Stewart Precalc 6e Chapter 1 Focus on Modeling: Fitting Lines to Data

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Stewart Precalc 6e Chapter 1 Focus on Modeling: Fitting Lines to Data 0/16
1 Linear Regression - Application · Level 3
*Femur Length and Height* Anthropologists use a linear model that relates femur length to height. The model allows an anthropologist to determine the height of an individual when only a partial skeleton (including the femur) is found. The data on femur length and height (both in cm) for eight males are: (50.1, 178.5), (48.3, 173.6), (45.2, 164.8), (44.7, 163.7), (44.5, 168.3), (42.7, 165.0), (39.5, 155.4), (38.0, 155.8). *(a)* Make a scatter plot of the data. *(b)* Find and graph a linear function that models the data. *(c)* An anthropologist finds a femur of length 58 cm. How tall was the person?
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2 Linear Regression - Application · Level 3
*Demand for Soft Drinks* A convenience store manager notices that sales of soft drinks are higher on hotter days, so he assembles data relating temperature (in \(^{\circ}\)F) to soft drink sales. *(a)* Make a scatter plot of the data. *(b)* Find and graph a linear function that models the data.
3 Exercise - Linear Regression Application · Level 2
Tree Diameter and Age. To estimate ages of trees, forest rangers use a linear model that relates tree diameter to age. The model is useful because tree diameter is much easier to measure than tree age (which requires special tools for extracting a representative cross section of the tree and counting the rings). To find the model, use the data in the table, which were collected for a certain variety of oaks.
(a) Make a scatter plot of the data.
(b) Find and graph a linear function that models the data.
(c) Use the model to estimate the age of an oak whose diameter is 18 in. Data (Diameter in inches, Age in years): (2.5, 15), (4.0, 24), (6.0, 32), (8.0, 56), (9.0, 49), (9.5, 76), (12.5, 90), (15.5, 89).

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4 Exercise - Linear Regression Application · Level 2
Carbon Dioxide Levels. The Mauna Loa Observatory, located on the island of Hawaii, has been monitoring carbon dioxide (CO₂) levels in the atmosphere since 1958. The table lists the average annual CO₂ levels measured in parts per million (ppm) from 1984 to 2006.
(a) Make a scatter plot of the data.
(b) Find and graph the regression line.
(c) Use the linear model in part (b) to estimate the CO₂ level in the atmosphere in 2005. Compare your answer with the actual CO₂ level of 379.7 that was measured in 2005. Data (Year, CO₂ level in ppm): (1984, 344.3), (1986, 347.0), (1988, 351.3), (1990, 354.0), (1992, 356.3), (1994, 358.9), (1996, 362.7), (1998, 366.5), (2000, 369.4), (2002, 372.0), (2004, 377.5), (2006, 380.9).

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5 Exercise - Linear Regression Application · Level 2
Temperature and Chirping Crickets. Biologists have observed that the chirping rate of crickets of a certain species appears to be related to temperature. The table shows the chirping rates for various temperatures.
(a) Make a scatter plot of the data.
(b) Find and graph the regression line.
(c) Use the linear model in part (b) to estimate the chirping rate at 100°F. Data (Temperature in °F, Chirping rate in chirps/min): (50, 20), (55, 46), (60, 79), (65, 91), (70, 113), (75, 140), (80, 173), (85, 198), (90, 211).

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6 Exercise - Linear Regression Application · Level 2
Extent of Arctic Sea Ice. The National Snow and Ice Data Center monitors the amount of ice in the Arctic year round. The table gives approximate values for the sea ice extent in millions of square kilometers from 1980 to 2006, in two-year intervals.
(a) Make a scatter plot of the data.
(b) Find and graph the regression line.
(c) Use the linear model in part (b) to estimate the ice extent in the year 2010. Data (Year, Ice extent in million km²): (1980, 7.9), (1982, 7.4), (1984, 7.2), (1986, 7.6), (1988, 7.5), (1990, 6.2), (1992, 7.6), (1994, 7.1), (1996, 7.9), (1998, 6.6), (2000, 6.3), (2002, 6.0), (2004, 6.1), (2006, 5.7).

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7 Exercise - Linear Regression Application · Level 2
Mosquito Prevalence. The table lists the relative abundance of mosquitoes (as measured by the mosquito positive rate) versus the flow rate (measured as a percentage of maximum flow) of canal networks in Saga City, Japan.
(a) Make a scatter plot of the data.
(b) Find and graph the regression line.
(c) Use the linear model in part (b) to estimate the mosquito positive rate if the canal flow is 70% of maximum. Data (Flow rate in %, Mosquito positive rate in %): (0, 22), (10, 16), (40, 12), (60, 11), (90, 6), (100, 2).

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8 Exercise - Linear Regression with Correlation · Level 3
Noise and Intelligibility. Audiologists study the intelligibility of spoken sentences under different noise levels. Intelligibility, the MRT score, is measured as the percent of a spoken sentence that the listener can decipher at a certain noise level in decibels (dB). The table shows the results of one such test.
(a) Make a scatter plot of the data.
(b) Find and graph the regression line.
(c) Find the correlation coefficient. Is a linear model appropriate?
(d) Use the linear model in part (b) to estimate the intelligibility of a sentence at a 94-dB noise level. Data (Noise level in dB, MRT score in %): (80, 99), (84, 91), (88, 84), (92, 70), (96, 47), (100, 23), (104, 11).

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9 Exercise - Linear Regression Application · Level 2
Life Expectancy. The average life expectancy in the United States has been rising steadily over the past few decades, as shown in the table.
(a) Make a scatter plot of the data.
(b) Find and graph the regression line.
(c) Use the linear model you found in part (b) to predict the life expectancy in the year 2006.
(d) Search the Internet or your campus library to find the actual 2006 average life expectancy. Compare to your answer in part (c). Data (Year, Life expectancy): (1920, 54.1), (1930, 59.7), (1940, 62.9), (1950, 68.2), (1960, 69.7), (1970, 70.8), (1980, 73.7), (1990, 75.4), (2000, 76.9).

