Concept explainers
Toughness and fibrousness of asparagus are major determinants of quality. This was the focus of a study reported in “Post-Harvest Glyphosphate Application Reduces Toughening, Fiber Content, and Ligniflcation of Stored Asparagus Spears” (J. of the Amer. Soc. of Hort. Science, 1988: 569–572). The article reported the accompanying data (read from a graph) on x = shear force (kg) and y = percent fiber dry weight.
x | 46 | 48 | 55 | 57 | 60 | 72 | 81 | 85 | 94 |
y | 2.18 | 2.10 | 2.13 | 2.28 | 2.34 | 2.53 | 2.28 | 2.62 | 2.63 |
x | 109 | 121 | 132 | 137 | 148 | 149 | 184 | 185 | 187 |
y | 2.50 | 2.66 | 2.79 | 2.80 | 3.01 | 2.98 | 3.34 | 3.49 | 3.26 |
- a. Calculate the value of the sample
correlation coefficient . Based on this value, how would you describe the nature of the relationship between the two variables? - b. If a first specimen has a larger value of shear force than does a second specimen, what tends to be true of percent dry fiber weight for the two specimens?
- c. If shear force is expressed in pounds, what happens to the value of r? Why?
- d. If the simple linear regression model were fit to this data, what proportion of observed variation in percent fiber dry weight could be explained by the model relationship?
- e. Carry out a test at significance level .01 to decide whether there is a positive linear association between the two variables.
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Chapter 12 Solutions
Probability and Statistics for Engineering and the Sciences
- Foot ulcers are a common problem for people with diabetes. Higher skin temperatures on the foot indicate an increased risk of ulcers. The article "An Intelligent Insole for Diabetic Patients with the Loss of Protective Sensation" (Kimberly Anderson, M.S. Thesis, Colorado School of Mines), reports measurements of temperatures, in °F, of both feet for 181 diabetic patients. The results are presented in the following table. Left Foot Right Foot 80 80 85 85 75 80 88 86 89 87 87 82 78 78 88 89 89 90 76 81 89 86 87 82 78 78 80 81 87 82 86 85 76 80 88 89 Construct a scatterplot of the right foot temperature (y) versus the left foot temperature (x). Verify that a linear model is appropriate. b. Compute the least-squares line for predicting the right foot temperature from the left foot temperature. If the left foot temperatures of two patients differ by 2 degrees, by how much would you predict their right foot temperatures to differ? Predict the right foot temperature for a patient whose left…arrow_forwardEstriol Level and Birth Weight. J. Greene and J. Touchstone conducted a study on the relationship between the estriol levels of pregnant women and the birth weights of their children. Their findings, “Urinary Tract Estriol: An Index of Placental Function,” were published in the American Journal of Obstetrics and Gynecology (Vol. 85(1), pp. 1–9). The data from the study are provided on the WeissStats site, where estriol levels are in mg/24 hr and birth weights are in hectograms. a. Decide whether finding a regression line for the data is reasonable. If so, then also do parts (b)–(d). b. Obtain the coefficient of determination. c. Determine the percentage of variation in the observed values of the response variable explained by the regression, and interpret your answer. d. State how useful the regression equation appears to be for making predictions.arrow_forwardThe depth of wetting of a soil is the depth to which water content will increase owing to extemal factors. The article "Discussion of Method for Evaluation of Depth of Wetting in Residential Areas" (J. Nelson, K. Chao, and D. Overton, Journal of Geotechnical and Geoenvironmental Engineering, 2011:293-296) discusses the relationship between depth of wetting beneath a structure and the age of the structure. The article presents measurements of depth of wetting, in meters, and the ages, in years, of 21 houses, as shown in the following table. Age Depth 7.6 4 4.6 6.1 9.1 3 4.3 7.3 5.2 10.4 15.5 5.8 10.7 4 5.5 6.1 10.7 10.4 4.6 7.0 6.1 14 16.8 10 9.1 8.8 Compute the least-squares line for predicting depth of wetting (y) from age (x). b. Identify a point with an unusually large x-value. Compute the least-squares line that results from deletion of this point. Identify another point which can be classified as an outlier. Compute the least-squares line that results from deletion of the outlier,…arrow_forward
