Correlation facts II If we assume that the conditions for

Association Suppose you were to collect data for each pair of variables. You want to make a scatterplot. Which variable would you use as the explanatory variable and which as the response variable? Why? What would you expect to see in the scatterplot? Discuss the likely direction, form, and strength. a) Apples: weight in grams, weight in ounces b) Apples: circumference (inches), weight (ounces) c) College freshmen: shoe size, grade point average d) Gasoline: number of miles you drove since filling up, gallons remaining in your tank

Association II Suppose you were to collect data for each pair of variables. You want to make a scatterplot. Which variable would you use as the explanatory variable and which as the response variable? Why? What would you expect to see in the scatterplot? Discuss the likely direction, form, and strength. a) T-shirts at a store: price each, number sold b) Scuba diving: depth, water pressure c) Scuba diving: depth, visibility d) All elementary school students: weight, score on a reading test

Problem 3E Bookstore sales Consider the following data from a small bookstore. a) Prepare a scatterplot of Sales against Number of sales people working. b) What can you say about the direction of the association? c) What can you say about the form of the relationship? d) What can you say about the strength of the relationship? e) Does the scatterplot show any outliers?

Problem 4E Disk drives 2014 Disk drives have been getting larger. Their capacity is now often given in terabytes (TB) where 1 TB = 1000 gigabytes, or about a trillion bytes. A search of prices for external disk drives on Amazon.com in early 2014 found the following data: a) Prepare a scatterplot of Price against Capacity. b) What can you say about the direction of the association? c) What can you say about the form of the relationship? d) What can you say about the strength of the relationship? e) Does the scatterplot show any outliers?

Correlation facts If we assume that the conditions for correlation are met, which of the following are true? If false, explain briefly. a) A correlation of -0.98 indicates a strong, negative association. b) Multiplying every value of x by 2 will double the correlation. c) The units of the correlation are the same as the units of y.

Correlation facts II If we assume that the conditions for correlation are met, which of the following are true? If false, explain briefly. a) A correlation of 0.02 indicates a strong positive association. b) Standardizing the variables will make the correlation 0. c) Adding an outlier can dramatically change the correlation.

Problem 7E Bookstore sales again A larger firm is considering acquiring the bookstore of Exercise. An analyst for the firm, noting the relationship seen in Exercise, suggests that when they acquire the store they should hire more people because that will drive higher sales. Is his conclusion justified? What alternative explanations can you offer? Use appropriate statistics terminology. Exercise Bookstore sales Consider the following data from a small bookstore. Number of Sales People Working Sales (in $1000) 2 10 3 11 7 13 9 14 10 18 10 20 12 20 15 22 16 22 20 26 = 10.4 = 17.6 SD(x) = 5.64 SD(y) = 5.34 a) Prepare a scatterplot of Sales against Number of sales people working. ________________ b) What can you say about the direction of the association? ________________ c) What can you say about the form of the relationship? ________________ d) What can you say about the strength of the relationship? ________________ e) Does the scatterplot show any outliers?

Problem 8E Blizzards A study finds that during blizzards, online sales are highly associated with the number of snow plows on the road; the more plows, the more online purchases. The director of an association of online merchants suggests that the organization should encourage municipalities to send out more plows whenever it snows because, he says, that will increase business. Comment.

Problem 9E Salaries and logs For an analysis of the salaries of your company, you plot the salaries of all employees against the number of years they have worked for the company. You find that plotting the base-10 logarithm of salary makes the plot much straighter. A part-time shipping clerk, who has worked at the company for one year earns $10,000. A manager earns $100,000 after 15 years with the firm. The CEO, who founded the company 30 years ago, receives $1,000,000. What are the values you will plot? Will the plot of these three points be straight enough?

Problem 10E Dexterity scores Scores on a test of dexterity are recorded by timing a subject who is inverting pegs by picking them up with one hand, manipulating them to turn them over, and then placing them back in a frame. A typical 4-year-old needs about 2.5 seconds to invert a peg, but a 9-year-old takes 2 seconds and a 12-year-old can do it in 1.5 seconds. Plot these three points. Now try the reciprocal re-expression (1/y) vs. age. Which version is straighter?

