Chapter-Quizzes-28ymha7

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Quiz 3.1A AP Statistics Name: The scatterplot below shows the fuel efficiency (in miles per gallon) and weight (in pounds) of twenty 2009 subcompact cars. Fuel efficiency and Car weight 50 MPG 40 Fuel Efficiency,
30 20 10 2500 3000 3500 4000 4500
Car weight Is there a clear explanatory variable and response variable in this setting? If so, tell which is which. If not, explain why not. Does the scatterplot show a positive association, negative association, or neither? Explain why this makes sense.
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3. How would you describe the form of the relationship?
Which of the following is closest to the correlation between car weight and fuel efficiency for these 20 vehicles? Explain. r = – 0.9 r = – 0.6 r = 0 r = 0.4 5. There is one ―unusual point‖ on the graph. Explain what is ―unusual‖ about this car. What effect would removing the ―unusual point‖ have on the correlation? Justify your answer.
If we converted the car weights to metric tons (1 metric ton ≈ 2,205 pounds). How would the correlation change? Explain.
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Quiz 3.1B AP Statistics Name: Below is a scatterplot relating systolic blood pressure and age for 14 men from 42 to 67 years old. Scatterplot of Systolic Blood Pressure vs Age 210 200 190
180 SBP 170 160
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150 140 130
120 40 45 50 55 60 65 70 AGE Is there a clear explanatory variable and response variable in this setting? If so, tell which is which. If not, explain why not.
Does the scatterplot show a positive association, negative association, or neither? What does this tell you about the relationship between age and systolic blood pressure? 3. How would you describe the form of the relationship?
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Which of the following is closest to the correlation between systolic blood pressure and age for this group of 14 men? Explain. r = 0.9 r = 0.5 r = 0.2 r = 0.2 5. There is one ―unusual point‖ on the graph. Explain what is ―unusual‖ about this subject. What effect would removing the ―unusual point‖ have on the correlation? Justify your answer.
Suppose we rescaled the ages so that they were expressed as number of years above (+) or below (–) 50 years old. That is, suppose we subtract 50 from each value. How would the correlation change? Explain.
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Quiz 3.1C AP Statistics Name: A student wonders if tall women tend to date taller men. She measures herself, her dormitory roommate, and the women in the adjoining rooms; then she measures the next man each woman dates. Here are the data (heights in inches): Women 66 64 66 65 70 65 Men 72 68 70 68 71 65 Is there a clear explanatory variable and response variable in this setting? If so, tell which is which. If not, explain why not. Make a well-labeled scatterplot of these data.
Based on the scatterplot, describe the pattern, if any, in the relationship between the heights of women and the heights of the men they date.
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Use your calculator to find the correlation r between the heights of the men and women. Do the data show any evidence that taller women tend to date taller men? Explain. How would r change if all the men were 6 inches shorter than the heights given in the table?
heights were measured in centimeters rather than inches? (There are 2.54 centimeters in an inch.) Suppose another 70-inch-tall female who dated a 73-in-tall male were added to the data set. How would this influence r ?
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Quiz 3.2A AP Statistics Name: The table and scatterplot below show the relationship between student enrollment (in thousands) and total number of property crimes (burglary and theft) in 2006 for eight colleges and universities in a certain U.S. state. Enrollment (in No. of Property 1000s) ( x ) Crimes ( y ) 16 201 2 6 9 42 10 141 14 138 26 601 21 230 19 294 Scatterplot of burglary vs enrollment 600 2006 500
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crimes, 400 property 300 200 of Number 100 0 -100
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0 5 10 15 20 25 Student enrollment (in 1000s) The equation of the least-squares regression line is y ˆ   112.58 21.83 x , where y ˆ = predicted number of property crimes and x = student enrollment in thousands. Interpret the slope of the least-squares line in the context of the problem. How many crimes would you predict on a campus with enrollment of 14 thousand students? Show your work.
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Find the residual for the campus with 14 thousand students and 138 property crimes. Show your work. Interpret the value of the residua in the context of the problem.
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Use the scatterplot to make a rough sketch of the residual plot for these data. (No calculations are necessary).
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Would the slope of the regression line change if the point (26, 601) were removed from the data set? In what direction? The value of r 2 for these data is 0.801. Interpret this value in the context of the problem. Is the given least-squares regression line a good model for these data? Support your answer with appropriate evidence from your answers above.
