9. The table below shows the estimated population of Ghana (in millions) rounded to three digits. \begin{tabular}{lllll} \hline Year & 2000 & 2005 & 2010 & 2015 \\ \hline Population & 227 & 237 & 249 & 262 \\ \hline \end{tabular} i. Transform the equation \( p(t)=\beta_{0} e^{\beta_{1} t} \), where \( \mathrm{t} \) is the time and \( \mathrm{p}(\mathrm{t}) \) is the population at time \( t \) into a suitable linear model. ii. Use matrix methods to find the least square regression estimate of \( \beta_{o} \) and \( \beta_{1} \) for the population data above. Write down the complete population model. iii. Predict the population of Ghana in 2030. iv. Find the standard error of estimate s associated with the least square regression line.
Added by Purificaci-N A.
Close
Step 1
This yields: \[ \ln(p(t)) = \ln(\beta_0 e^{\beta_1 t}) \] \[ \ln(p(t)) = \ln(\beta_0) + \ln(e^{\beta_1 t}) \] \[ \ln(p(t)) = \ln(\beta_0) + \beta_1 t \] Let \( y = \ln(p(t)) \), \( a = \ln(\beta_0) \), and \( x = t \). The linear model becomes: \[ y = a + Show more…
Show all steps
Your feedback will help us improve your experience
Supreeta N and 71 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Population Problem Population per Square Mile The table shows the populations per square mile P of land in the United States for selected years from 1790 to 2010. A model for the data is given by P = 4.110e^{0.0144t}, 0 ≤ t ≤ 220, where t is the year with t = 0 corresponding to 1790. (Source: U.S. Census Bureau) Year | Population per square mile, P 1890 | 17.8 1900 | 21.5 1910 | 26.0 1920 | 29.9 1930 | 34.7 1940 | 37.2 1950 | 42.6 1960 | 50.6 1970 | 57.5 1980 | 64.0 1990 | 70.3 2000 | 87.4 (a) Plot the data. (b) Calculate population per square mile using the model equation, and create a table that compares the actual data values with the values given by the model. (c) Does it appear that the model is a good fit for the data? Explain your reasoning. (d) Would you use the model to predict the population per square mile for future years? Explain your reasoning using both your graph and your calculations. (e) Use the model to predict the population per square mile in 2030. Does your answer seem reasonable?
Donna D.
Sri K.
The following data give the approximate population of China for selected years from 1900 until 2010: Year: 1900, 1950, 1970, 1980, 1990, 2000, 2010 Population (millions): 400, 557, 825, 981, 1135, 1266, 1370 Assume that the population growth can be modeled with an exponential function p = be^mx, where x is the year and p is the population in millions. Write the equation in a linear form (Section 6.3), and use linear least-squares regression to determine the constants b and m for which the function best fits the data. Use the equation to estimate the population in the year 1985.
T. L.
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD