1. Measurements and Linear Regression 1.1 Introduction The objective of this lab assignment is to represent measurement data in graphical form in order to illustrate experimental data and uncertainty visually. It is often convenient to represent experimental data graphically, not only for reporting results but also to compute or measure several physical parameters. For example, consider two physical quantities represented by x and y that are linearly related according to the algebraic relationship, y=mx+b, (1.1) where m is the slope of the line and b is the y-intercept. In order to assess the linearity between y and x, it is convenient to plot these quantities in a y versus x graph, as shown in Figure 1.1. Datapoints Line of regression Figure 1.1: Best fit line example. Once the data points are plotted, it is necessary to draw a "best fit line" or "regression line" that describes the data. A best fit line is a straight line that is the best approximation of the given set of data, and it is used to study the nature of the relation between the two variables. A best fit line can be roughly determined using an eyeball method by drawing a straight line on a scatter plot so that the number of points above the line and below the line is about equal. To calculate the slope, two random points along the best fit line, with coordinates (x1,y1) and (x2,y2), must be chosen. The slope of the line is equal to the ratio of the vertical and horizontal intervals, m= Ay y-yı Axx-x (1.2) Points should be chosen relatively far apart on the line. For best results, points corresponding to data points should not be chosen, even if they appear to lie on the line. The best fit line and its slope can also be determined automatically using computer software like Excel or similar. See the following video for instructions: https://www.youtube.com/watch?v=d65jx4BhslA Chapter 1. Measurements and Linear Regression 1.2 Procedure According to Hubble's Law, galaxies are moving away from the Earth at velocities proportional to their distance. In other words, the further they are the faster they are moving away from Earth. Mathematically, Hubble's law can be expressed as, v=Hod, (1.3) where v is the speed of the galaxy, d is its distance from Earth and Ho is the proportionality constant, commonly known as Hubble constant. Equation (1.3) can be plotted as a straight line in the general form y=mx, where yv and x-d. The slope of the line is Hubble constant, m = Ho. Table 1.1 contains the measured speed and distance data for different galaxies. The speed is expressed in km/s and the distance is expressed in megaparsec (1 Mpc=3.09 x 10 km). As a result, Hubble constant is most frequently quoted in (km/s)/Mpc. 1. Plot v versus d and determine the best fit line. Attach the graph in a separate page. 2. Calculate the slope of the best fit line, Le. the experimental value of Hubble constant. 3. Calculate the percent error of your result. The theoretical value of Hubble constant is 73.9 (km/s)/Mpc. 1.3 Data Distance d Galaxy (Mpc) Speed v (km/s) UGC 7412 14 1140 UGC 9358 25 1903 UGC 3834 29 2033 UGC 858 36 2375 UGC 1633 52 4247 Table 1.1: Galaxy velocity and distance data. 1.4 Results • Experimental result for Hubble constant: H₁ = Percent error of the result for Hubble constant: Percent Error Experimental Theoretical | x 100% Theoretical 1. Measurements and Linear Regression 1.1 Introduction The objective of this lab assignment is to represent measurement data in graphical form in order to illustrate experimental data and uncertainty visually. It is often convenient to represent experimental data graphically, not only for reporting results but also to compute or measure several physical parameters. For example, consider two physical quantities represented by x and y that are linearly related according to the algebraic relationship, y=mx+b, (1.1) where m is the slope of the line and b is the y-intercept. In order to assess the linearity between y and x, it is convenient to plot these quantities in a y versus x graph, as shown in Figure 1.1. Datapoints Line of regression Figure 1.1: Best fit line example. Once the data points are plotted, it is necessary to draw a "best fit line" or "regression line" that describes the data. A best fit line is a straight line that is the best approximation of the given set of data, and it is used to study the nature of the relation between the two variables. A best fit line can be roughly determined using an eyeball method by drawing a straight line on a scatter plot so that the number of points above the line and below the line is about equal. To calculate the slope, two random points along the best fit line, with coordinates (x1,y1) and (x2,y2), must be chosen. The slope of the line is equal to the ratio of the vertical and horizontal intervals, m= Ay y-yı Axx-x (1.2) Points should be chosen relatively far apart on the line. For best results, points corresponding to data points should not be chosen, even if they appear to lie on the line. The best fit line and its slope can also be determined automatically using computer software like Excel or similar. See the following video for instructions: https://www.youtube.com/watch?v=d65jx4BhslA Chapter 1. Measurements and Linear Regression 