Lesson 6.2.3 HW: 6-85 to 6-90 Learning Target: Scholars will understand that cause and effect cannot be determined from a study that reports an association. A study found that the more hours students spent on activities outside school, the higher their grades tended to be. Does that mean if you go sign up for more activities, your grades will go up? Another study found a link between how often you brush your teeth and a reduction in heart disease. Does that mean if you brush your teeth twice a day, your heart will be healthier? As a consumer of statistical information, you need to be aware of the difference between association and causation. Today’s investigation will explore that difference. 6-79. Fire hoses come in different diameters. How far the hose can throw water depends on the diameter of the hose. The Smallville Fire Department collected data about their fire hoses. The residual plot for the data is shown at right. 1. What does the residual plot tell you about the LSRL model the fire department used? 2. Find the worst prediction made with the LSRL. How different was the worst prediction from what was actually observed? Explain why in context. 3. Make a conjecture about what the original scatterplot might have looked like and sketch it. Label both axes. 6-80. The mayor of Smallville finds the following graph in the town’s annual financial report. 1. Describe the association, if any, in the scatterplot. 2. The mayor immediately orders the fire department to send fewer firefighters to each fire so that there is less damage. Why do you think the mayor said this? Do you agree with the mayor’s decision? Explain why or why not. 6-81. A dietician studying the benefits of eating spinach surveyed a large sample of individuals. She recorded the amount of spinach they ate and their physical strength. The dietician found the spinach eaters to be much stronger than the non-spinach eaters. The next day the newspaper headline was, “Popeye was right! Eating spinach makes you stronger!” 1. Do you agree with the newspaper? Do you agree that if you eat more spinach, you will grow stronger muscles and increase your strength? 2. The dietician correctly found an association. What could explain this association other than spinach makes you stronger? 6-82. A lurking variable is a hidden variable that was not part of the study. The size of the fire in problem 6-80, and the amount people work out in problem 6-81, are lurking variables. A medical study found a strong link between the numbers of hours high school students wear a helmet and the number of concussions (head injuries). However, it is unlikely that wearing helmets causes head injuries. Can you think of a lurking variable that might explain this association? 6-83. A web of associated variables can get complex and be difficult to unravel. Consider a medical study focused on hearing loss. It may associate variables like eating prunes to hearing loss as strongly as it associates an actual cause like long-term noise exposure to hearing loss. Here are some newspaper headlines from actual observational studies. Each of them found an association. Some even imply a cause and effect relationship. Determine at least one plausible lurking variable that could explain the actual cause. 1. 2. 3. 4. 5. “Calcium in diet may cut risk for some cancers, study finds” “Study: Family time declines as Web use booms” “Chocolate is linked to depression” “Study: Kids who were spanked have lower IQs” “Lack of Health Insurance Kills 45,000 a Year” 6-85. A human resources manager recorded the experience and hourly wage for a sample of 10 technology workers. 1. Sketch a scatterplot showing the association between the wage and the years of experience. Describe the association. 2. Sketch the residual plot. Is a linear model appropriate? 3. What is the correlation coefficient? What does it tell you? 6-86. Marissa went with her friends to the amusement park on a beautiful spring day. The park was crowded. Marissa wondered if there was an association between the weather and attendance. From data she received at the theme park office, Marissa randomly picked ten Saturdays and analyzed the data. 4. Marissa calculated the least squares regression line a = –14 + 0.41t, where a is the attendance (in 1000s) and t is the high temperature (°F) that day. Interpret the slope in this context. 5. The residual plot Marissa created is shown at right. On days when temperatures were in the 80s, would you expect the predictions made by Marissa’s model to be too high, too low, or pretty accurate? 