3.3.11.

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Feb 20, 2024

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Prac±ce: Regression Lines and Bivariate Sta±s±cs 1. The following table contains informa±on about the robbery rate (number of robberies per 100,000 popula±on) and percentage of urban popula±on (percentage of people living in urban as opposed to rural areas) across a set of states: State Percentage of Urban Popula±on. Robberies per 100,000 People Massachuse²s 91 193, Wisconsin 67 73, South Dakota 28 16 Virginia 72 106, South Carolina 60 99, Texas 81 240, Arizona 75 169, California 96 343, Arkansas 44 88, Hawaii 77 106 Conduct a full regression analysis using robbery rate as the response variable: A. Sketch the sca²erplot with the least-squares line, and sketch the residual plot. Interpret your sketch. Remember, use robbery rate as the response variable, since that's the value you're trying to predict. (1 point) The data plot reveals that the associa±on between the percentage of urban popula±on and the robbery rate is posi±ve and linear. In addi±on, the overall pa²ern of the points closely follows a line (although there is one outlier in the upper right). B. Write and interpret the correla±on coeﬃcient. (.5 points) r=0.821, which indicates a strong posi±ve rela±onship between % of urban popula±on and robbery rate. C. Write the regression equa±on and interpret the regression coeﬃcient. (1 point) yˆ = -117.031 + 3.767 x. The regression coeﬃcient is 3.767, which means an increase of one percent of

urban popula±on will increase the robbery rate by 3.767. D. Show and interpret the coeﬃcient of determina±on. (.5 points) r^2=0.673, which means 67.3% of the total varia±on in the robbery rate is explained by the % of urban popula±on. E. State whether you think there is a rela±onship between the two variables, and jus±fy your answer. (1 point) Because the r=0.821 correla±on value is posi±ve and very high, I believe there is a strong posi±ve linear associa±on between the percentage of urban popula±on and the robbery rate. Furthermore, the coeﬃcient of determina±on demonstrates that the percentage of urban popula±on explains 67.3% of the en±re varia±on in the robbery rate. F. We know that the percentage of urban popula±on in Idaho is 35%. We also know that the percentage of urban popula±on in Florida is 78%. Predict the robbery rates in each of these states. Are these extrapola±ons or interpola±on, and are they valid predic±ons? (1 point) yˆ = -117.031 + 3.767 x Idaho Y= -117.031 + 3.767 x 35= 14.814 Florida Y= -117.031 + 3.767 x 78= 176.795 These are interpola±on ( in the range from 28 to 96), so these are valid predic±ons. 2. Geothermal power is an important source of energy. Since the amount of energy contained in 1 pound of water is a func±on of its temperature, you might wonder whether water obtained from deeper wells contains more energy per pound. Loca±on of Well Average (max.) Drill Hole Depth (m) Average (max.) Temperature (C°) El Tateo, Chile 650 230 Ahuachapan, El Salvador 1,000 230 Nama³all, Iceland 1,000 250 Lardarello (region), Italy 600 200 Matsukawa, Japan 1,000 220

Cerro Prieto, Mexico 800 300 Wairakei, New Zealand 800 230 Kizildere, Turkey 700 190 The Geysers, United States 1,500 250 The data in the table are reproduced from an ar±cle on geothermal systems by A.J. Ellis. A. Following steps A-E in the previous ques±on, do a full regression analysis, using hole depth as the explanatory variable. (4 points) 1.) The data plot shows the rela±onship between the hole depth and the temperature is posi±ve linear. Also the overall pa²ern of the points follows a line closely (though there is one outliner in the top). 2.) r=0.330, which indicates a weak posi±ve rela±onship. 3.) yˆ = 198.93 + 0.0385 x. The regression coeﬃcient is 0.0385, which means an increase of one 100 meters depth will increase the water temperature by 3.85 4.) r^2=0.109, which means 10.9% of the total varia±on in the temperature is explained by the hole depth 5.) I think there is a weak posi±ve linear rela±onship between the hole depth and the temperature, since the r=0.330 correla±on coeﬃcient is posi±ve but low. Also the coeﬃcient of determina±on shows 10.9% of the total varia±on in the temperature is explained by the hole depth B. Using the regression equa±on, predict the temperature of the water in a well with an average drill hole depth of 2000m. Is this a reliable predic±on? (1 point) yˆ = 198.93 + 0.0385 x 2000 = 275.93 (C°) This is not a reliable predic±on, since 2000m is the extrapola±on (out of the range). 3. Use the potency and temperature data below. An experiment was conducted to observe the effect of an increase in temperature on the potency of an an±bio±c. Three 1-ounce por±ons of the an±bio±c were stored for equal lengths of ±me at each of these