Problem 11E Association III Suppose you were to collect data for each pair of variables. You want to make a scatterplot. Which variable would you use as the explanatory variable and which as the response variable? Why? What would you expect to see in the scatterplot? Discuss the likely direction, form, and strength. a) When climbing mountains: altitude, temperature b) For each week: ice cream cone sales, air-conditioner sales c) People: age, grip strength d) Drivers: blood alcohol level, reaction time

Problem 15E Performance IQ scores vs. brain size A study examined brain size (measured as pixels counted in a digitized magnetic resonance image [MRI] of a cross section of the brain) and IQ (4 Performance scales of the Weschler IQ test) for college students. The scatterplot shows the Performance IQ scores vs. the brain size. Comment on the association between brain size and IQ.

Problem 13E Scatterplots Which of the scatterplots below show a) little or no association? ________________ b) a negative association? ________________ c) a linear association? ________________ d) a moderately strong association? ________________ e) a very strong association? (1) (2) (3) (4)

Scatterplots II Which of these scatterplots show a) little or no association? b) a negative association? c) a linear association? d) a moderately strong association? e) a very strong association?

Kentucky Derby 2014 The fastest horse in Kentucky Derby history was Secretariat in 1973. The scatterplot shows speed (in miles per hour) of the winning horses each year. What do you see? In most sporting events, performances have improved and continue to improve, so surely we anticipate a positive direction. But what of the form? Has the performance increased at the same rate throughout the past 140 years?

Firing pottery A ceramics factory can fire eight large batches of pottery a day. Sometimes a few of the pieces break in the process. In order to understand the problem better, the factory records the number of broken pieces in each batch for 3 days and then creates the scatterplot shown. a) Make a histogram showing the distribution of the number of broken pieces in the 24 batches of pottery examined. b) Describe the distribution as shown in the histogram. What feature of the problem is more apparent in the histogram than in the scatterplot? c) What aspect of the company’s problem is more apparent in the scatterplot?

Problem 12E Association IV Suppose you were to collect data for each pair of variables. You want to make a scatterplot. Which variable would you use as the explanatory variable and which as the response variable? Why? What would you expect to see in the scatterplot? Discuss the likely direction, form, and strength. a) Legal consultation time, cost b) Lightning strikes: distance from lightning, time delay of the thunder c) A streetlight: its apparent brightness, your distance from it d) Cars: weight of car, age of owner

Matching Here are several scatterplots. The calculated correlations are -0.923, -0.487, 0.006, and 0.777. Which is which?

Problem 21E Politics A candidate for office claims that “there is a correlation between television watching and crime.” Criticize this statement on statistical grounds.

Problem 22E Car thefts The National Insurance Crime Bureau reports that Honda Accords, Honda Civics, and Toyota Camrys are the cars most frequently reported stolen, while Ford Tauruses, Pontiac Vibes, and Buick LeSabres are stolen least often. Is it reasonable to say that there’s a correlation between the type of car you own and the risk that it will be stolen?

Problem 20E Matching Here and on the next page are several scatterplots. The calculated correlations are -0.977, -0.021, 0.736, and 0.951. Which is which? (a) (b) (c) (d)

Problem 23E Roller coasters 2014 Most roller coasters get their speed by dropping down a steep initial incline, so it makes sense that the height of that drop might be related to the speed of the coaster. Here’s a scatterplot of top Speed and largest Drop for 107 roller coasters around the world. a) Does the scatterplot indicate that it is appropriate to calculate the correlation? Explain. b) In fact, the correlation of Speed and Drop is 0.937. Describe the association.

Problem 18E Coffee sales Owners of a new coffee shop tracked sales for the first 20 days and displayed the data in a scatterplot (by day). a) Make a histogram of the daily sales since the shop has been in business. ________________ b) State one fact that is obvious from the scatterplot, but not from the histogram. ________________ c) State one fact that is obvious from the histogram, but not from the scatterplot.

Antidepressants A study compared the effectiveness of several antidepressants by examining the experiments in which they had passed the FDA requirements. Each of those experiments compared the active drug with a placebo, an inert pill given to some of the subjects. In each experiment some patients treated with the placebo had improved, a phenomenon called the placebo effect. Patients’ depression levels were evaluated on the Hamilton Depression Rating Scale, where larger numbers indicate greater improvement. (The Hamilton scale is a widely accepted standard that was used in each of the independently run studies.) The scatterplot at the top of the next column compares mean improvement levels for the antidepressants and placebos for several experiments. a) Is it appropriate to calculate the correlation? Explain. b) The correlation is 0.898. Explain what we have learned about the results of these experiments.

