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Word Problems: Linear Regression
Linear Regression is a process by which the equation of a line is found that “best fits” a given set of data. The line of best fit approximates the best linear representation for your data. One very important aspect of a regression line is the relationship between the equation and the “science quantity” often represented by the slope of the line.
 
In order to solve problems involving linear regression, it is necessary to
 
 
Let's look at an example of linear regression by examining the data in the following table to discover the relationship between temperatures measured in Celsius (Centigrade) and Fahrenheit. [Remember that lines are named using the convention y vs. x whereas data tables are constructed as x | y.]
 
Fahrenheit
degrees
(ºF)
Celsius
degrees
(ºC)
32
0
68
20
86
30
122
50
158
70
194
90
212
100
 
Based on this data:
 
  1. interpolate the equivalent temperature in degrees Celsius of our body temperature, 98.6 °F
  2. relate the linear equation of your model with its associated science formula to determine the "physical" meaning of the slope of this data's trend line
 
Step 1: First we will plot the data using a TI-83 graphing calculator. We will enter the data measured in degrees Fahrenheit in L1 and the temperatures measured in degrees Celsius in L2. Once the data is entered, your screen should look like the following:
 
 
After entering the data into the calculator, graph the data. The Fahrenheit data, listed in L1, represents the x-axis, and the Celsius data, listed in L2, represents the y-axis. Your screen should look like the following:
 
 
Step Two: Now we need to find a linear equation that models the data we have plotted. According to the calculator, our equation has the following properties:
 
 
Our linear equation (rounded to thousandths) that best fits our data is
 
 
Step Three: The graph of the function is shown below.
 
 
Based on the graph and the equation information listed above, our correlation coefficient (r) is equal to 1. That means that our data perfectly models a linear function.
 
Step Four: Using the model from step two and the graph on our calculator from step three, we can trace along the graph and determine what temperature in degrees Celsius equals 98.6 °F, our body temperature.
 
 
This screen capture shows us that 98.6 °F (x-value) is equivalent to 37 °C (y-value).
 
Step Five: Consider our equation . The accepted formula used to convert Fahrenheit degrees to Celsius degrees is typically written as
 
 
If we distribute the fraction , we have
 


 
Expressed in this form we can clearly see that our model's equation is indeed the same equation conventionally used to convert temperatures between these two measuring scales. Since our line's slope [] is a decimal, we know that the size of a "degree" on the two temperature scales is not the same; that is, these scales are not in a one-to-one correspondence -- 1 Celsius degree (Cº) does not equal 1 Fahrenheit degree (Fº).
 
Examining the slope in fraction form [] we can clearly see that the relationship between the two scales is such that from a given point on the line, you move up five degrees on the Celsius scale and right nine degrees on the Fahrenheit scale to arrive at the next point on the line. Or equivalently, when the temperature changes 9 Fº it only changes 5 Cº.
 
Additionally, the y-intercept for our equation which tells us that when the temperature is 0 °F the temperature is -27.778 °C. The y-intercept is the point where the graph crosses the y-axis

Let's Practice

Question #1
Audio
Obesity is a risk factor for the development of medical problems such as high blood pressure, high cholesterol, heart disease, and diabetes. How much a person can safely weigh depends on his or her height. Body Mass Index (BMI) is a method of examining a person's weight in accordance with his or her height.
 
 
 
According to Roche Pharmaceuticals, in Australia, a BMI of 30 or greater can create an increased risk of developing medical problems associated with obesity. The chart below shows height (in inches) and weight (in pounds) for individuals who have a BMI of 30.
 
Height
(inches)
Weight
(pounds)
61
160
63
170
65
180
67
190
69
200
72
220
73
230
 
Based on this data:
 
  1. extrapolate the weight of a 75-inch tall person who has a BMI = 30
  2. discover the "physical" meaning of the slope of the data's trend line



Question #2
Audio
The table below lists distances in megaparsecs (3.09 x 1022 meters) and velocities in km/sec (1 x 103 m/sec) of four galaxies moving rapidly away from the earth:
 
Distance
(Mpc)
Velocity
(km/sec)
Virgo
15
1600
Ursa Minor
200
15,000
Corona Borealis
290
24,000
Bootes
520
40,000
Source: Astronomical Methods and Calculations (1994)
 
  1. Use this data to extrapolate the velocity of Hydra, a galaxy located 776 megaparsecs from earth.
  2. Use the equation of your trend line and dimensional analysis to ascertain the "physical" meaning of its slope.

 



Try These
Question #1
Audio
A ball is rolled down a hallway and its position is recorded at five different times. Use the data given in the table shown below to predict the location of the ball at 12 seconds.
 
 
Time
(seconds)
Position
(meters)
1
9
2
12
4
17
6
21
8
26


Question #2
Audio
The data in the table below represents the apparent temperature (ºF) vs. the relative humidity (%) in a room whose actual temperature is 72 °F. Use this data to predict the apparent temperature when the relative humidity reaches 110%.
 
Relative
Humidity
(%)
Apparent Temperature
(ºF)
0
64
10
65
20
67
30
68
40
70
50
71
60
72
70
73
80
74
90
75
100
76



As you can see, this type of problem requires that you use a graphing calculator and a modeling approach. You must correctly enter the data into your calculator, graph the data, calculate an equation that best fits the data, graph that equation, and then make the prediction asked for in the problem. To better understand the data's behavior, you should endeavor to find the relationship between the equation of the line and the “physical” meaning if its slope. At the conclusion of the problem, you should always check for the reasonableness of your solutions.

D Saye

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