Exemplo 2 de Algebra and Trigonometry, 6.8 Fitting Exponential Models to Data

Using Logarithmic Regression to Fit a Model to Data

Problema e solução em inglês, como no livro original.

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.

Table 3 shows the average life expectancies, in years, of Americans from 1900–201010.

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

Table 3

ⓐ 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.
ⓑ Use the model to predict the average American life expectancy for the year 2030.

Resolva no papel primeiro. Depois abra a solução um passo de cada vez e pare assim que conseguir terminar sozinho.

1
Solução
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 Figure 3:
  
  Figure 3
  
  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 Figure 4:
  
  Figure 4
2. ⓑ To predict the life expectancy of an American in the year 2030, substitute $x = 14$ for the in the model and solve for $y :$
  $\begin{array}{l} y & = 42.52722583 + 13.85752327 \ln (x) & Use the regression model found in part (a) . \\ = 42.52722583 + 13.85752327 \ln (14) & Substitute 14 for x . \\ \approx 79 .1 & Round to the nearest tenth. \end{array}$
  If life expectancy continues to increase at this pace, the average life expectancy of an American will be 79.1 by the year 2030.

Sua vez

O Try It do livro logo depois deste exemplo: mesma ideia, outros números. Só a resposta é dada.

Sales of a video game released in the year 2000 took off at first, but then steadily slowed as time moved on. Table 4 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	-

Table 4

ⓐ 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.
ⓑ If games continue to sell at this rate, how many games will sell in 2015? Round to the nearest thousand.

Como foi?

Sua vez

O Try It do livro logo depois deste exemplo: mesma ideia, outros números. Só a resposta é dada.

Sales of a video game released in the year 2000 took off at first, but then steadily slowed as time moved on. Table 4 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	-

Table 4

ⓐ 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.
ⓑ If games continue to sell at this rate, how many games will sell in 2015? Round to the nearest thousand.

Solução

Sua vez

Solução

Sua vez