An accurate assessment of oxygen consumption provides important information for determining energy expenditure requirements for physically demanding tasks. The paper “Oxygen Consumption During Fire Suppression: Error of Heart Rate Estimation” (Ergonomics [1991]: 1469–1474) reported on a study in which x = Oxygen consumption (in milliliters per kilogram per minute) during a treadmill test was determined for a sample of 10 firefighters. Then y = Oxygen consumption at a comparable heart rate was measured for each of the 10 individuals while they performed a fire-suppression simulation. This resulted in the following data and
- a. Does the scatterplot suggest an approximate linear relationship?
- b. The investigators fit a least-squares line. The resulting Minitab output is given in the following:
The regression equation is firecon = 211. 4 + 1. 09 treadcon
Predict fire-simulation consumption when treadmill consumption is 40.
- c. How effectively does a straight line summarize the relationship?
- d. Delete the first observation, (51.3, 49.3), and calculate the new equation of the least-squares line and the value of r2. What do you conclude? (Hint: For the original data, Σx = 388.8, Σy = 310 .3, Σx2 = 15,338.54, Σxy = 12,306.58, and Σy2 = 10,072.41.)
a.
Discuss whether the scatterplot indicates an approximate linear relationship.
Answer to Problem 78CR
No, the scatterplot does not indicate an approximate linear relationship.
Explanation of Solution
The data relates the oxygen consumption (milliliters per kilogram per minute) of 10 firefighters during a fire-suppression simulation, y to that during a treadmill test, x. The scatterplot between the two variables is given.
Denote the estimated response variable as
A careful inspection of the given scatterplot shows that the points do not fall on a straight line. Rather, the points are scattered almost in a random manner, without showing any pattern in particular. However, there is one extreme point, which is far away from the remaining points. This extreme point appears to provide an impression that there might be a linear relationship between the two variables. Once this point is ignored, it is clear that no such relationship can be determined.
Thus, the scatterplot does not indicate an approximate linear relationship.
b.
Predict the fire-simulation oxygen consumption, if the treadmill oxygen consumption is 40.
Answer to Problem 78CR
The fire-simulation oxygen consumption, when the treadmill oxygen consumption is 40 is 32.254 milliliters per kilogram per minute.
Explanation of Solution
Calculation:
The MINITAB output for the fitting of a least-squares regression line to the given data is given.
In the given output, the column of “Coef” gives the coefficients corresponding to the variables given in the column of “Predictor”. The term “Constant” under the column of ‘Predictor’ gives the intercept of the equation; the term “treadcon” denotes the oxygen consumption of during the treadmill test, x.
Using the values in the output, the equation of the least-squares regression line is
For a treadmill oxygen consumption of 40 milliliters per kilogram per minute,
Thus, the fire-simulation oxygen consumption, when the treadmill oxygen consumption is 40 is 32.254 milliliters per kilogram per minute.
c.
Explain the effectivity of the straight line to summarize the relationship between the variables.
Explanation of Solution
In the given output, the value of
Now,
Thus, it can be interpreted that the oxygen consumption during the treadmill test can predict about 59.5% of the variability in the oxygen consumption during the fire-suppression simulation.
This suggests that the straight line is moderately effective in summarizing the relationship between the variables.
d.
Find the equation of the least-squares line and the value of
Answer to Problem 78CR
The equation of the least-squares line after deleting the first observation, (51.3, 49.3) is
The value of
Explanation of Solution
Calculation:
It is given that, for the original data set,
For the first observation,
Now, the lest-squares regression line is of the form:
Using this formula and the values obtained above, b and a are respectively obtained as follows:
Now,
Thus,
Using the values of a and b obtained above, the equation of the least-squares line after deleting the first observation, (51.3, 49.3) is
Now, it is known that the slope for the least-squares regression of y on x, that is, b can be given by the formula:
Now, it can be shown that:
Similarly,
Thus,
Using the values obtained above, the value of r can be calculated as follows:
It is known that
Hence, the value of
Now,
Now,
Thus, it can be interpreted that the oxygen consumption during the treadmill test can predict about 2% of the variability in the oxygen consumption during the fire-suppression simulation, which is a very low percentage.
Thus, the model 9is not a very good fit for the data.
Want to see more full solutions like this?
