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Chapter 10 Review Problems

# Chapter 10 Review Problems - Chapter 10 Re~expressing Data...

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Unformatted text preview: Chapter 10 Re~expressing Data: It's Easter Than Ypu Think (Review) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) For the model; = 1.10 + 1.7x, predict y when x = 2. Round to two decimal places. Y A) 4.50 B) 0.22 ‘ C) 43.43 D) —2.30 2) For the model 37 a 1.74 + 4.6\/;, predict y when x = 2. Round to two decimal places. A) 8.25 B) 20.14 C) 0.00 D) “4.77 3) For the model y : 2.52 + 2.3x, predict y when x m 2. Round to two decimal places. A) «~7.12 B) 7.69 C) "7.69 D) 6.39 4) For the model?: 4.4 x02, predict y when x m 6. Round to two decimal places. A) 5.18 B) 8.45 C) 26.4 D) 6.3 5) For the model log y = 3.3 + 0.6x, predict y when x = 2. Round to two decimal places. A)065 B)3L63148 C)3L622y8 13)6&09&7345 E)3L63798 E)018 E) 10.94 E)712 E)52s 1) 2) 3) 4) i’rovide an appropriate response. 6) One of the important factors determining a car's fuel efficiency is its weight. This relationship is examined for 11 cars, and the association is shown in the scatterplot beiow. NMNW (DJ-106:0 Fuel Efﬁciency (mpg) I—l ON Describe the association between these variables shown in the scatterplot. A) Fairly Quadratic, positive, weak relationship B) Fairly linear, positive, strong relationship C) Fairly linear, negative, weak relationship D) Fairly Exponential, positive, strong relational E) Fairly linear, negative, strong relationship 7) One of the important factors determining a car's fuei efﬁciency is its weight. This relationship is examined for 11 cars, and the association is shown in the scatterplot below. Fuei Bﬁiciency (mpg) .... m N M N W 9’ on N c‘ c: a. 00 N m & if a linear model is considered, the regression analysis is as follows: Dependent variable: MPG Rwsquared = 84.7% VARIABLE COEFFICIENT Intercept 47.1181 Weight \$34614 The residuals plot is: 6) Residuals (mpg) -3.5 '0 Predicted (mpg) Based upon the residuals plot, do you think that this linear model is appropriate? A) No, residuals Show a curved pattern. B) No, residuals show no pattern. (I) Yes, residuals Show a curveti pattern. D) Yes, residuals show no pattern. ' I?) Yes, residuals show a linear pattern. 8) The relationship between two quantities x and y is examined, and the association is shown in the 8) Scatterplot below. X Describe the association between these variables shown in the scatterpiot. A) Fairly linear, strong relationship B) Fairly linear, weak relationship C) Fairly exponential, weak relationship 1)) Fairly quadratic, weak relationship E) Fairly exponential, strong relationship 9) The reiationship between two quantities x and y is examined, and the association is shown in the 9) scatterplot below. NWAU‘Q‘JOOW ,_. 9,5 1 1.5 2 2.5 3 3.5 4 4.5 5 7: Describe the association between these variables shown in the scatterplot. A) Fairly linear, positive, strong relationship B) Fairly linear, negative, strong relationship C) Fairly linear, negative, weak relationship D) Fairly linear, positive, weak relationship E) Fairly exponential, strong relationship 10) The relationship between two quantities X and Y is examined, and the association is shown in the 10) scatterplot below. Should you re—express these data before trying to fit a model? Explain. A) Dara should be re—expressed. Scatterplot shows linear relationship. B) Data should be re~expressed. Scatterplot shows upward curvature and increasing scatter as X increases. C) Data should be rewexpressed. Scatterplot shows linear relationship and uniform scatter. D) Data does not need to be reuexpressed. Scatterplot shows upward curvature and increasing scatter as X increases. E) Data does not need to be rte-«expressed. Scatterpiot shows linear relationship Create an appropriate model for the data. 11) Consider the data listed in the following table. 11) Create an appropriate model. What re~expression of Y does this model involve? 1 1 1 A_._., Em C D“. E 2 )Y )4; )W )y )Y 12) Consider the date listed in the following table. 12) Create an appropriate model. What Eta—expression of Y does this model involve? 1 1 1 A)—-- B)--— C)——-—— D)\/_ E) 2 y y «I; y y 13) Consider the data listed in the following table. 13) Create an appropriate model. What re—expression of Y does this model involve? I 1 1 A — B w C _.... D 2 E 4" ) y ) «I ) y )y ) y 14) Consider the data listed in the following table. 14) Create an appropriate model, Estimate the value of y when x = 39. Round your answer to four decimal places. A) 0.0023 B) 0.0002 C) —0.0059 D) 4.5921 E) 0.7131 15) Consider the data Eisted in the foﬁowing table. 