1.2 Procedure According to Hubble's Law, galaxies are moving away from the Earth at velocities proportional to their distance. In other words, the further they are the faster they are moving away from Earth. Mathematically, Hubble's law can be expressed as, v=Hod, (1.3) where v is the speed of the galaxy, d is its distance from Earth and Ho is the proportionality constant, commonly known as Hubble constant. Equation (1.3) can be plotted as a straight line in the general form y=mx, where yv and x-d. The slope of the line is Hubble constant, m = Ho. Table 1.1 contains the measured speed and distance data for different galaxies. The speed is expressed in km/s and the distance is expressed in megaparsec (1 Mpc=3.09 x 10 km). As a result, Hubble constant is most frequently quoted in (km/s)/Mpc. 1. Plot v versus d and determine the best fit line. Attach the graph in a separate page. 2. Calculate the slope of the best fit line, Le. the experimental value of Hubble constant. 3. Calculate the percent error of your result. The theoretical value of Hubble constant is 73.9 (km/s)/Mpc. 1.3 Data Distance d Galaxy (Mpc) Speed v (km/s) UGC 7412 14 1140 UGC 9358 25 1903 UGC 3834 29 2033 UGC 858 36 2375 UGC 1633 52 4247 Table 1.1: Galaxy velocity and distance data. 1.4 Results • Experimental result for Hubble constant: H₁ = Percent error of the result for Hubble constant: Percent Error Experimental Theoretical | x 100% Theoretical
1. Measurements and Linear Regression 1.1 Introduction The objective of this lab assignment is to represent measurement data in graphical form in order to illustrate experimental data and uncertainty visually. It is often convenient to represent experimental data graphically, not only for reporting results but also to compute or measure several physical parameters. For example, consider two physical quantities represented by x and y that are linearly related according to the algebraic relationship, y=mx+b, (1.1) where m is the slope of the line and b is the y-intercept. In order to assess the linearity between y and x, it is convenient to plot these quantities in a y versus x graph, as shown in Figure 1.1. Datapoints Line of regression Figure 1.1: Best fit line example. Once the data points are plotted, it is necessary to draw a "best fit line" or "regression line" that describes the data. A best fit line is a straight line that is the best approximation of the given set of data, and it is used to study the nature of the relation between the two variables. A best fit line can be roughly determined using an eyeball method by drawing a straight line on a scatter plot so that the number of points above the line and below the line is about equal. To calculate the slope, two random points along the best fit line, with coordinates (x1,y1) and (x2,y2), must be chosen. The slope of the line is equal to the ratio of the vertical and horizontal intervals, m= Ay y-yı Axx-x (1.2) Points should be chosen relatively far apart on the line. For best results, points corresponding to data points should not be chosen, even if they appear to lie on the line. The best fit line and its slope can also be determined automatically using computer software like Excel or similar. See the following video for instructions: https://www.youtube.com/watch?v=d65jx4BhslA Chapter 1. Measurements and Linear Regression 1.2 Procedure According to Hubble's Law, galaxies are moving away from the Earth at velocities proportional to their distance. In other words, the further they are the faster they are moving away from Earth. Mathematically, Hubble's law can be expressed as, v=Hod, (1.3) where v is the speed of the galaxy, d is its distance from Earth and Ho is the proportionality constant, commonly known as Hubble constant. Equation (1.3) can be plotted as a straight line in the general form y=mx, where yv and x-d. The slope of the line is Hubble constant, m = Ho. Table 1.1 contains the measured speed and distance data for different galaxies. The speed is expressed in km/s and the distance is expressed in megaparsec (1 Mpc=3.09 x 10 km). As a result, Hubble constant is most frequently quoted in (km/s)/Mpc. 1. Plot v versus d and determine the best fit line. Attach the graph in a separate page. 2. Calculate the slope of the best fit line, Le. the experimental value of Hubble constant. 3. Calculate the percent error of your result. The theoretical value of Hubble constant is 73.9 (km/s)/Mpc. 1.3 Data Distance d Galaxy (Mpc) Speed v (km/s) UGC 7412 14 1140 UGC 9358 25 1903 UGC 3834 29 2033 UGC 858 36 2375 UGC 1633 52 4247 Table 1.1: Galaxy velocity and distance data. 1.4 Results • Experimental result for Hubble constant: H₁ = Percent error of the result for Hubble constant: Percent Error Experimental Theoretical | x 100% Theoretical 1. Measurements and Linear Regression 1.1 Introduction The objective of this lab assignment is to represent measurement data in graphical form in order to illustrate experimental data and uncertainty visually. It is often convenient to represent experimental data graphically, not only for reporting results but also to compute or measure several physical parameters. For example, consider two physical quantities represented by x and y that are linearly related according