Question

the u s census bureau takes a census every 10 years. the population in 2000 was estimated at 281.4 million people. a model created at the time predicted that the population grew at about 2% per year.
a how many people would the census bureau have expected to count in the 2005 census?
b if the census bureau had conducted a count in 2010, how many people would it have expected to count?
c the actual 2010 census put the population at 309 million people. what is the residual? what does it mean?

Explanation:

Step1: Identify the compound - growth formula

The compound - growth formula is $P = P_0(1 + r)^t$, where $P_0$ is the initial population, $r$ is the annual growth rate as a decimal, and $t$ is the number of years. Here, $P_0=281.4$ million, $r = 0.02$.

Step2: Calculate the expected population in 2005 (a)

For the year 2005, $t = 2005 - 2000=5$. Substitute the values into the formula: $P=281.4\times(1 + 0.02)^5$.
$P=281.4\times(1.02)^5$
$(1.02)^5=1.02\times1.02\times1.02\times1.02\times1.02 = 1.1040808032$.
$P=281.4\times1.1040808032\approx310.69$ million.

Step3: Calculate the expected population in 2010 (b)

For the year 2010, $t = 2010 - 2000 = 10$. Substitute into the formula: $P=281.4\times(1 + 0.02)^{10}$.
$(1.02)^{10}\approx1.218994419$.
$P=281.4\times1.218994419\approx343.92$ million.

Step4: Calculate the residual in 2010 (c)

The residual is the actual value minus the predicted value. The actual value in 2010 is $309$ million and the predicted value is approximately $343.92$ million.
Residual $=309 - 343.92=- 34.92$ million. It means that the model over - predicted the population in 2010 by about $34.92$ million people.

Answer:

a. Approximately $310.69$ million
b. Approximately $343.92$ million
c. Residual: $-34.92$ million. It means the model over - predicted the 2010 population by about $34.92$ million people.