15) Create an appropriate model. What rewexpression of Y does this model involve? 1 1 1 A— 3-... C- D 2 E )y )y )4; )y N? SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question Draw the requested scatterplot. 16) The consumer price index (CPI) is a measure of the relative cost of goods in the US. for a 16) particular year. The table below shows the CPI for various years beginning in 1920. Re-express the CPI by taking the log of the data, then plot the rte—expressed data as a scatterplot. 19313 1948 ”£958 1950 19?!) 1988 1990 200i] 201%] Year 17) A psychology graduate student is studying the effect of alcohol intoxication on a person‘s ability to perform manual tasks. In a very simple experiment, a subject's blood alcohoi levei is brought to a certain level and then the subject is presented with 10 rope knots. The subject then attempts to untie as many knots as he or she can in 30 seconds. The results of the experiments are summarized in the table below. cap :30 N01 0 O \J ‘99 . CDC) mm CD CD 01 Re—express the number of knots untied by taking the inverse of each count, then plot the rte-«expressed data as a scatterpiot. 110:1 knots untied} Blood alcohoi Level 17) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 18) The consumer price index (CPI) is a measure of the relative cost of goods in the a given country for 18) a particular year. The table below shows the CPI for a country for the stated years beginning in 1940. 1990 130.7 2000 172.2 Create an appropriate model to re—express the CPI. What ire—expression of the CPI does this model involve? A) log(CP1) B) ”E- C) «ICPI D) ““1 E) 103 (1cm) CPI « I CPI 19) The consumer price index (CPI) is a measure of the relative cost of goocis in the a given country for 19) a particular year. The table below shows the CPI for a country for the stated years beginning in 1940. 1990 130.7 2000 172.2 Rewexpress the CPI. Then determine the regression equation and correlation coefficient for the res—expressed data. Use your model to predict the CPI 1112017. A) 354.3 13) 342 C) 361.5 13349.1 E) 326.2 2.0) Dioxins are a class of long—lived and highly toxic pollutants. The topsoil concentration in parts per 20) million (ppm) are shown in the table below as a function of distance from the dump. Distance from Dioxin dump (meters) concentration Create an appropriate model to Ice-«express the concentration (ppm). Then determine the regression equation and correlation coefficient for the re—expressed data. Round all ﬁgures to 4 clecimal places. . 1 A) Re—expressmn: :lmm (13191“) B) Rewexpression: 7—565)— ; y : .0012x + 0.0218, r2 a 0.9970 ; y m .0012x + .0500, r2 a 1 C) Rte—expression: .........l......... ; y = .0012x + 0.0233, :2 = 0.9940 thpm) . 1 D) Re—express1on:———-—-——- «1319111) B) Rea-«expression: ———1——; y : .0012x + 0.0251, r2 a 0.9857 40313111) ; y u .0012x + 0.0277, r2 a 0.9688 10 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question Answer the question. 21) The consumer price index (CPI) is a measure of the relative cost of goods in the US. for a 21} particular year. The table below shows the C391 for various years beginning in 1940. a) Re—express the CPI by taking the log of the data, then plot the Ice—expressed data as a scatterplot. 1930 192313 19513 1950 3933 1980 19330 2880 2B1!) Yea: ‘0) Determine the regression equation and correlation coefficient for the Ice—expressed data. c) Use your modei from part (b) to predict the CPI in 2010. 11 22) A psychology graduate student is studying the effect of alcohol intoxication on a person’s 22) ability to perform manual tasks. In a very simple experiment, a subject's blood alcohol level is brought to a certain level and then the subfect is presented with 10 rope knots. The subject then attempts to untie as many knots as he or she can in 30 seconds. The results of the experiments are summarized in the table below. a) Remexpress the number of knots untied by taking the inverse of each count, then plot the ire-expressed data as a scatterplot. “(if knots untied) 0 .34 Blood Alcohol Laue! b) Determine the regression equation and correlation coefficient for the renexptessecl data. c) interpret the ymintercept of your model. Does it have meaning? MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 23) A company‘s sales increase by the same amount each year. This growth is . . . 23) A) exponential 13) linear C) logarithmic D) quadratic E) power 12 24) A scatterplot of T}: vs. x shows a strong positive iinear pattern. It is probabiy true that 24) Y A) the scatterpiot of Y vs X also shows a linear pattern. 13) accurate predictions can be made for Y even if extrapolation is involved. C) the correlation between X and Y is near +1.0. D) the residuals plot for regression of Y on X shows a curved pattern. 