Ch 4 Homework
The accompanying data represent the weights of various domestic cars and their gas mileages in the city. The linear correlation coefficient between the weight of a car and its miles per gallon in the city is r = -0.976. The least-squares regression line treating weight as the explanatory variable and miles per gallon as the response variable is y = -0.0067x + 43.6850. Complete parts (a) and (b). (a) What proportion of the variability in miles per gallon is explained by the relation between weight of the car and miles per gallon? (b) Interpret the coefficient of determination.
(a) 95.3% (b) 95.3% of the variance in gas mileage is explained by the linear model.
An author of a book discusses how statistics can be used to judge both a baseball player's potential and a team's ability to win games. One aspect of this analysis is that a team's on-base percentage is the best predictor of winning percentage. The on-base percentage is the proportion of time a player reaches a base. For example, an on-base percentage of 0.3 would mean the player safely reaches bases 3 times out of 10, on average. For a certain baseball season, winning percentage, y, and on-base percentage, x, are linearly related by the least-squares regression equation y = 2.92x - 0.4877. (a) Interpret the slope. Choose the correct answer below. (b) For this baseball season, the lowest on-base percentage was 0.320 and the highest on-base percentage was 0.362. Does it make sense to interpret the y-intercept? (c) Would it be a good idea to use this model to predict the winning percentage of a team whose on-base percentage was 0.230? (d) A certain team had an on-base percentage of 0.322 and a winning percentage of 0.548. What is the residual for that team? How would you interpret this residual?
(a) For each percentage point increase in on-base percentage, the winning percentage will increase by 2.92 percentage points, on average. (b) no (c) No, it would be a bad idea. (d) 0.0955 This residual indicates that the winning percentage of the team is above average for teams with an on-base percentage of 0.322.
An engineer wanted to determine how the weight of a car affects gas mileage. The accompanying data represent the weights of various domestic cars and their gas mileages in the city for a certain model year. Suppose that we add Car 12 to the original data. Car 12 weighs 3,305 pounds and gets 19 miles per gallon. Complete parts (a) through (f) below. (a) Draw the scatter diagram with Car 12 included. (b) Compute the linear correlation coefficient with Car 12 included. . (c) The linear correlation coefficient for the data without Car 12 included is r = -0.941. Compare the results of parts (a) and (b) to the scatter diagram and linear correlation coefficient without Car 12 included. Why are the results here reasonable? The absolute value of the correlation coefficient ____________ and the sign of the correlation coefficient ______________. The results here are reasonable because Car 12 ________________. (d) Now suppose that Car 13 (a hybrid car) is added to the original data (remove Car 12). Car 13 weighs 2,890 pounds and gets 60 miles per gallon. Redraw the scatter diagram with Car 13 included. (e) Recompute the linear correlation coefficient with Car 13 included. How did this new value affect your result? The linear correlation coefficient with Car 13 included is r = _____. .The absolute value of the correlation coefficient __________________ and the sign of the correlation coefficient ___________. (f) Why does this observation not follow the pattern of the data?
(a) Graph A (b) r = -0.917 (c) did not change significantly; did not change.; follows the overall pattern of the data (d) Graph A (e) -0.493; became significantly closer to 0, did not change (f) Car 13 is a hybrid car, while the other cars likely are not.
The accompanying data represent the annual rates of return of two companies' stock for the past 12 years. Complete parts (a) through (i). (a) Draw a scatter diagram of the data treating the rate of return of Company 1 as the explanatory variable. Choose the correct graph below. (b) Determine the correlation coefficient between rate of return of Company 1 and Company 2. (c) Based on the scatter diagram and correlation coefficient, is there a linear relation between rate of return of Company 1 and Company 2? (d) Find the least-squares regression line treating the rate of return of Company 1 as the explanatory variable. Choose the correct answer below. (e) Predict the rate of return of Company 2 if the rate of return of Company 1 is 0.15 (15%). (f) If the actual rate of return for Company 2 was 30.2% when the rate of return for Company 1 was 15%, was the performance of Company 2 above or below average among all years the returns of Company 1 were 15%? (g) Interpret the slope. Choose the correct answer below. (h) Interpret the y-intercept. Choose the correct answer below. (i) What proportion of the variability in the rate of return of Company 2 is explained by the variability in the rate of return of Company 1?
(a) Graph B (b) 0.948 (c) Yes (d) y = 1.5343x. + 0.0314 (e) 0.2615 (f) Above Average (g) For each percentage point increase in the rate of return for Company 1, the rate of return of Company 2 will increase by about 1.53 percentage points, on average. (h) The y-intercept indicates that the rate of return for Company 2 will be 0.0314 when there is no change to Company 1. (i) 89.9%
Complete parts (a) through (h) for the data below. (a) By hand, draw a scatter diagram treating x as the explanatory variable and y as the response variable. Choose the correct scatter diagram below. (b) Find the equation of the line containing the points (50,78) and (80,53). (c) Graph the line found in part (b) on the scatter diagram. Choose the correct graph below. (d) By hand, determine the least-squares regression line. (e) Graph the least-squares regression line on the scatter diagram. Choose the correct graph below. (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). Consider the two diagrams to the right. Compare the number of points that each line passes through. The line with the least sum of square residuals is the better fit. Compare the values to determine which line best fits the data.
