartments in the borough of Queens and Nassau County, New York. Problem 3 (a) On the same graph, draw a scatter diagram for both Queens and Nassau County apartments treating square footage as the explanatory variable. Be sure to use a different plotting symbol for each group. (b) Compute the linear correlation coefficient between square footage and rent for each location. (c) Does a linear relation exist between the two variables for each location? (d) Does location appear to be a factor in rent? Problem 5 Using the Queens data and results from Problem 3, do the following: (a) Find the least-squares regression line, treating square footage as the explanatory variable. (b) Interpret the slope and y-intercept, if appropriate. (c) Is the rent on the 825-square-foot apartment in the data above or below average among 825-square-foot apartments?

Problem 9AYU dParty AffiliationIs there an association between party affiliation and gender? The following data represent the gender and party affiliation of registered voters based on a random sample of 802 adults Female Male Republican 105 115 Democrat 150 103 Independent 150 179 (a) Construct a frequency marginal distribution. ________________ (b) Construct a relative frequency marginal distribution. ________________ (c) What proportion of registered voters considers themselves to be Independent? ________________ (d) Construct a conditional distribution of party affiliation by gender. ________________ (e) Draw a bar graph of the conditional distribution found in part (d). ________________ (f ) Is gender associated with party affiliation? If so, how?

Shark Attacks. The correlation between the number of visitors to the state of Florida and the number of shark attacks since 1990 is 0.946. Should the number of visitors to Florida be reduced in an attempt to reduce shark attacks? Explain your reasoning.

Feelings on Abortion The Pew Research Center for the People and the Press conducted a poll in which it asked about the availability of abortion. The table is based on the results of the survey (a) Construct a frequency marginal distribution. (b) Construct a relative frequency marginal distribution. (c) What proportion of college graduates feel that abortion should never be permitted? (d) Construct a conditional distribution of people’s feelings about the availability of abortion by level of education. (e) Draw a bar graph of the conditional distribution found in part (d). (f ) Is level of education associated with opinion on the availability of abortion? If so, how?

Problem 10RE New versus Used Car Satisfaction Are you more likely to be satisfied with your automobile purchase when it is new or used? The following data represent the level of satisfaction of the buyer for both new and used cars New Used Total Not too satisfied 11 25 36 Pretty satisfied 78 79 157 Extremely satisfied 118 85 203 Total 207 189 396 (a) How many were extremely satisfied with their automobile purchase? ________________ (b) Construct a relative frequency marginal distribution. ________________ (c) What proportion of consumers was extremely satisfied with their automobile purchase? ________________ (d) Construct a conditional distribution of satisfaction by purchase type (new or used). ________________ (e) Draw a bar graph of the conditional distribution found in part (d). ________________ (f ) Do you think that the purchase type (new versus used) is associated with satisfaction?

Health and Happiness. The General Social Survey asks questions about one’s happiness and health. One would think that health plays a role in one’s happiness. Use the data in the table to determine whether healthier people tend to also be happier. Treat level of health as the explanatory variable.

Unemployment Rates Recessions are an economic phenomenon that are often defined as two consecutive quarters of reduced national output. One measure to assess the severity of a recession is the rate of unemployment. The table shows the number of employed and unemployed residents of the United States at the peak of each recession (in thousands) (a) Determine the unemployment rate for each recession. Which recession appears worse as measured by unemployment rate? Note: Unemployment rate = unemployed/(employed + unemployed). (b) Determine the unemployment rate for each level of education for both recessions. (c) Draw a bar graph of the conditional distribution from part (b). (d) Write a report that suggests the recession of 2009 is worse than that of 1982.

Problem 12AYU Happy in Your Marriage?The General Social Survey asks questions about one’s happiness in marriage. Is there an association between gender and happiness in marriage? Use the data in the table to determine if gender is associated with happiness in marriage. Treat gender as the explanatory variable Male Female Total Very happy 7,609 7,942 15,551 Pretty happy 3,738 4,447 8,185 Not too happy 259 460 719 Total 11,606 12,849 24,455

Problem 12RE (a) The correlation between number of married residents and number of unemployed residents in 2008 for the 50 states and Washington, DC, is 0.922. A scatter diagram of the data drawn in MINITAB is shown. What type of relation appears to exist between number of marriages and number unemployed? ________________ (b) A lurking variable is in the relation presented in part (a). Use the following correlation matrix to explain how population is a lurking variable. ________________ (c) The correlation between unemployment rate (number unemployed divided by population size) and marriage rate (number married divided by population size) in 2008 for the 50 states and Washington, DC, is 0.050. A scatter diagram between unemployment rate and marriage rate drawn in MINITAB is shown next. What type of relation appears to exist between unemployment rate and marriage rate? ________________ (d) Write a few sentences to explain the danger in using correlation to conclude that a relation exists between two variables without considering lurking variables.

