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6.8 Fitting exponential models to data  (Page 4/12)

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Using logarithmic regression to fit a model to data

Due to advances in medicine and higher standards of living, life expectancy has been increasing in most developed countries since the beginning of the 20th century.

[link] shows the average life expectancies, in years, of Americans from 1900–2010 Source: Center for Disease Control and Prevention, 2013 .

Year 1900 1910 1920 1930 1940 1950
Life Expectancy(Years) 47.3 50.0 54.1 59.7 62.9 68.2
Year 1960 1970 1980 1990 2000 2010
Life Expectancy(Years) 69.7 70.8 73.7 75.4 76.8 78.7
  1. Let   x   represent time in decades starting with   x = 1   for the year 1900,   x = 2   for the year 1910, and so on. Let   y   represent the corresponding life expectancy. Use logarithmic regression to fit a model to these data.
  2. Use the model to predict the average American life expectancy for the year 2030.
  1. Using the STAT then EDIT menu on a graphing utility, list the years using values 1–12 in L1 and the corresponding life expectancy in L2. Then use the STATPLOT feature to verify that the scatterplot follows a logarithmic pattern as shown in [link] :
    Graph of a scattered plot.

    Use the “LnReg” command from the STAT then CALC menu to obtain the logarithmic model,

    y = 42.52722583 + 13.85752327 ln ( x )

    Next, graph the model in the same window as the scatterplot to verify it is a good fit as shown in [link] :

    Graph of a scattered plot with an estimation line.
  2. To predict the life expectancy of an American in the year 2030, substitute   x = 14   for the in the model and solve for   y :
    y = 42.52722583 + 13.85752327 ln ( x ) Use the regression model found in part (a) . = 42.52722583 + 13.85752327 ln ( 14 ) Substitute 14 for  x . ≈ 79 .1 Round to the nearest tenth.

    If life expectancy continues to increase at this pace, the average life expectancy of an American will be 79.1 by the year 2030.

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Sales of a video game released in the year 2000 took off at first, but then steadily slowed as time moved on. [link] shows the number of games sold, in thousands, from the years 2000–2010.

Year 2000 2001 2002 2003 2004 2005
Number Sold (thousands) 142 149 154 155 159 161
Year 2006 2007 2008 2009 2010 -
Number Sold (thousands) 163 164 164 166 167 -
  1. Let   x   represent time in years starting with   x = 1   for the year 2000. Let   y   represent the number of games sold in thousands. Use logarithmic regression to fit a model to these data.
  2. If games continue to sell at this rate, how many games will sell in 2015? Round to the nearest thousand.
  1. The logarithmic regression model that fits these data is   y = 141.91242949 + 10.45366573 ln ( x )  
  2. If sales continue at this rate, about 171,000 games will be sold in the year 2015.
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Building a logistic model from data

Like exponential and logarithmic growth, logistic growth increases over time. One of the most notable differences with logistic growth models is that, at a certain point, growth steadily slows and the function approaches an upper bound, or limiting value . Because of this, logistic regression is best for modeling phenomena where there are limits in expansion, such as availability of living space or nutrients.

It is worth pointing out that logistic functions actually model resource-limited exponential growth. There are many examples of this type of growth in real-world situations, including population growth and spread of disease, rumors, and even stains in fabric. When performing logistic regression analysis , we use the form most commonly used on graphing utilities:

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
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what is the change in momentum of a body?
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Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
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A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
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8m/s²
Aishat
What is Thermodynamics
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velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
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50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
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I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
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Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
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What is specific heat capacity
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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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