to the algebraic relationship, y=mx+b, (1.1) where m is the slope of the line and b is the y-intercept. In order to assess the linearity between y and x, it is convenient to plot these quantities in a y versus x graph, as shown in Figure 1.1. Datapoints Line of regression Figure 1.1: Best fit line example. Once the data points are plotted, it is necessary to draw a "best fit line" or "regression line" that describes the data. A best fit line is a straight line that is the best approximation of the given set of data, and it is used to study the nature of the relation between the two variables. A best fit line can be roughly determined using an eyeball method by drawing a straight line on a scatter plot so that the number of points above the line and below the line is about equal. To calculate the slope, two random points along the best fit line, with coordinates (x1,y1) and (x2,y2), must be chosen. The slope of the line is equal to the ratio of the vertical and horizontal intervals, m= Ay y-yı Axx-x (1.2) Points should be chosen relatively far apart on the line. For best results, points corresponding to data points should not be chosen, even if they appear to lie on the line. The best fit line and its slope can also be determined automatically using computer software like Excel or similar. See the following video for instructions: https://www.youtube.com/watch?v=d65jx4BhslA Chapter 1. Measurements and Linear Regression 1.2 Procedure According to Hubble's Law, galaxies are moving away from the Earth at velocities proportional to their distance. In other words, the further they are the faster they are moving away from Earth. Mathematically, Hubble's law can be expressed as, v=Hod, (1.3) where v is the speed of the galaxy, d is its distance from Earth and Ho is the proportionality constant, commonly known as Hubble constant. Equation (1.3) can be plotted as a straight line in the general form y=mx, where yv and x-d. The slope of the line is Hubble constant, m = Ho. Table 1.1 contains the measured speed and distance data for different galaxies. The speed is expressed in km/s and the distance is expressed in megaparsec (1 Mpc=3.09 x 10 km). As a result, Hubble constant is most frequently quoted in (km/s)/Mpc. 1. Plot v versus d and determine the best fit line. Attach the graph in a separate page. 2. Calculate the slope of the best fit line, Le. the experimental value of Hubble constant. 3. Calculate the percent error of your result. The theoretical value of Hubble constant is 73.9 (km/s)/Mpc. 1.3 Data Distance d Galaxy (Mpc) Speed v (km/s) UGC 7412 14 1140 UGC 9358 25 1903 UGC 3834 29 2033 UGC 858 36 2375 UGC 1633 52 4247 Table 1.1: Galaxy velocity and distance data. 1.4 Results • Experimental result for Hubble constant: H₁ = Percent error of the result for Hubble constant: Percent Error Experimental Theoretical | x 100% Theoretical
Glencoe Physics: Principles and Problems, Student Edition
1st Edition
ISBN:9780078807213
Author:Paul W. Zitzewitz
Publisher:Paul W. Zitzewitz
Chapter1: A Physics Toolkit
Section1.3: Measurement
Problem 19SSC
Related questions
Question
Transcribed Image Text:1. Measurements and Linear Regression
1.1 Introduction
The objective of this lab assignment is to represent measurement data in graphical form in order to
illustrate experimental data and uncertainty visually. It is often convenient to represent experimental
data graphically, not only for reporting results but also to compute or measure several physical
parameters. For example, consider two physical quantities represented by x and y that are linearly
related according to the algebraic relationship,
y=mx+b,
(1.1)
where m is the slope of the line and b is the y-intercept. In order to assess the linearity between y
and x, it is convenient to plot these quantities in a y versus x graph, as shown in Figure 1.1.
Datapoints
Line of
regression
Figure 1.1: Best fit line example.
Once the data points are plotted, it is necessary to draw a "best fit line" or "regression line" that
describes the data. A best fit line is a straight line that is the best approximation of the given set of
data, and it is used to study the nature of the relation between the two variables. A best fit line can
be roughly determined using an eyeball method by drawing a straight line on a scatter plot so that
the number of points above the line and below the line is about equal. To calculate the slope, two
random points along the best fit line, with coordinates (x1,y1) and (x2,y2), must be chosen. The
slope of the line is equal to the ratio of the vertical and horizontal intervals,
m=
Ay y-yı
Axx-x
(1.2)
Points should be chosen relatively far apart on the line. For best results, points corresponding
to data points should not be chosen, even if they appear to lie on the line. The best fit line and its
slope can also be determined automatically using computer software like Excel or similar. See the
following video for instructions: https://www.youtube.com/watch?v=d65jx4BhslA
Chapter 1. Measurements and Linear Regression
1.2 Procedure
According to Hubble's Law, galaxies are moving away from the Earth at velocities proportional
to their distance. In other words, the further they are the faster they are moving away from Earth.