6. What was the actual attendance on the day when the temperature was 95°F? 7. Marissa drew the upper boundary line at a = –7 + 0.41t and the lower boundary line at a = –21 + 0.41t. What are the upper and lower bounds for the predicted number of people attending when the temperature is 80°F? 8. Would you rely on this model to make predictions? Why or why not? 6-87. Here is an alternative way to define a recursive sequence. a1 = 5 an = an−1 + 6 Therefore: a1 = 5 a2 = a1 + 6= 5 + 6 = 11 a3 = a2 + 6 = 11 + 6 = 17 Now continue the sequence: 1. a4 = a? + ? = ? 2. a5 = a? + ? = ? 3. The first 5 terms of the sequence are: 6-88. Multiply each pair of polynomials. 4. (2a + b)(a − 3b) 5. (x + 2)(x2 − 2x + 5) 6-89. Given the sequence 2, 10, 50, 250, … complete parts (a) through (c) below. 6. What kind of sequence is it? 7. Describe the shape of the graph. 8. Give an explicit equation for the sequence. 6-90. Solve each equation. 9. 3x + 2 = 10 − 4(x − 1) 10. 4(x − 1) − 2(3x + 5) = −3x + 1 Lesson 6.2.3 6-79. See below: 1. A line may not have been the best way to model this data. The residual plot has a definite U-shaped pattern. 2. The worst prediction was for a 3.5" hose. Residual was –24 feet. The model expected the water to be thrown about 24 feet more than it actually was. 3. See sample solution below. 6-80. See below: 1. There is a positive linear association of moderate strength with no outliers. 2. The mayor said this because the more firefighters, the more damage at the fire. I do not agree with the mayor’s decision. The reason there are more firefighters—and more damage—is probably because those fires are bigger; the firefighters are not causing the damage, the size of the blaze is. 6-81. See below: 1. There could be other variables that would cause someone who eats spinach to be physically strong. 2. Sample answer: If you are health conscious, you eat healthy things, but you also work out. It could be the working out that is causing the increased strength. 6-82. The number of hours you play in a contact sport; the more hours you play in a contact sport means more chance of a concussion, but also more hours that you wear a helmet. 6-83. See below: 1. It is possible that young people have more calcium in their diet and young people get less cancer. Also, the kind of person that has more calcium in their diet may be more health conscious, get more exercise, and avoid smoking, etc. 2. As times change so do many variables that may or may not be connected. The world is becoming more technologically oriented. However, fast food consumption is also booming. Perhaps it is causing fewer families to eat together and so they spend less time together. Maybe families with lots of activities use more technology and have less time to spend together so the activities are the cause. 3. It is more likely that depressed people are eating chocolate than chocolate is causing depression. Chocolate is in many foods that can cause weight gain. Perhaps weight gain is a significant cause of depression. 4. Kids who are prone to misbehavior may get spanked more than more compliant children and have a more difficult time learning and testing in school environments. Spanking may have cultural or economic links that are also connected with the amount of emphasis or value placed on education in the home. 5. People who live in poverty or are unemployed cannot afford health insurance and also cannot afford the good nutrition and healthcare which leads to less risk of dying younger. It may be that people without heath insurance work (without benefits) in more dangerous jobs. 6-85. See below: 1. A very strong positive linear association with no outliers. See graph below. 2. See plot below. Yes, the residual plot shows random scatter with no apparent pattern. 3. r = 0.998 a very strong positive linear association. 6-86. See below: 1. With each additional degree of temperature, we predict an increase of 410 park visitors. 2. The residuals are positive, so we expect the actual values to be greater than the predicted values. The predictions from the model may be too low. 3. The residual is about 17 thousand people; the LSRL predicts 24.95 thousand people. actual –24.95 = –7; the actual number of people in attendance was about 17,900. 4. The predicted attendance is between 11,800 and 25,800 people. 5. The residual plot shows a clear curve; the linear model is not appropriate. For temperatures in the 80s the model predicts too low an attendance; for temperatures in the mid 90s, the model predicts too high an attendance. The range for predicted attendance in part (d) is very large and therefore not useful. 6-87. See below: 1. a4 = a3 + 6 = 23 2. a5 = a4+ 6 = 29 6-88. See below: 1. 2a2 – 5ab – 3b2 2. x3 + x + 10 6-89. See below: 1. geometric 2. curved 3. t(n) = (5)n 6-90. See below: 1. x = 2. x = 15