12) large values of X are associated with large Values of Y. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 25) A company manufactures of polypropylene rope in six different sizes. To assess the 25) strength of the ropes they test two sampies of each size to see how much force (in kilograms) the ropes will hold without breaking. The table shows the results of the tests. We want to create a model for predicting the breaking strength from the diameter of the rope. Find a model that uses are—expressed data to straighten the scatterpiot. Diameter Strength 26) A company manufactures of polypropylene rope in six different sizes. To assess the 26) strength of the ropes they test two samples of each size to see how much force (in kilograms) the ropes will hold without breaking. The table shows the results of the tests. We want to create a model for predicting the breaking strength from the diameter of the rope. The company is thinking of introducing a new 25mm rope. How strong should it be? (Write a sentence in context based on one of your models.) Diameter Strength 13 27) Doctors studying how the human body assimilates medication inject some patients with 27) penicillin, and then monitor the concentration of the drug (in units/cc) in the patients blood for seven hours. The data are shown in the scatterpiot. First they tried to ﬁt a linear model. The regression analysis and residuals plot are shown. Using this model, estimate What the concentration of penicillin will be after 4 hours. Dependent variants is: Concentration Ne Seiector R squared = 30.8% Fl squared {adjusted} = 90.8% e = 3.47'2 with 43 - 2 = 41 degrees of freedom Source Sum of Squares cit Mean Square F-ra'i‘io Regression 4908.55 1 4980.55 8387’ Residual 494.199 41 12.0538 Variable Coefficient 5.5:. of Ernst! tvretio prob Constant 40.3288 1 .205 31.1 0 0.0001 Time 5.95950 0.2050 ~20? 0 0.0001 wt! 10 20 3O nredieteclﬁlr’l) 14 28) Doctors studying how the human body assimilates medication inject some patients with 28) penicillin, and then monitor the concentration of the drug (in units/cc) in the patients‘ blood for seven hours. The data are shown in the scatterplot. First they tried to fit a linear model. Now the researchers try a new model, using the ire—expression log(Concentration). Examine the regression analysis and the residuals plot below. Explain why you think this model is better than the original linear model. Dependent variable is: Log Cnn No Selector R squared = 88.0% R squared facilitated) = 98.0% s = 0.0451 with 43 - 2 = 41 degrees of freedom Source Sum crE Squares df Heart Square F-ratio Regression 4.1 1395 1 4.11 335 2022 Residual 3.083412 41 0.002034 Variabie Coefficient se. of Coeff t-ratto prob Constant 1 80184 0.01 88 10? E 0.0001 Time -0.‘l were 0.0030 "45.0 é 0.0001 0.?5 1 .25 predicteddﬁj MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 29) Which of the following is not a goal of renexpressing data? 29) A) Make the scatter in a scatterplot spread Out evenly rather than following a fan shape. B) Make the distribution of a variable more symmetric. C) Make the form of a scatterplot more nearly linear. D) Make the spread of several groups more alike. E) All of the above are goals of rewexpressing data. 30) The model Vdistance z 3.30 + 0.235(speed) can be used to predict the stopping distance (in feet) for 30) a car traveling at a speciﬁc speed (in mph). According to this model, about how much distance will a car going 65 mph need to stop? A) 18.6 feet B) 27.0 feet C) 729.0 feet D) 4.3 feet E) 345.0 feet 15 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 31) QuarkNet, a project hmded by the National Science Foundation and the US. Department 31) of Energy, poses the foiiowing problem on its website: "Last year, deep Within the Soudan mine, QuarkNet teachers began a long—term experiment to measure the amount of carbonwlti remaining in an initial 100.. gram sampie at 2000—year intervais. The experiment will be complete in the year 32001. Fortunately, a method for sending information backwards in time will be discovered in the year 29998, so, although the experiment is far from over, the results are in." Here is a portion of the data: Time(yr) 0 2000 4000 6000 8000 10,000 12,000 14,000 16,000 18,000 Mass (g) 100 '76 61 47 36 29 22 1‘7 13 10 A scatterplot of these data looks like: Scatterplut {If Mass {9) on Time arr} 10000 i 5000 20000 Time {yr} Use your modei to estimate what the mass will be after 7500 years. MULTIPLE CHOICE. Choose the one atternative that best completes the statement or answers the question, 32) Which statement about rte-expressing data is not true? 32) I. Unirnodai distributions that are skewed to the left wili be made more symmetric by