Chapter 5 Solutions
Introduction To Statistics And Data Analysis
Additional Math Textbook Solutions
Statistics for Business & Economics, Revised (MindTap Course List)
Elementary Statistics
Basic Business Statistics, Student Value Edition
Introductory Statistics (2nd Edition)
Elementary Statistics: A Step By Step Approach
Essentials of Statistics, Books a la Carte Edition (5th Edition)
- A deficiency of the trace element selenium in the diet can negatively impact growth, immunity, muscle and neuromuscular function, and fertility. The introduction of selenium supplements to dairy cows is justified when pastures have low selenium levels. Authors of a research paper supplied the following data on milk selenium concentration (mg/L) for a sample of cows given a selenium supplement (the treatment group) and a control sample given no supplement, both initially and after a 9-day period. Initial Measurement Treatment Control 11.2 9.1 9.6 8.7 10.1 9.7 8.5 10.8 10.3 10.9 10.6 10.6 11.7 10.1 9.7 12.3 10.8 8.8 10.3 10.4 10.4 10.9 11.2 10.4 9.4 11.6 10.6 10.9 10.7 8.4 After 9 Days Treatment Control 138.3 9.3 104 8.7 96.4 8.7 89 10.1 88 9.6 103.8 8.6 147.3 10.2 97.1 12.2 172.6 9.3 146.3 9.5 99 8.2 122.3 8.9 103 12.5 117.8 9.1 121.5 93 (a) Use the given data for the treatment group to determine if there…arrow_forwardA methodological study had established values for the MIC on a scale that measured physical function: The MIC for improvement (higher scores) was 4.0, and the MIC for deterioration (lower scores) was 3.0. Lawrence studied clinically significant change in physical functioning over a 1-year period for a sample of 100 patients with COPD. Some change score information is presented below for 10 patients. Which patients experienced clinically significant change in physical function in the 12-month period between assessments? Patient Baseline Score* 12-Month Score* 1 19 15 2 12 10 3 16 14 4 17 16 5 9 10 6 11 12 7 13 17 8 15 13 9 18 14 10 16 9 *Higher scores = higher level of physical function Which patients had clinically significant deterioration? Which patients had clinically significant improvement? Which patients had no clinically significant change?arrow_forwardA study published by Babcock and Marks (2010) showed that the average full-time U.S. college student studied for μ = 14 hours per week (SD = 4.8 hours per week) in 2005. We want to know if this average has changed in the past 15 years. In other words, we are going to do a study in which we try to determine whether there has been an impact of the passage of time on the amount of time college students spend studying. We selected a sample of n = 64 of today’s college students and find that they spent an average of M = 12.5 hours per week studying. Does this sample indicate a significant change in the number of hours spent studying? Use a two-tailed test (this means non-directional hypothesis) with α = .05.arrow_forward
- Researchers compared protein intake among three groups of postmenopausal women: (1) women eating a standard American diet (STD), (2) women eating a lacto- ovo-vegetarian diet (LAC), and (3) women eating a strict vegetarian diet (VEG). The mean ± 1 sd for protein intake (mg) is presented in Table 12.29. Table 12.29 Protein intake (mg) among three dietary groups of postmenopausal women Group Мean sd STD 75 9 10 LAC 57 13 10 VEG 47 17 *12.1 Perform a statistical procedure to compare the means of the three groups using the critical-value method. *12.2 What is the p-value from the test performed in Prob- lem 12.1?arrow_forwardAn experiment was conducted to investigate leaking current in a SOS MOSFETS device. The purpose of the experiment was to investigate how leakage current varies as the channel length changes. Four channel lengths were selected. For each channel length, five different widths were also used, and width is to be considered a nuisance factor. The data are as follows: (a) Test the hypothesis that mean leakage voltage does not depend on the channel length using a= 0.05. (b) Analyze the residuals from this experiment. Comment on the residual plots. (c) The observed leakage voltage for channel length 4 and width 5 was erroneously recorded. The correct observation is 4.0. Analyze the corrected data from this experiment. Is there evidence to conclude that mean leakage voltage increases with channel length?arrow_forwardJ 2arrow_forward