Streams and hard water In a study of streams in the Adirondack Mountains, the following relationship was found between the water’s pH and its hardness (measured in grains): Is it appropriate to summarize the strength of association with a correlation? Explain.

Cold nights Is there an association between time of year and the nighttime temperature in North Dakota? A researcher assigned the numbers 1–365 to the days January 1–December 31 and recorded the temperature at 2:00 a.m. for each. What might you expect the correlation between DayNumber and Temperature to be? Explain.

Association V A researcher investigating the association between two variables collected some data and was surprised when he calculated the correlation. He had expected to find a fairly strong association, yet the correlation was near 0. Discouraged, he didn’t bother making a scatterplot. Explain to him how the scatterplot could still reveal the strong association he anticipated.

Traffic headaches A study of traffic delays in 68 U.S. cities found the following relationship between Total Delays (in total hours lost) and Mean Highway Speed: Is it appropriate to summarize the strength of association with a correlation? Explain.

Problem 30E More predictions Hurricane Katrina’s hurricane force winds extended 120 miles from its center. Katrina was a big storm, and that affects how we think about the prediction errors. Suppose we add 120 miles to each error to get an idea of how far from the predicted track we might still find damaging winds. Explain what would happen to the correlation between Prediction Error and Year, and why.

Problem 31E Correlation errors Your Economics instructor assigns your class to investigate factors associated with the gross domestic product (GDP) of nations. Each student examines a different factor (such as Life Expectancy, Literacy Rate, etc.) for a few countries and reports to the class. Apparently, some of your classmates do not understand Statistics very well because you know several of their conclusions are incorrect. Explain the mistakes in their statements below. a) “My very low correlation of -0.772 shows that there is almost no association between GDP and Infant Mortality Rate.” ________________ b) “There was a correlation of 0.44 between GDP and Continent.”

Problem 29E Prediction units The errors in predicting hurricane tracks (examined in this chapter) were given in nautical miles. An ordinary mile is 0.86898 nautical miles. Most people living on the Gulf Coast of the United States would prefer to know the prediction errors in miles rather than nautical miles. Explain why converting the errors to miles would not change the correlation between Prediction Error and Year.

Problem 33E Height and reading A researcher studies children in elementary school and finds a strong positive linear association between height and reading scores. a) Does this mean that taller children are generally better readers? ________________ b) What might explain the strong correlation?

Problem 34E Smart phones and life expectancy A survey of the world’s nations in 2014 shows a strong positive correlation between percentage of the country using smart phones and life expectancy in years at birth. a) Does this mean that smart phones are good for your health? b) What might explain the strong correlation?

Problem 36E Correlation conclusions II The correlation between Fuel Efficiency (as measured by miles per gallon) and Price of 150 cars at a large dealership is r = -0.34. Explain whether or not each of these possible conclusions is justified: a) The more you pay, the lower the fuel efficiency of your car will be. ________________ b) The form of the relationship between Fuel Efficiency and Price is moderately straight. ________________ c) There are several outliers that explain the low correlation. ________________ d) If we measure Fuel Efficiency in kilometers per liter instead of miles per gallon, the correlation will increase.

Problem 35E Correlation conclusions I The correlation between Age and Income as measured on 100 people is r = 0.75. Explain whether or not each of these possible conclusions is justified: a) When Age increases, Income increases as well. ________________ b) The form of the relationship between Age and Income is straight. ________________ c) There are no outliers in the scatterplot of Income vs. Age. ________________ d) Whether we measure Age in years or months, the correlation will still be 0.75.

Problem 37E Baldness and heart disease Medical researchers followed 1435 middle-aged men for a period of 5 years, measuring the amount of Baldness present (none = 1, little = 2, some = 3, much = 4, extreme = 5) and presence of Heart Disease (No = 0, Yes = 1). They found a correlation of 0.089 between the two variables. Comment on their conclusion that this shows that baldness is not a possible cause of heart disease.