(a) Graph C (b) y=-5/6x+359/3 (c) Graph A (d) y=-0.700x+(112.400) (e) Graph D (f) 51.667 (g) 31.200 (h) (b); (d); minimizes
One of the biggest factors in determining the value of a home is the square footage. The accompanying data represent the square footage and selling price (in thousands of dollars) for a random sample of homes for sale in a certain region. Complete parts (a) through (h) below. (a) Which variable is the explanatory variable? (b) Draw a scatter diagram of the data. Choose the correct scatter diagram below. (c) Determine the linear correlation coefficient between square footage and asking price. (d) Is there a linear relation between square footage and asking price? (e) Find the least-squares regression line treating square footage as the explanatory variable. (f) Interpret the slope. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. (g) Is it reasonable to interpret the y-intercept? Why? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. (h) One home that is 1495 square feet sold for $220 thousand. Is this home's price above or below average for a home of this size?
(a) Square Footage (b) Graph D (c) r = 0.902 (d) yes (e) y = 0.153x + 22 (f) For every additional square foot, the selling price increases by 0.153 thousand dollars, on average. (g) No -- a house of 0 square feet is not possible and outside the scope of the model. (h) below; 251
Researchers initiated a long-term study of the population of American black bears. One aspect of the study was to develop a model that could be used to predict a bear's weight (since it is not practical to weigh bears in the field). One variable thought to be related to weight is the length of the bear. The accompanying data represent the lengths and weights of 12 American black bears. Complete parts (a) through (d) below. (a) Which variable is the explanatory variable based on the goals of the research? (b) Draw a scatter diagram of the data. Choose the correct graph below. (c) Determine the linear correlation coefficient between weight and length. (d) Does a linear relation exist between the weight of the bear and its length? The variables weight of the bear and length of the bear are _____ associated because r is ______ and the absolute value of the correlation coefficient, ___, is _____ than the critical value, ____.
(a) The length of the bear (b) Graph A (c) r=0.743 (d) positively; positive; 0.743; greater; 0.576
Match the linear correlation coefficient to the scatter diagram. The scales on the x- and y-axis are the same for each scatter diagram. (a) r=-0.049, (b) r=-0.810, (c) r=-1
(a) diagram 3 (b) diagram 1 (c) diagram 2
Match the linear correlation coefficient to the scatter diagram. The scales on the x- and y-axis are the same for each scatter diagram. (a) r=1, (b) r=0.946, (c) r=0.523
(a) diagram 3 (b) diagram 1 (c) diagram 2
An engineer wants to determine how the weight of a car, x, affects gas mileage, y. The following data represent the weights of various cars and their miles per gallon. (a) Find the least-squares regression line treating weight as the explanatory variable and miles per gallon as the response variable. (b) Interpret the slope and intercept, if appropriate. (c) Predict the miles per gallon of car B and compute the residual. Is the miles per gallon of this car above average or below average for cars of this weight? (d) Draw the least-squares regression line on the scatter diagram of the data and label the residual.
(a) y = -0.00562x + 40.1 (b) The slope indicates the mean change in miles per gallon for an increase of 1 pound in weight. It is not appropriate to interpret the y-intercept because it does not make sense to talk about a car that weighs 0 pounds. (c) The predicted value is 23.55 miles per gallon. The residual is -1.37 miles per gallon. It is below average. (d) Graph C
The data below represent the number of days absent, x, and the final grade, y, for a sample of college students at a large university. Complete parts (a) through (e) below. (a) Find the least-squares regression line treating the number of absences, x, as the explanatory variable and the final grade, y, as the response variable. (b) Interpret the slope and y-intercept, if appropriate. (c) Predict the final grade for a student who misses five class periods and compute the residual. Is the observed final grade above or below average for this number of absences? (d) Draw the least-squares regression line on the scatter diagram of the data. Choose the correct graph below. (e) Would it be reasonable to use the least-squares regression line to predict the final grade for a student who has missed 15 class periods? Why or why not?
(a) y = -3.121x + 88.133 (b) For every day absent, the final grade falls by 3.121, on average. For zero days absent, the final score is predicted to be 88.133. (c) The predicted final grade is 72.5. This observation has a residual of -0.1, which indicates the final grade is below average. (d) Graph A (e) No -- 15 missed class periods is outside the scope of the model.
The following data represent the time between eruptions and the length of eruption for 8 randomly selected geyser eruptions. The coefficient of determination is found to be 80.3%. Interpret this result.
80.3% of the variation in length of eruption is explained by the least-squares regression equation.
What is a residual? What does it mean when a residual is positive?
A residual is the difference between an observed value of the response variable y and the predicted value of y. If it is positive, then the observed value is greater than the predicted value.
What does it mean if r = 0?
No linear relationship exists between the variables.
Determine whether the scatter diagram indicates that a linear relation may exist between the two variables. If the relation is linear, determine whether it indicates a positive or negative association between the variables. Do the two variables have a linear relationship? Do the two variables have a positive or a negative association?
The data points have a linear relationship because they lie mainly in a straight line. The two variables have a positive association.
What does it mean to say that two variables are positively associated? Negatively associated?
There is a linear relationship between the variables, and whenever the value of one variable increases, the value of the other variable increases. There is a linear relationship between the variables, and whenever the value of one variable increases, the value of the other variable decreases.
Will the following variables have positive correlation, negative correlation, or no correlation? number of children in the household under the age of 3 and expenditures on diapers
positive
