Introduction to the Practice of Statistics Sixth Edition Moore, McCabe

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1 Introduction to the Practice of Statistics Sixth Edition Moore, McCabe Section 2.1 Homework Answers 2.6 Relationship between first test and final exam. How strong is the relationship between the score on the first test and the score on the final exam in an elementary statistics course? Here are data for eight students from such a course: First test Score Final exam Score (a) Which variable should play the role of the explanatory variable in describing this relationship? The explanatory variable should be the first exam score since it should help explain the variation in the final exam score. (b) Make a scatterplot and describe the relationship. Relationship Between First Exam and Final Exam Final Exam Score First Exam Score (c) Give some possible reasons why this relationship so weak. The graph is showing that some students are improving while others are getting worse. The line drawn represents a person that scores the same on the final exam as on the first exam. If you are above the line then that person did better on the final than on the first exam. Likewise if you are below the line you did worse on the final than on the first exam. You can see that five students did better on the final than on the first exam, with one outlier student that did the worst on the first exam but received the best score on the final; certainly this student is an outlier due to the extreme turn around.

2 2.7 Relationship between second test and final exam. Refer to the previous exercise. Here are the data for the second test and the final exam for the same students: Second- test Score Final-exam Score (a) Explain why you should use the second-test score as the explanatory variable. The explanatory variable should be the second exam score since it should help explain the variation in the final exam score. (b) Make a scatterplot and describe the relationship. Final Exam Score The relationship seems to be positive. As second exam scores increase so do final exam scores on average Second Exam Score (c) Why do you think the relationship between the second-test score and the final-exam score is stronger than the relationship between the first-test score and the final-exam score? One possible explanation is that by the second exam (during the first exam the student may still be trying to get their bearings) the student is showing how much understanding they have of the subject matter, thus by the time the final exam is taken, the student will typically score around their true average (which indicates their understanding of the material) which is hopefully approximated by the biased second exam score (Why bias? As an instructor you hope that the second exam score is consistently an underestimate of the final potential of the student. That is you hope that the student will continue to grow in their understanding).

3 2.11 Reading ability and IQ. A study of reading ability in schoolchildren chose 60 fifth-grade children at random from a school. The researchers had the children's scores on an IQ test and on a test of reading ability. 8 Figure 2.6 (on page 96) plots reading test score (response) against IQ score (explanatory). (a) Explain why we should expect a positive association between IQ and reading score for children in the same grade. Does the scatterplot show a positive association? Unfortunately I have never seen an actual IQ test, but have only seen examples of questions for some article usually criticizing the test. Thus I can only answer this question from what I think the test does. The IQ test is supposed to measure intelligence. A child in second grade is suppose to be able to read at their grade level. You would expect that ability is somewhat related to the child s intelligence (It can not be 100% since many factors can lead to a poor or good reading ability. For example a child that is very intelligent, but comes from poor family environment where education is not emphasized might score lower than say a child with below average intelligence, but is constantly challenged at home to improve their reading). But overall you would expect a high IQ to be on average associatted with a high reading ability and vice versa. (b) A group of four points appear to be outliers. In what way do these children's IQ and reading scores deviate from the overall pattern? The cross-hairs superimposed on the graph indicate approximately (I eye balled it) the location of the sample mean reading score and the sample mean IQ score. You can see that the four students in question scored about the sample average as far as their IQ score is concerned, yet the consitently are way below the sample average reading score. (c) Ignoring the outliers, is the association between IQ and reading scores roughly linear? Is it very strong? Explain your answers. I would say it is not too strong, since there is still quite a bit of variation from the linear trend. Look at a child that scored approximately 100 on the IQ test, for example. The redbox shows that the approximate range for a reading score is 22 to 80, which is not too different from the overall range of the reading scores of 10 to 99.

4 2.16 City and highway gas mileage. Table 1.10 (page 31) gives the city and highway gas mileages for minicompact and two-seater cars. We expect a positive association between the city and highway mileages of a group of vehicles. We have already seen that the Honda Insight is a different type of car, so omit it as you work with these data. (a) Make a scatterplot that shows the relationship between city and highway mileage, using city mileage as the explanatory variable. Use different plotting symbols for the two types of cars. 70 Removing the Outlier Highway MPG Hwy Type T Hwy Type M Hwy Type T Hwy Type M City MPG City MPG (b) Interpret the plot. Is there a positive association? Is the form of the plot roughly linear? Is the form of the relationship similar for the two types of car? What is the most important difference between the two types? Above we see the result of removing the outlier so we can get a better view of the data. You can see that the association is very strong for both types of vehicles.

5 2.22 Fuel consumption and speed. How does the fuel consumption of a car change as its speed increases? Below are data for a British Ford Escort. Speed is measured in kilometers per hour, and fuel consumption is measured in liters of gasoline used per 100 kilometers.traveled. 14 (a) Make a scatterplot. (Which variable should go on the x-axis?) Speed Fuel It would make sense to put speed on the x-axis since this might help explain the variation in the fuel (histogram below shows how the fuel used is distributed and the boxplot shows the variation) (b) Describe the form of the relationship. In what way is it not linear? Explain why the form of the relationship makes sense. It is obviously not linear since it does not follow a linear trend. It makes sense that as you increase vehicle speed the momentum increases (The vehicle s momentum can help continue the movement and overcome the friction associated with contact with the road). At some point however, two other forms of resistance start to increase: air resistance, mechanical resistance from the mechanics of the design of the motor and transmission. Air resistance starts to become a huge factor as the vehicle attempts to increase speed; thus more fuel is needed to overcome this extra resistance. The transmission and engine have limits as to how fast the wheels can spin; I am sure heat generated (I am totally guessing here) also plays a huge role and more fuel is needed to make those wheels spin faster. (c) It does not make sense to describe the variables as either positively associated or negatively associated. Why not? You can see that at some instance of the interval the association is negative; around 60 km/h that trend changes to a positive association.

6 (d) Is the relationship reasonably strong or quite weak? Explain your answer. It is clear that the points are following some pattern. The ordered pairs are not showing any (as far as the eye can see) deviation from that pattern. Thus, the relationship is very, very, very strong. For example if the scatterplot graph had been the one shown below, then we recognize the pattern is strong, but with the variation from the patter is is visible