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10 Exercise - Restricted Domain Regression · Level 3
Olympic Pole Vault. The graph in Figure 7 indicates that in recent years the winning Olympic men's pole vault height has fallen below the value predicted by the regression line in Example 2. This might have occurred because when the pole vault was a new event, there was much room for improvement in vaulters' performances, whereas now even the best training can produce only incremental advances. Let's see whether concentrating on more recent results gives a better predictor of future records.
(a) Use the data in Table 2 to complete the table of winning pole vault heights. (Note that we are using \(x = 0\) to correspond to the year 1972, where this restricted data set begins.)
(b) Find the regression line for the data in part (a).
(c) Plot the data and the regression line on the same axes. Does the regression line seem to provide a good model for the data?
(d) What does the regression line predict as the winning pole vault height for the 2008 Olympics? Compare this predicted value to the actual 2008 winning height of 5.96 m, as described on page 133. Has this new regression line provided a better prediction than the line in Example 2? Partial data (Year, \(x\), Height in m): (1972, 0, 5.64), (1976, 4, fill in from Table 2), (1980, 8, fill in), (1984, 12, fill in), (1988, 16, fill in), (1992, 20, fill in), (1996, 24, fill in), (2000, 28, fill in), (2004, 32, fill in).

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11 Exercise - Comparative Regression · Level 3
Olympic Swimming Records. The tables give the gold medal times in the men's and women's 100-m freestyle Olympic swimming event.
(a) Find the regression lines for the men's data and the women's data.
(b) Sketch both regression lines on the same graph. When do these lines predict that the women will overtake the men in the event? Does this conclusion seem reasonable? Men's data (Year, Time in s): (1908, 65.6), (1912, 63.4), (1920, 61.4), (1924, 59.0), (1928, 58.6), (1932, 58.2), (1936, 57.6), (1948, 57.3), (1952, 57.4), (1956, 55.4), (1960, 55.2), (1964, 53.4), (1968, 52.2), (1972, 51.22), (1976, 49.99), (1980, 50.40), (1984, 49.80), (1988, 48.63), (1992, 49.02), (1996, 48.74), (2000, 48.30), (2004, 48.17), (2008, 47.21). Women's data (Year, Time in s): (1912, 82.2), (1920, 73.6), (1924, 72.4), (1928, 71.0), (1932, 66.8), (1936, 65.9), (1948, 66.3), (1952, 66.8), (1956, 62.0), (1960, 61.2), (1964, 59.5), (1968, 60.0), (1972, 58.59), (1976, 55.65), (1980, 54.79), (1984, 55.92), (1988, 54.93), (1992, 54.64), (1996, 54.50), (2000, 53.83), (2004, 53.84), (2008, 53.12).

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12 Exercise - Survey and Correlation · Level 1
Shoe Size and Height. Do you think that shoe size and height are correlated? Find out by surveying the shoe sizes and heights of people in your class. (Of course, the data for men and women should be separate.) Find the correlation coefficient.
13 Exercise - Linear Demand Equation · Level 2
Demand for Candy Bars. In this problem you will determine a linear demand equation that describes the demand for candy bars in your class. Survey your classmates to determine what price they would be willing to pay for a candy bar.
(a) Make a table of the number of respondents who answered "yes" at each price level.
(b) Make a scatter plot of your data.
(c) Find and graph the regression line \(y = m p + b\) which gives the number of respondents \(y\) who would buy a candy bar if the price were \(p\) cents. This is the demand equation. Why is the slope \(m\) negative?
(d) What is the \(p\)-intercept of the demand equation? What does this intercept tell you about pricing candy bars? Survey prices: 30¢, 40¢, 50¢, 60¢, 70¢, 80¢, 90¢, \$1.00, \$1.10, \$1.20.

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14 Example - Regression Line · Level 2
*Regression Line for U.S. Infant Mortality Rates* Table 1 lists U.S. infant mortality rates per 1000 live births from 1950 to 2000. *(a)* Find the regression line for the infant mortality data in Table 1. *(b)* Graph the regression line on a scatter plot of the data. *(c)* Use the regression line to estimate the infant mortality rates in 1995 and 2006.
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15 Example - Regression Line · Level 2
*Regression Line for Olympic Pole Vault Records* Table 2 gives the men's Olympic pole vault records (gold medal winning heights, in meters) up to 2004. Let \(x = \) year \(- 1900\), so 1896 corresponds to \(x = -4\), 1900 to \(x = 0\), 1904 to \(x = 4\), and so on through 2004 \((x = 104)\). *(a)* Find the regression line for the data. *(b)* Make a scatter plot of the data, and graph the regression line. Does the regression line appear to be a suitable model for the data? *(c)* What does the slope of the regression line represent? *(d)* Use the model to predict the winning pole vault height for the 2008 Olympics.
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16 Example - Regression Line · Level 2
*Regression Line for Links Between Asbestos and Cancer* When laboratory rats are exposed to asbestos fibers, some of the rats develop lung tumors. Table 3 lists the results of several experiments by different scientists, giving asbestos exposure (in fibers/mL) and the percent of rats that develop lung tumors. The data are: (50, 2), (400, 6), (500, 5), (900, 10), (1100, 26), (1600, 42), (1800, 37), (2000, 28), (3000, 50). *(a)* Find the regression line for the data. *(b)* Make a scatter plot and graph the regression line. Does the regression line appear to be a suitable model for the data? *(c)* What does the \(y\)-intercept of the regression line represent?
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