Problem 32E More correlation errors Students in the Economics class discussed in Exercise also wrote these conclusions. Explain the mistakes they made. a) “There was a very strong correlation of 1.22 between Life Expectancy and GDP.” ________________ b) “The correlation between Literacy Rate and GDP was 0.83. This shows that countries wanting to increase their standard of living should invest heavily in education.” Exercise Correlation errors Your Economics instructor assigns your class to investigate factors associated with the gross domestic product (GDP) of nations. Each student examines a different factor (such as Life Expectancy, Literacy Rate, etc.) for a few countries and reports to the class. Apparently, some of your classmates do not understand Statistics very well because you know several of their conclusions are incorrect. Explain the mistakes in their statements below. a) “My very low correlation of -0.772 shows that there is almost no association between GDP and Infant Mortality Rate.” ________________ b) “There was a correlation of 0.44 between GDP and Continent.”

Problem 38E Sample survey A polling organization is checking its database to see if the two data sources it used sampled the same zip codes. The variable Datasource = 1 if the data source is MetroMedia, 2 if the data source is DataQwest, and 3 if it’s RollingPoll. The organization finds that the correlation between five-digit zip code and Datasource is -0.0229. It concludes that the correlation is low enough to state that there is no dependency between Zip Code and Source of Data. Comment.

Problem 39E Income and housing The Office of Federal Housing Enterprise Oversight (www.fhfa.gov) collects data on various aspects of housing costs around the United States. Here is a scatterplot of the Housing Cost Index versus the Median Family Income for each of the 50 states. The correlation is 0.65. a) Describe the relationship between the Housing Cost Index and the Median Family Income by state. b) If we standardized both variables, what would the correlation coefficient between the standardized variables be? c) If we had measured Median Family Income in thousands of dollars instead of dollars, how would the correlation change? d) Washington, DC, has a housing cost index of 548 and a median income of about $45,000. If we were to include DC in the data set, how would that affect the correlation coefficient? e) Do these data provide proof that by raising the median family income in a state, the housing cost index will rise as a result? Explain. *f) For these data Kendall’s tau is 0.51. Does that provide proof that by raising the median income in a state, the Housing Cost Index will rise as a result? Explain what Kendall’s tau says and does not say.

Drug abuse A survey was conducted in the United States and 10 countries of Western Europe to determine the percentage of teenagers who had used marijuana and other drugs. The results are summarized in the table. a) Create a scatterplot. b) What is the correlation between the percent of teens who have used marijuana and the percent who have used other drugs? c) Write a brief description of the association. d) Do these results confirm that marijuana is a “gateway drug,” that is, that marijuana use leads to the use of other drugs? Explain.

Burgers II In the previous exercise you analyzed the association between the amounts of fat and sodium in fast food hamburgers. What about fat and calories? Here are data for the same burgers: a) Analyze the association between fat content and calories using correlation and scatterplots. *b) Repeat your analysis using Spearman’s rho. Explain any differences you see.

Problem 45E Attendance 2013 American League baseball games are played under the designated hitter rule, meaning that pitchers, often weak hitters, do not come to bat. Baseball owners believe that the designated hitter rule means more runs scored, which in turn means higher attendance. Is there evidence that more fans attend games if the teams score more runs? Data collected from American League games during the 2013 season indicate a correlation of 0.384 between runs scored and the number of people at the game. (espn.go.com/mlb/attendance) a) Does the scatterplot indicate that it’s appropriate to calculate a correlation? Explain. b) Describe the association between attendance and runs scored. c) Does this association prove that the owners are right that more fans will come to games if the teams score more runs?

Problem 40E Interest rates and mortgages 2013 Since 1985, average mortgage interest rates have fluctuated from a low of nearly 3% to a high of over 14%. Is there a relationship between the amount of money people borrow and the interest rate that’s offered? Here is a scatterplot of Mortgage Loan Amount in the United States (in trillions of 2013 dollars) versus yearly Interest Rate since 1985. The correlation is -0.80. a) Describe the relationship between Mortgage Loan Amount and Interest Rate. b) If we standardized both variables, what would the correlation coefficient between the standardized variables be? c) If we were to measure Mortgage Loan Amount in billions of dollars instead of trillions of dollars, how would the correlation coefficient change? d) Suppose that next year, interest rates were 11% and mortgages totaled $60 trillion. How would including that year with these data affect the correlation coefficient? e) Do these data provide proof that if mortgage rates are lowered, people will take out larger mortgages? Explain. *f) For these data Kendall’s tau is -0.65. Does that provide proof that if mortgage rates are lowered, people will take out more mortgages? Explain what Kendall’s tau says and does not say.