Problem 13AYU Smoking Is Healthy?Could it be that smoking actually increases survival rates among women? The following data represent the 20-year survival status and smoking status of 1314 English women who participated in a cohort study from 1972 to 1992 Smoking Status Smoker (S) Nonsmoker (NS) Total Dead 139 230 369 Alive 443c 502 945 Total 582 732 1314 (a) What proportion of the smokers was dead after 20 years? What proportion of the nonsmokers was dead after 20 years? What does this imply about the health consequences of smoking?The data in the table above do not take into account a variable that is strongly related to survival status, age. The data shown next give the survival status of women and their age at the beginning of the study. For example, 14 women who were 35 to 44 at the beginning of the study were smokers and dead after 20 years 18–24 25–34 35–44 45–54 55–64 65–74 75 or older S NS S NS S NS S NS S NS S NS S NS Dead 2 1 3 15 14 7 27 12 51 40 29 101 13 64 Alive 53 61 121 152 95 114 103 66 64 81 7 28 0 0 (b) Determine the proportion of 18- to 24-year-old smokers who were dead after 20 years. Determine the proportion of 18- to 24-yearold nonsmokers who were dead after 20 years. ________________ (c) Repeat part (b) for the remaining age groups to create a conditional distribution of survival status by smoking status for each age group. ________________ (d) Draw a bar graph of the conditional distribution from part (c). ________________ (e) Write a short report detailing your findings.

List the eight properties of the linear correlation coefficient.

Problem 14AYU Treating Kidney Stones Researchers conducted a study to determine which of two treatments, A or B, is more effective in the treatment of kidney stones. The results of their experiment are given in the table Treatment A Treatment B Total Effective 273 289 562 Not effective 77 61 138 Total 350 350 700 (a) Which treatment appears to be more effective? Why? The data in the table above do not take into account the size of the kidney stone. The data shown next indicate the effectiveness of each treatment for both large and small kidney stones Small Stones Large Stones A B A B Effective 81 234 273 55 Not effective 6 36 77 25 (b) Determine the proportion of small kidney stones that were effectively dealt with using treatment A. Determine the proportion of small kidney stones that were effectively dealt with using treatment B. ________________ (c) Repeat part (b) for the large stones to create a conditional distribution of effectiveness by treatment for each stone size. ________________ (d) Draw a bar graph of the conditional distribution from part (c). ________________ (e) Write a short report detailing your findings

Analyzing a Newspaper Article In a newspaper article written in the Chicago Tribune on September 29, 2002, it was claimed that poorer school districts have shorter school days. (a) The following scatter diagram was drawn using the data supplied in the article. In this scatter diagram, the response variable is length of the school day (in hours) and the explanatory variable is percent of the population that is low income. The correlation between length and income is ? 0.461. Do you think that the scatter diagram and correlation coefficient support the position of the article? (b) The least-squares regression line between length, y, and income, x, is \(\hat{y}=-0.0102 x+7.11\). Interpret the slope of this regression line. Does it make sense to interpret the y-intercept? If so, interpret the y-intercept. (c) Predict the length of the school day for a district in which 20% of the population is low-income by letting x = 20. (d) This same article included average Prairie State Achievement Examination (PSAE) scores for each district. The article implied that shorter school days result in lower PSAE scores. The correlation between PSAE score and length of school day is 0.517. A scatter diagram treating PSAE as the response variable is shown next. Do you believe that a longer school day is positively associated with a higher PSAE score? (e)The correlation between percentage of the population that is low income and PSAE score is -0.720. A scatter diagram treating PSAE score as the response variable is shown next. Do you believe that percentage of the population that is low is negativelv associated with PSAE score? (f) Can you think of any lurking variables that are playing a role in this study?