Chapter 5 EPS Due 10/30/15 42 Points A regression line is a ________ line that describes how a __________ variable y changes as an ____________ variable x changes. You can use a regression line to predict the value of y for any value of x by substituting this x into the equation of the line. The slope b of a regression line = a + bx is the ________ at which the predicted response _____ changes along the line as the explanatory variable x changes. Specifically, b is the change in ______ when x increases by 1.
The intercept a of a regression line = a + bx is the predicted response when the explanatory variable x=______. This prediction is of no statistical interest unless x can actually take values near 0. The most common method of fitting a line to a scatterplot is ___________. The leastsquares regression line is the straight line = a + bx that ___________ the sum of the squares of the vertical distances of the observed points from the line. The least-squares regression line of y on x is the line with slope b = _________ and intercept a = _________ . This line always passes through the point (____,____ ). Correlation and regression are closely connected. The correlation _____ is the slope of the leastsquares regression line when we measure both x and y in standardized units. The square of the
correlation ______ is the fraction of the variation in one variable that is explained by least-squares regression on the other variable. Correlation and regression must be interpreted with caution. Plot the data to be sure the relationship is roughly ________ and to detect __________ and influential _________. A plot of the residuals makes these effects easier to see. Look for influential observations, individual points that substantially ______ the correlation or the regression line. ___________in the x direction are often influential for the regression line. Be aware of ecological correlation, the tendency for correlations based on _____________ to be stronger than correlations based on _____________. Be careful not to misinterpret correlations
based on averages as applying to individuals. Avoid ________________, the use of a regression line for prediction for values of the explanatory variable far outside the range of the data from which the line was calculated. _____________________ variables may explain the relationship between the explanatory and response variables. Correlation and regression can be misleading if you ignore important ___________ variables. Most of all, be careful not to conclude that there is a ________-and-__________ relationship between two variables just because they are strongly associated. High
correlation does not imply ________________. The best evidence that an association is due to ______________ comes from an experiment in which the explanatory variable is directly changed and other influences on the response are controlled. 1 point each 5.30 Penguins diving. A study of king penguins looked for a relationship between how deep the penguins dive to seek food and how long they stay underwater.14 For all
but the shallowest dives, there is a linear relationship that is different for different penguins. The study report gives a scatterplot for one penguin titled The relation of dive duration (DD) to depth (D). Duration DD is measured in minutes, and depth D is in meters. The report then says, The regression equation for this bird is: DD = 2.69 + 0.0138D. (a)What is the slope of the regression line? Explain in specific language what this slope says about this penguins dives.
(b)According to the regression line, how long does a typical dive to a depth of 200 meters last? (c)The dives varied from 40 meters to 300 meters in depth. Use the regression equation to determine DD for D = 40 and D = 300 and then plot the regression line from D = 40 to D = 300. 2 points
5.31 The price of diamond rings. A newspaper advertisement in the Straits Times of Singapore contained pictures of diamond rings and listed their prices, diamond weight (in carats), and gold purity. Based on data for only the 20-carat gold ladies rings in the advertisement, the least-squares regression line for predicting price (in Singapore dollars) from the weight of the diamond (in carats) is: Price = 259.63 + 3721.02 Carats (a)What does the slope of this line say about the relationship between price and number of carats? (b)What is the predicted price when number of carats = 0? How would you interpret this price?