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Quiz 3.2B AP Statistics Name: The table and scatterplot below describe the relationship between latitude and average July temperature in the twelve largest U.S. cities. Latitude July City ( x ) Temp ( y ) New York 40 77 Los Angeles 34 74 Chicago 42 75 Houston 29 84 Philadelphia 40 77 Phoenix 33 94 San Diego 32 71 San Antonio 29 85 Dallas 32 86 San Jose 37 70 Detroit 42 74 Indianapolis 39 75 Latitude and summer temperatures for major cities 95
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90 Temp 85 July 80 75 70
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30 32 34 36 38 40 42 Latitude The equation of the least-squares regression line is y ˆ 106.5 0.782 x , where y ˆ = July temperature in Fº and x = latitude. 1. Interpret the slope of the least-squares line in the context of the problem. 2. Predict the average July temperature for a city at a latitude of 42 degrees. Show your work.
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Find the value of the residual for Detroit. Show your work. Interpret the value of the residual in the context of the problem.
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Use the scatterplot to make a rough sketch of the residual plot for these data. (No calculations are necessary). Phoenix has a very large positive residual. How would the slope of the regression line change if it were removed from the data set?
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6. The value of r 2 for these data is 0.277. Interpret this value in the context of the problem. Is the given least-squares regression line a good model for using latitude to predict average July temperature of U.S. cities? Support your answer with appropriate evidence from your answers above.
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Quiz 3.2C AP Statistics Name: Below is some data on the relationship between the price of a certain manufacturer’s flat-panel LCD televisions and the area of the screen. We would like to use these data to predict the price of televisions based on size. Screen Area Price (sq. inches) (dollars) 154 250 207 265 289 330 437 375 584 575 683 650 Scatterplot of Price vs area 700 600
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500 Price 400 300 200 100 200 300 400 500 600 700 area (a) Use your calculator to find the equation of the least-squares regression equation. Write the equation below, defining any variables you use.
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Explain what is meant by ―least squares‖ in the expression ―least-squares regression line.‖ This manufacturer also produces a television with a screen size of 943 square inches. Would it be reasonable to use this equation to predict the price of that television? Explain. Calculate the residual for the television that has a screen area of 437 square inches. What does this number suggest about the cost of this television, relative to the others?
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Alana’s favorite exercise machine is a stair climber. On the ―random‖ setting, it changes speeds at regular intervals, so the total number of simulated ―floors‖ she climbs varies from session to session. She also exercises for different lengths of time each session. She decides to explore the relationship between the number of minutes she works out on the stair climber and the number of floors it tells her that she’s climbed. She records minutes of climbing time and number of floors climbed for six exercise sessions. Computer output and a residual plot from a linear regression analysis of the data are shown below. Predictor Coef SE Coef T P Constant -3.822 5.458 -0.70 0.522 Minutes 5.2150 0.2779 18.76 0.000 S = 2.34720 R-Sq = 98.9% R-Sq(adj) = 98.6% Residuals Versus Minutes 3 2
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Residual 1 0 -1 -2 15.0 17.5 20.0 22.5 25.0 Minutes What is the equation of the least-squares line? Be sure to define any variables you use.
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Is a line an appropriate model for these data? Justify your answer. Interpret the value of s (S = 2.3472) in the context of this problem.