taking the square root of the variable. II. A curve that is descending as the explanatory variabie increases may be straightened by iooking at the reeiprocai of the response variable. III. One goal of re—expression may be to make the variability of the response variable more uniform. A) Hand III B) I only C) I, H, and HE D) Hi only ‘13) H only 1.6 33) Over the past decade a farmer has been able to increase his wheat production by about the same 33) percentage each year. His most useful predictive model is probably... A) exponential B) linear C) power D) quadratic E) logarithmic 34) A scatterplot of log(Y) vs. log(X) reveals a linear pattern with very little scatter. It is probabiy true 34.) that A) the caicuiator‘s LnReg function wiii model the association between X and Y. B) the scatterplot of Y vs X shOWs a ﬁnear association. C) the correlation between X and Y is near 0. D) the correlation between X and Y is near +1. E) the residuals plot for regression of Y on X shows a curved pattern. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 35) Doctors studying how the human body assimilates medication inject a patient with 35) penicillin, and then monitor the concentration of the drug in the patient's blood for severe} hours. The data are shown in the table. Time elapsed Concentration (Hours) (Units! cc) 1 42 2 28 3 19 4 13 5 9 6 6 7 4 Use your model to estimate what the concentration of penicillin will be after 8 hours. 17 36) A forester would like to know how big a maple tree might be at age 50 years. She gathers 36) data from some trees that have been cut down, and plots the diameters {in inches) of the trees against their ages (in years). First she makes a linear model. The scatterplot and residuals plot are shown. Do you think the linear model is appropriate? Explain. 6 B 1 2 1 5 orecﬁictedcmﬂ.) UPKUMW ~91: saw-mm "‘ 18 37) A forester would like to know how big a maple tree might be at age 50 years. She gathers 3'7) data from some trees that have been cut. down, and plots the diameters {in inches) of the trees against their ages (in years). First she makes a linear model. Now she ire—expresses the data, using the logarithm of age to try to predict the diameter of the tree. Here are the regression analysis and the residuals plot. Explain why you think this is a better model. Dependem mele i5: Bism R5 amused = 6’4. 3% Vawiabl'e C’oeﬁciem‘ 5e. OfC’oej Cms rant m «5’. {it}??? 3. 68} Engage) 35. 0%)} 1299 predictedefL} 19 Answer Key Testname: CHAPTER 10 RE—EXPRESSING DATA — IT'S EASTER THAN YPU THINK (REVIEW) 1) B 2) A 3) E 4) D 5) c 6) E 7) A 8) E 9) A 10) 13 11) D 12) A 13) E 14) B 15) C 16) \$830 igéﬂ 1958 1950 19?U 1980 \$390 2880 2030 Year 20 Answer Key Testname: CHAPTER 10 RE—EXPRESSING DATA - IT'S EASTER THAN YPU THINK (REVIEW) Blood No. of Alcohol Knots Untied Pm: mg E :1: CD o 0-: 0003-3 CRL-‘U‘l N .0 9.09. c: 000 {.10 \‘IHN .0 025 03- 3; m 11.8?5xu {3.0808 R3 m 8.??47 1m knots untied [1.92 {MM £3.06 13.88 [3.1 Blood Aicohal Level 18) A 19) 1:) 20) E 21 Answer Key Testname: CHAPTER 10 REuEXPRESSING DATA — IT'S EASTER THAN YPU THINK (REVIEW) 1938 1940 195E 1968 19?U 1980 1333 2080 2010 Year b) y m 0.0185x - 34771, R2 2 0.9794 c) y(2010) = Iog(CPI) a (10185-2010 ... 34,771 = 2.414. So, CPI 3 102.424 2 25% 22 Answer Key Tes’mame: CHAPTER 10 RE—EXPRESSING DATA ~ IT‘S EASTER THAN YPU THINK (REVIEW) 22) a) #knots m. m m“— _0_07 ——-m_ M-— m- o_-25 0-05 —. y 8 11.0883- 0.0800 R2 m 0.774? 103 knots untied 8.32 [3.04 8.06 0.88 0.1 Blood alcohol Level b) y = 11.875x .. 0.0808, R2 .._.. 07747 c) The constant term from the model is equal to '410808. Thus, the model predicts that a person with a blood alcohol level of 0.00 will untie 1/(—0.0808) m 42.4 knots. Clearly, this is not physical and has no meaning. 23) B 24) D 25) ﬁr = 2.64 + 1.37(dia) (among other possibilities) 26) The model estimates this rope will have an ayproximate breaking strength of 1367 (1396) kg, but this extrapolation should be viewed with caution. 2'7) 16.5 units/co 28) The residuals show a random pattern with no curvature. 29) E 30) E 31) log(Mass) = 2.00143 — 00000550500) 2 1.58893, so MQSS 2 7.015889% = 38.800883 remaining. 32) A 33) A 34) E 35) When Time = 8, log(Conc) m 1.789 — 0.169(8) = 0.437; Cont: = 100-437 = 2.74 units/CC 23 Answer Key Testname: CHAPTER 10 RE—EXPRESSING DATA ~ IT‘S EASTER THAN YPU THINK (REVIEW) 36) No, the plot of residuals shows an obvious pattern. Trees with diameters iess than 6 inches have negative residuals, trees with diameters between 9 and 14 inches have positive residuals, and trees with diameters larger than 15 inches have negative residuals. 37) There is no obvious pattern to the residual plot. 24 ...
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