2 points 5.33 Toucans beak. Exercise 4.45 (page 125) gives data on beak heat loss, as a percent of total body heat loss from all sources, at various temperatures. The data show that beak heat loss is higher at higher temperatures and that the relationship is roughly linear. Figure 5.12 shows Minitab regression output for these data. (a)What is the equation of the least-squares regression line for predicting beak heat loss, as a percent of total body heat loss
from all sources, from temperature? Explain in specific language what this slope says about the relationship between beak heat loss and temperature. (b)Use the equation of the least-squares regression line to predict beak heat loss, as a percent of total body heat loss from all sources, at a temperature of 25 degrees Celsius. (c)What percent of the variation in beak heat loss is explained
by the straight-line relationship with temperature? (d)Use the information in Figure 5.12 to find the correlation r between beak heat loss and temperature. How do you know whether the sign of r is positive or negative? 2 points 5.35 Whats my grade? In Professor Krugmans economics course, the correlation between the students total scores prior to the final examination and their final-examination scores is r = 0.5. The pre-exam totals for all students in the course have mean 280
and standard deviation 40. The final-exam scores have mean 75 and standard deviation 8. Professor Krugman has lost Julies final exam but knows that her total before the exam was 300. He decides to predict her final-exam score from her pre-exam total. (a)What is the slope of the least-squares regression line of final-exam scores on pre-exam total scores in this course? What is the intercept? Interpret the slope in the context of the problem. (b)Use the regression line to predict Julies final-exam score. (c)Julie doesnt think this method accurately predicts how well she did on the final exam. Use r2 to argue that her actual score could have been much higher (or much lower) than the predicted value
2 points 5.36 Going to class. A study of class attendance and grades among first-year students at a state university showed that, in general, students who attended a higher percent of their classes earned higher grades. Class attendance explained 16% of the variation in grade index among the students. What is the numerical value of the correlation between percent of classes attended and grade index?
2 points 5.37 Sisters and brothers. How strongly do physical characteristics of sisters and brothers correlate? Here are data on the heights (in inches) of 12 adult pairs (a)Use your calculator or software to find the correlation and the equation of the least-squares line for predicting sisters height from brothers height. Make a scatterplot of the data, and add the regression line to your plot.
(b)Damien is 70 inches tall. Predict the height of his sister Tonya. Based on the scatterplot and the correlation r, do you expect your prediction to be very accurate? Why? 2 points 5.39 Sparrowhawk colonies. One of natures patterns connects the percent of adult birds in a colony that return from the previous year and the number of new adults that join the colony. Here are data for 13 colonies of sparrowhawks:
You saw in Exercise 4.28 that there is a moderately strong linear relationship, with correlation r = 0.748. (a)Find the least-squares regression line for predicting y from x. Make a scatterplot and draw your line on the plot. (b)Explain in words what the slope of the regression line tells us. (c)An ecologist uses the line, based on 13 colonies, to predict how
many new birds will join another colony, to which 60% of the adults from the previous year return. What is the prediction? 2 points 5.42 Always plot your data! Table 5.1 presents four sets of data prepared by the statistician Frank Anscombe to illustrate the dangers of calculating without first plotting the data: (a)Without making scatterplots, find the correlation and the least-squares regression
line for all four data sets. What do you notice? Use the regression line to predict y for x = 10. (b)Make a scatterplot for each of the data sets, and add the regression line to each plot. (c)In which of the four cases would you be
willing to use the regression line to describe the dependence of y on x? Explain your answer in each case. 8 points 5.44 The effect of changing units. The equation of a regression line, unlike the correlation, depends on the units we use to measure the explanatory and response variables. Return to the data on percent body fat and preferred amount of salt in Exercise 4.23:
In calculating the preferred amount of salt, the weight of the salt was in milligrams. (a)Find the equation of the regression line for predicting percent body fat from preferred amount of salt when weight is in milligrams. (b)A mad scientist decides to measure weight in tenths of milligrams. The same data in these units are Find the equation of the regression line for predicting percent body fat from preferred amount of salt when weight is in tenths of milligrams. (c)Use both lines to predict the percent body fat from preferred amount of salt for a child with preferred amount of salt 0.7
when weight is measured in milligrams, which is the same as 7 when weight is in tenths of milligrams. Are the two predictions the same (up to any roundoff error)? 4 points 5.49 Learning online. Many colleges offer online versions of courses that are also taught in the classroom. It often happens that the students who enroll in the online version do better than the classroom students on the course exams. This does not show that online instruction is more effective than classroom teaching, because the people who sign up for online courses are often quite different from the classroom students. Suggest some differences between online and classroom students that might
explain why online students do better. 2 points 5.52Grade inflation and the SAT. The effect of a lurking variable can be surprising when individuals are divided into groups. In recent years, the mean SAT score of all high school seniors has increased. But the mean SAT score has decreased for students at each level of high school grades (A, B, C, and so on). Explain how grade inflation in high school (the lurking variable) can account for this pattern. 2 points
5.53 Workers incomes. Here is another example of the group effect cautioned about in the previous exercise. Explain how, as a nations population grows older, median income can go down for workers in each age group, yet still go up for all workers. 2 points
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