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- 5.25. Representative data on x = carbonation depth (in millimeters) and y = strength (in megapascals) for a sample of concrete core specimens taken from a particular building were read from a plot in the article “The Carbonation of Concrete Structures in the Tropical Environment of Singapore” (Magazine of Concrete Research [1996]: 293-300): Depth, x 8.0 20.0 20.0 30.0 35.0 40.0 50.0 55.0 65.0 Strength, y 22.8 17.1 21.1 16.1 13.4 12.4 11.4 9.7 6.8 a. Construct a scatterplot. Does the relationship between carbonation depth and strength appear to be linear? Yes, the relationship between carbonation depth and strength appears to be linear however it is a negative linear relation. b. Find the equation of the of the least-squares line.c. What would you predict for strength when carbonation depth is 25 mm?d. Explain why it would not be reasonable to use the least-squares line to predict strength when carbonation depth…arrow_forwardThe article "Lead Dissolution from Lead Smelter Slags Using Magnesium Chloride Solutions" (A. Xenidis, T. Lillis, and I. Hallikia, The AusIMM Proceedings, 1999:37-14) discusses an investigation of leaching rates of lead in solutions of magnesium chloride. The data in the following table (read from a graph) present the percentage of lead that has been extracted at various times (in minutes). Time (t) 4 8 16 30 60 120 Percent extracted (v) |1.2 1.6 2.3 2.8 3.6 4.4 a. The article suggests fitting a quadratic model y = Bo + B,t + Bz² + ɛ to these data. Fit this model, and compute the standard deviations of the coefficients. b. The reaction rate at time t is given by the derivative dy/dt = B, + 2B,t. Estimate the time at which the reaction rate will be equal to 0.05. c. The reaction rate at t = Oisequal to B1. Find a 95% confidence interval for the reaction rate at t = 0. d. Can you conclude that the reaction rate is decreasing with time? Explain.arrow_forwardLecture(8.8): The amount of time people engage in physical activity mat be related to health outcome. Those who report that they spend more than 15 hours are put into one while those who spend less 10 were put into another group. (Those who fall in between 10 and 15 were left out of the study); The reaearcher then ask the participants to wear a monitor for one month. The average time in minutes is recorded and shown below . Is there any evidence that on average people who watch less than 10 hours watching televsion spend more time on physical activity.?. Test the hypotheses at (alpha=0.05) using the 5 step procedure <10 hours 75 63 118 35 82 >15 hours 62 6 78 43 22 33arrow_forward
- A sales manager wants to examine the relationship between the number of daily customers (x) and the revenue generated (y). For this purpose, he made observations in a randomly chosen 6 days for a store and observed the number of daily customers and the revenue (1000 TL). The personally created data set for the x and y variables is included in the attached “Homework data” file. Using the data set defined on your behalf; x1 x2 x3 x4 x5 x6 y1 y2 y3 y4 y5 y6 sb1 Burak data 74 86 96 105 110 124 290 307 331 406 421 481 0.491 a) Create the regression equation b) Interpret b0 and b1 values c) Test and interpret whether there is a linear relationship between the number of daily customers and the income obtained at the level of α = 0.05 significance. Could you explain this question again and in more detail especially not in excel imagearrow_forwardA sales manager wants to examine the relationship between the number of daily customers (x) and the revenue generated (y). For this purpose, he made observations in a randomly chosen 6 days for a store and observed the number of daily customers and the revenue (1000 TL). The personally created data set for the x and y variables is included in the attached “Homework data” file. Using the data set defined on your behalf; x1 x2 x3 x4 x5 x6 y1 y2 y3 y4 y5 y6 sb1 Burak data 74 86 96 105 110 124 290 307 331 406 421 481 0.491 a) Create the regression equation b) Interpret b0 and b1 values c) Test and interpret whether there is a linear relationship between the number of daily customers and the income obtained at the level of α = 0.05 significance. d) Establish and interpret the 95% confidence interval for β1.arrow_forwardA sales manager wants to examine the relationship between the number of daily customers (x) and the revenue generated (y). For this purpose, he made observations in a randomly chosen 6 days for a store and observed the number of daily customers and the revenue (1000 TL). The personally created data set for the x and y variables is included in the attached “Homework data” file. Using the data set defined on your behalf; x1 x2 x3 x4 x5 x6 y1 y2 y3 y4 y5 y6 sb1 Burak data 74 86 96 105 110 124 290 307 331 406 421 481 0.491 d) Establish and interpret the 95% confidence interval for β1.arrow_forward
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