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temperatures: 30°, 50°, 70°, and 90°. The potency readings observed at each temperature of the experimental period are listed here: Potency Readings, y 38, 43, 29 32, 26, 33 19, 27, 23 14, 19, 21 Temperature, x 30° 50° 70° 90° As before, conduct a full regression analysis (see 1A-1E) to answer this ques±on: Within the sample data, is a one-degree increase in storage temperature of an±bio±cs associated with a decrease in potency? If so, how much? In this case, you'll need to determine which is the response variable in your analysis. Looking at the ques±on, which variable would you say the researcher is more interested in predic±ng? The predicted variable is the response variable. (You can also look at the research ques±on as an interpreta±on of a regression coeﬃcient—when you write a statement that interprets a regression coeﬃcient, which variable goes first?) (5 points) I use the temperature as the explanatory variable, as we are trying to predict the potency of an an±bio±c (response variable). The data plot reveals that the associa±on between temperature and an±bio±c potency is nega±ve linear. In addi±on, the overall arrangement of the points closely follows a line (although there are some outliers in 30C°). r=-0.872, which indicates a strong nega±ve rela±onship. yˆ = 46 - 0.3167 x. The regression coeﬃcient is -0.3167, which means an one-degree increase in storage temperature will decrease the potency by 0.3167 r^2=0.760, which means 76% of the total varia±on in the potency is explained by the temperature I think there is a strong nega±ve linear rela±onship between the temperature and the potency of an±bio±c is nega±ve linear, because the r= -0.872 correla±on coeﬃcient is nega±ve and very high.

Also the coeﬃcient of determina±on shows 76% of the total varia±on in the potency is explained by the temperature. Therefore, we can predict a one-degree increase in storage temperature will decrease the potency by 0.3167. 4. Below is some hypothe±cal data that shows the Ra±ng (an index of how many people watch the show) and Average Cost for a 30-second adver±sement for a number of game shows: GAME SHOW Ra±ng Average Cost for a 30 Second Ad ($) Is That Real Hair? 18.2 $55,000 Feng Shui Happy Booth 8.4 $20,000 Love Signals! 13.6 $26,000 How Many Worms? 11.9 $39,000 Spin the Bo²le 9.0 $11,000 Wake that Possum! 22.6 $75,000 Sit on a Potato 3.5 $10,000 Name that Fruit! 2.1 $5,000 The $2.00 Ques±on 4.6 $19,000 As before, conduct a full (five step) regression analysis to answer this ques±on: Within the sample data, is an increase in ra±ngs associated with an increase the average cost for a 30-second ad? If so, how much? (5 points) The data plot shows the rela±onship between the ra±ng and the cost of a 30-second adver±sement is posi±ve linear.

r=0.937, which indicates a strong posi±ve rela±onship. yˆ = -4235.8+3174.9 x. The regression coeﬃcient is 3174.9, which means an increase of the ra±ng will increase the cost of 30 sec ad by $3,174.9 r^2=0.878, which means 87.8% of the total varia±on in the cost of 30 sec ad is explained by the ra±ng. Because the r= 0.937 correla±on coeﬃcient is posi±ve and very high, I believe there is a significant posi±ve linear link between the ra±ng and the cost of a 30-second commercial. The ra±ng also explains 87.8% of the en±re difference in the cost of a 30-second commercial, according to the coeﬃcient of determina±on. As a result, we can forecast an increase of the ra±ng will increase the cost of 30 sec ad by $3,174.9. 5. Demographers o´en examine the rela±onship between income measures and popula±on factors such as births, deaths, marriages, or migra±on rates. You have the following data on Per Capita Income in 1987 and the % Births that are "Low Birth Weight" across a sample of states: STATE Per Capita Income (as of 1987) % of Births of "Low Birth Weight" (as of 1988) Alaska $13,263 5.0 Colorado $12,271 7.8 Delaware $12,785 7.4 Georgia $11,406 8.4 Iowa $11,198 5.4 Louisiana $8,961 8.8 Maine $10,478 4.8 Minnesota $12,281 5.0 Nebraska $11,139 5.5 New York $13,167 7.8 Ohio $11,323 6.9 Oregon $11,045 5.2 South Dakota $8,910 4.7 Utah $9,288 5.7 Wisconsin $11,417 5.4 As before, conduct a full regression analysis to answer this ques±on:

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Within the sample data, is an increase in per capita income associated with a decrease in the percentage of low birth weights? If so, how much? (5 points) The data plot shows the rela±onship between the per capita income and the percentage of low birth weight is posi±ve linear. r=0.110, which indicates a very weak posi±ve rela±onship. yˆ = 4.9879+ 0.0001 x. The regression coeﬃcient is 0.0001, which means an increase of the per capita income $10000 will increase the percentage of low birth weight by 1%. r^2=0.012, which means only 1.2% of the total the percentage of low birth weight is explained by the per capita income. Within the sample data, I believe there is a weak posi±ve linear rela±onship. An increase in per capita capital will not reduce the percentage of low birth weight. Instead, we believe it will likely increase the percentage of babies born with low birth weight. (However, this is a very shaky rela±onship.

3.3.11.

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