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Chapter 3 Solutions Quiz 3.1A 1 . Car weight is the explanatory variable, Fuel efficiency is response. We expect a car’s weight to be a major factor in determining its fuel efficiency. 2. The association is negative. This makes sense because it takes more fuel to move a heavier car. 3. The form is linear, with the exception of one outlier in the y (fuel efficiency) direction. 4. r = – 0.6. The relationship is negative, but because of the outlier it is not as high as –0.9. 5. This car has remarkably high fuel efficiency— far better than any other car in the data set (In fact, it’s a gas-electric hybrid). 6. Removing this point would make the correlation much closer to –1. 7. This would not change the correlation at all, since the units in which variables are expressed has no impact on the correlation. Quiz 3.1B
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1. Age is the explanatory variable, systolic blood pressure is response. We expect men’s age to be a factor in systolic blood pressure (SBP). SBP does not influence age! 2. The association is positive. This suggests that older men are more likely to suffer from high blood pressure. 3. The form is linear, with the exception of one outlier in the y (systolic blood pressure) direction. 4. r = 0.5—a moderate positive relationship. If not for the outlier, r might be as high as 0.9. 5. This man has unusually high systolic blood pressure—higher than for any other subject —even though he is not among the oldest men. 6. Removing this point would make the correlation much closer to 1. 7. This would not change the correlation at all, since subtracting 50 from each score would not change its distance from the mean. Quiz 3.1C 1. Since the student’s question is, ―Do taller women date taller men?‖ the implication is that the women’s heights explain the heights of their dates. 2 . See graph below. 3. Answers may vary. While the scatterplot appears to show that the relationship between the heights of women and the heights of the men is somewhat positive, it does not appear be a very strong relationship. Whether it is linear or not is difficult to determine with so few data points—especially since there are no women between 66 and 70 inches in height. 4. r = 0.566. Since r is positive, here is some evidence that tall women tend to date taller men. 5. Subtracting the same amount from each y value will not change the correlation, nor would multiplying each height by a constant to convert the heights into centimeters. 6. Adding this point to the data would reinforce the weak positive trend, thereby making the correlation much closer to 1. Scatterplot of Men vs Women 72
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71 70 Men 69
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68 67 66
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65 64 65 66 67 68 69 70 Women © 2011 BFW Publishers The Practice of Statistics, 4/e- Chapter 3 137
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Quiz 3.2A 1. For each 1000-student increase in enrollment, the predicted number of property crimes increases by 21.83. 2. y ˆ   112.58 21.83(14) 193.04 crimes. 3. y y ˆ 138 193.04   55.04 . The actual number of property crimes at this college is 55.04 fewer than the number of crimes predicted by this linear model. 4. See graph below. 5. That point has a high, positive residual, so it tends to ―pull‖ the line toward it. Removing it would reduce the slope. 6. 80.1% of the variation in property crimes can be accounted for by the regression of property crimes on enrollment. 7. Answers will vary and will depend on the appearance residual plot sketched in #4. Some students will suggest that there is enough of a ―U‖ shape in the residual plot to suggest that the linear model is not appropriate. Others will say the lack of pattern in residuals suggests that a linear fit is appropriate. Residuals versus Enrollment 150 100
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Residual 50 0 -50
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-100 0 5 10 15 20 25 Enrollment Quiz 3.2B 1. For each 1-degree increase in latitude, the predicted average July temperature decreases by 0.782 degrees. 2. y ˆ 106.5 0.782 42 ; y ˆ 73.66degrees. 3. y y ˆ 74 0.782 42 106.5 74 73.66 0.34 . The actual average July temperature
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in Detroit is 0.34 degrees higher that the average July temperature predicted by this linear model. 4. See graph below. 5. Since Phoenix’s residual is large and positive, and Phoenix is at a relatively low latitude, the slope of the line would increase (that is, get closer to 0). 6. 28% of the variability in average July temperature can be accounted for by the regression of average July temperature on latitude. 7. Answers will vary and will depend on the appearance residual plot sketched in #4. Some students will say that there is no distinctive pattern in the residuals, so the linear model is a good fit. Others may argue that the variability is much larger for smaller values of latitude than for higher values of latitude and therefore this model is not appropriate. Residuals Versus Latitude 15 10
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Residual 5 0 -5 -10
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30 32 34 36 38 40 42 Latitude 138 The Practice of Statistics, 4/e- Chapter 3 © 2011 BFW Publishers
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Quiz 3.2C 1. (a) y ˆ 105.74 0.769 x ; y ˆ = predicted price, x = screen area. (b) The least-squares regression line is the line that minimizes the sum of the squared deviations between observed prices and prices predicted by the linear model. (c) 943 sq. in. is well beyond the range of screen areas used to produce the regression line, so this would be extrapolation. We cannot be sure that the relationship described by this line holds outside the range of available data. (d) y y ˆ 375 (.769(437) 105.74) 375 441.79   66.79 . Since the residual is negative, the observed value is lower than the value predicted by the regression. This suggests that this particular television is a good buy! 2. (a) y ˆ   3.822 5.215 x ; x = minutes of exercise, y ˆ = predicted number of floors climbed. (b) Since there is no distinctive pattern in the residuals, the linear model is a good fit. (c) On average, the predicted values will be about 2.35 floors from the actual values
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