Burgers Fast food is often considered unhealthy because much of it is high in both fat and sodium. But are the two related? Here are the fat and sodium contents of several brands of burgers. a) Analyze the association between fat content and sodium using correlation and scatterplots. *b) Find Spearman’s rho for these data. Compare it with the Pearson correlation. Comment.

Problem 41E Fuel economy 2014 Here are advertised engine size (in liters) and gas mileage (estimated combined city and highway) for several 2014 vehicles. (www.kbb.com) a) Make a scatterplot for these data. b) Describe the direction, form, and strength of the plot. c) Find the correlation between engine size and miles per gallon. d) Write a few sentences telling what the plot says about fuel economy.

Problem 46E Second inning 2013 Perhaps fans are just more interested in teams that win. The displays below are based on American League teams for the 2013 season. (espn.go.com) (Use the data set Attendance_2013). a) Do winning teams generally enjoy greater attendance at their home games? Describe the association. b) Is attendance more strongly associated with winning or scoring runs? Explain. c) How strongly is scoring runs associated with winning games?

Thrills Since 1994, the Best Roller Coaster Poll (www.ushsho.com/bestrollercoasterpoll.htm) has been ranking the world’s best roller coasters. In 2011, Bizarro earned the top steel coaster rank for the sixth straight year. Here are data on the top 10 steel coasters from this poll: What do these data indicate about the Length of the track and the Duration of the ride you can expect?

Problem 48E Thrills 2011 Since 1994, the Best Roller Coaster Poll (www.ushsho.com/bestrollercoasterpoll.htm) has been ranking the world’s best roller coasters. In 2011, Bizarro earned the top steel coaster rank for the sixth straight year. Here are data on the top 10 steel coasters from this poll: What do these data indicate about the Length of the track and the Duration of the ride you can expect?

Problem 52E Flights 2012 Here are the number of domestic flights flown in each year from 2000 to 2012 (www.TranStats .bts.gov/Data_Elements.aspx?Data=2). a) Find the correlation of Flights with Year. b) Make a scatterplot and describe the trend. c) Why is the correlation you found in part a not a suitable summary of the strength of the association? d) Find Kendall’s tau for these data. Would that be an appropriate measure for summarizing their relationship? Explain.

Problem 49E Thrills II For the roller coaster data in Exercise: a) Examine the relationship between Initial Drop and Speed. ________________ b) Examine the relationship between Initial Drop and Height. ________________ c) What conclusions can you safely draw about the initial drop of a roller coaster? Is Initial Drop strongly correlated with other variables as well? Exercise Thrills 2011 Since 1994, the Best Roller Coaster Poll (www.ushsho.com/bestrollercoasterpoll.htm) has been ranking the world’s best roller coasters. In 2011, Bizarro earned the top steel coaster rank for the sixth straight year. Here are data on the top 10 steel coasters from this poll: What do these data indicate about the Length of the track and the Duration of the ride you can expect?

Problem 51E Planets (more or less) On August 24, 2006, the International Astronomical Union voted that Pluto is not a planet. Some members of the public have been reluctant to accept that decision. Let’s look at some of the data. Is there any pattern to the locations of the planets? The table shows the average distance of each of the traditional nine planets from the sun. a) Make a scatterplot and describe the association. (Remember: direction, form, and strength!) b) Why would you not want to talk about the correlation between a planet’s Position Number and Distance from the sun? c) Make a scatterplot showing the logarithm of Distance vs. Position Number. What is better about this scatterplot? d) Looking only at the scatterplot, what is Kendall’s tau for these data? Explain.

Vehicle weights The Minnesota Department of Transportation hoped that they could measure the weights of big trucks without actually stopping the vehicles by using a newly developed “weight-in-motion” scale. To see if the new device was accurate, they conducted a calibration test. They weighed several stopped trucks (Static Weight) and assumed that this weight was correct. * Then they weighed the trucks again while they were moving to see how well the new scale could estimate the actual weight. Their data are given in the table at the top of the next column. a) Make a scatterplot for these data. b) Describe the direction, form, and strength of the plot. c) Write a few sentences telling what the plot says about the data. (Note: The sentences should be about weighing trucks, not about scatterplots.) d) Find the correlation. e) If the trucks were weighed in kilograms, how would this change the correlation? (1 kilogram = 22 pounds) f) Do any points deviate from the overall pattern? What does the plot say about a possible recalibration of the weight-in-motion scale?