- A regression model to predict Y, the state burglary rate per 100,000 people, used the following four state predictors: X₁ = median age, X₂ = number of bankruptcies per 1.000 population, X3 = federal expenditures per capita (a leading predictor), and X4 = high school graduation percentage. Click here for the Excel Data File (a) Using the sample size of 50 people, calculate the calc and p-value in the table given below. (Negative values should be indicated by a minus sign. Leave no cells blank - be certain to enter "0" wherever required. Round your answers to 4 decimal places.) Predictor Intercept AgeMed Bankrupt FedSpend HSGrad% Answer is complete but not entirely correct. *calc 5.2526 -2.1764✔✔ 1.4101✔ Coefficient 4,198.5808 -27.3540 17.4893 -0.0124 -29.0314 SE 799.3395 12.5687 12.4033 0.0176 7.1268 -0.7045 -4.0736 p-value 0.0000 0.0348 0.2935 0.4848 0.0002arrow_forwardHoaglin, Mosteller, and Tukey (1983) presented data on blood levels of beta-endorphin as a function of stress. They took beta-endorphin levels for 19 patients 12 hours before surgery and again 10 minutes before surgery. The data are presented below, in fmol/ml Based on these data, what effect does increased stressed have on endorphin levels. include: the hypotheses tested (H0 and H1), the test-statistic and its df, the p-value of the test, and the conclusion as it relates to the research question. Participant 12 hours before 10 minutes before 1 10 6.5 2 6.5 14.0 3 8.0 13.5 4 12 18 5 5.0 14.5 6 11.5…arrow_forwardTo illustrate the effects of driving under the influence (DUI) of alcohol, a police officer brought a DUI simulator to a local high school. Student reaction time in an emergency was measured with unimpaired vision and also while wearing a pair of special goggles to simulate the effects of alcohol on vision. For a random sample of nine teenagers, the time (in seconds) required to bring the vehicle to a stop from a speed of 60 miles per hour was recorded. Complete parts (a) and (b). Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers. Click the icon to view the data table. (a) Whether the student had unimpaired vision or wore goggles first was randomly selected. Why is this a good idea in designing the experiment? A. This is a good idea in designing the experiment because it controls for any "learning" that may occur in using the simulator. B. This is a good idea in designing the experiment because…arrow_forward
- To illustrate the effects of driving under the influence (DUI) of alcohol, a police officer brought a DUI simulator to a local high school. Student reaction time in an emergency was measured with unimpaired vision and also while wearing a pair of special goggles to simulate the effects of alcohol on vision. For a random sample of nine teenagers, the time (in seconds) required to bring the vehicle to a stop from a speed of 60 miles per hour was recorded. Complete parts (a) and (b). Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers. Click the icon to view the data table. (a) Whether the student had unimpaired vision or wore goggles first was randomly selected. Why is this a good idea in designing the experiment? O A. This is a good idea in designing the experiment because the sample size is not large enough. B. This is a good idea in designing the experiment because it controls for any "learning"…arrow_forwardTo illustrate the effects of driving under the influence (DUI) of alcohol, a police officer brought a DUI simulator to a local high school. Student reaction time in an emergency was measured with unimpaired vision and also while wearing a pair of special goggles to simulate the effects of alcohol on vision. For a random sample of nine teenagers, the time (in seconds) required to bring the vehicle to a stop from a speed of 60 miles per hour was recorded. Complete parts (a) and (b). Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers. (a) Whether the student had unimpaired vision or wore goggles first was randomly selected. Why is this a good idea in designing the experiment? A.This is a good idea in designing the experiment because the sample size is not large enough. B. This is a good idea in designing the experiment because it controls for any "learning" that may occur in…arrow_forwardTo illustrate the effects of driving under the influence (DUI) of alcohol, a police officer brought a DUI simulator to a local high school. Student reaction time in an emergency was measured with unimpaired vision and also while wearing a pair of special goggles to simulate the effects of alcohol on vision. For a random sample of nine teenagers, the time (in seconds) required to bring the vehicle to a stop from a speed of 60 miles per hour was recorded. Complete parts (a) and (b). Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers. Click the icon to view the data table. (a) Whether the student had unimpaired vision or wore goggles first was randomly selected. Why is this a good idea in designing the experiment? A. This is a good idea in designing the experiment because reaction times are different. B. This is a good idea in designing the experiment because the sample size is not large enough. C.…arrow_forward
- Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw Hill