Mathematically, Hubble's law can be expressed as,
v=Hod,
(1.3)
where v is the speed of the galaxy, d is its distance from Earth and Ho is the proportionality constant,
commonly known as Hubble constant. Equation (1.3) can be plotted as a straight line in the general
form y=mx, where yv and x-d. The slope of the line is Hubble constant, m = Ho. Table 1.1
contains the measured speed and distance data for different galaxies. The speed is expressed in
km/s and the distance is expressed in megaparsec (1 Mpc=3.09 x 10 km). As a result, Hubble
constant is most frequently quoted in (km/s)/Mpc.
1. Plot v versus d and determine the best fit line. Attach the graph in a separate page.
2. Calculate the slope of the best fit line, Le. the experimental value of Hubble constant.
3. Calculate the percent error of your result. The theoretical value of Hubble constant is
73.9 (km/s)/Mpc.
1.3 Data
Distance d
Galaxy
(Mpc)
Speed v
(km/s)
UGC 7412
14
1140
UGC 9358
25
1903
UGC 3834
29
2033
UGC 858
36
2375
UGC 1633
52
4247
Table 1.1: Galaxy velocity and distance data.
1.4 Results
• Experimental result for Hubble constant:
H₁ =
Percent error of the result for Hubble constant:
Percent Error
Experimental Theoretical |
x 100%
Theoretical
Transcribed Image Text:1. Measurements and Linear Regression
1.1 Introduction
The objective of this lab assignment is to represent measurement data in graphical form in order to
illustrate experimental data and uncertainty visually. It is often convenient to represent experimental
data graphically, not only for reporting results but also to compute or measure several physical
parameters. For example, consider two physical quantities represented by x and y that are linearly
related according to the algebraic relationship,
y=mx+b,
(1.1)
where m is the slope of the line and b is the y-intercept. In order to assess the linearity between y
and x, it is convenient to plot these quantities in a y versus x graph, as shown in Figure 1.1.
Datapoints
Line of
regression
Figure 1.1: Best fit line example.
Once the data points are plotted, it is necessary to draw a "best fit line" or "regression line" that
describes the data. A best fit line is a straight line that is the best approximation of the given set of
data, and it is used to study the nature of the relation between the two variables. A best fit line can
be roughly determined using an eyeball method by drawing a straight line on a scatter plot so that
the number of points above the line and below the line is about equal. To calculate the slope, two
random points along the best fit line, with coordinates (x1,y1) and (x2,y2), must be chosen. The
slope of the line is equal to the ratio of the vertical and horizontal intervals,
m=
Ay y-yı
Axx-x
(1.2)
Points should be chosen relatively far apart on the line. For best results, points corresponding
to data points should not be chosen, even if they appear to lie on the line. The best fit line and its
slope can also be determined automatically using computer software like Excel or similar. See the
following video for instructions: https://www.youtube.com/watch?v=d65jx4BhslA
Chapter 1. Measurements and Linear Regression
1.2 Procedure
According to Hubble's Law, galaxies are moving away from the Earth at velocities proportional
to their distance. In other words, the further they are the faster they are moving away from Earth.
Mathematically, Hubble's law can be expressed as,
v=Hod,
(1.3)
where v is the speed of the galaxy, d is its distance from Earth and Ho is the proportionality constant,
commonly known as Hubble constant. Equation (1.3) can be plotted as a straight line in the general
form y=mx, where yv and x-d. The slope of the line is Hubble constant, m = Ho. Table 1.1
contains the measured speed and distance data for different galaxies. The speed is expressed in
km/s and the distance is expressed in megaparsec (1 Mpc=3.09 x 10 km). As a result, Hubble
constant is most frequently quoted in (km/s)/Mpc.
1. Plot v versus d and determine the best fit line. Attach the graph in a separate page.
2. Calculate the slope of the best fit line, Le. the experimental value of Hubble constant.
3. Calculate the percent error of your result. The theoretical value of Hubble constant is
73.9 (km/s)/Mpc.
1.3 Data
Distance d
Galaxy
(Mpc)
Speed v
(km/s)
UGC 7412
14
1140
UGC 9358
25
1903
UGC 3834
29
2033
UGC 858
36
2375
UGC 1633
52
4247
Table 1.1: Galaxy velocity and distance data.
1.4 Results
• Experimental result for Hubble constant:
H₁ =
Percent error of the result for Hubble constant:
Percent Error
Experimental Theoretical |
x 100%
Theoretical
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