Questions: The following table shows the personal income per capita for Georgia in for years since 1980 through 2000. Year GA income 3 10857 6 13917 9 16658 12 19139 15 22023 18 25680 21 28523 25 31121 30 35490 A scatter plot for the data with a graph of a linear model in red follows. Use the above scatter plot to decide whether a linear model fits the data. A linear model fits the data A linear model does not fit the data well. Let I(t) be the per capita income in GA in dollars for t years since 1980. A linear model for the data is I(t) = . Use three decimals in your answer. Estimate the per capita income in 2014. . Round your answer to dollars. Use the model to predict the year in which the income will be 38800.

The following table shows the personal income per capita for Georgia in for years since 1980 through 2000.

Year GA income
3 10857
6 13917
9 16658
12 19139
15 22023
18 25680
21 28523
25 31121
30 35490

A scatter plot for the data with a graph of a linear model in red follows.

Use the above scatter plot to decide whether a linear model fits the data.

A linear model fits the data

A linear model does not fit the data well.

Let I(t) be the per capita income in GA in dollars for t years since 1980. A linear model for the data is I(t) = . Use three decimals in your answer.

Estimate the per capita income in 2014. . Round your answer to dollars.

Use the model to predict the year in which the income will be 38800.

Transcript text: The following table shows the personal income per capita for Georgia in for years since 1980 through 2000. \begin{tabular}{|r|r|} \hline Year & GA income \\ \hline 3 & 10857 \\ \hline 6 & 13917 \\ \hline 9 & 16658 \\ \hline 12 & 19139 \\ \hline 15 & 22023 \\ \hline 18 & 25680 \\ \hline 21 & 28523 \\ \hline 25 & 31121 \\ \hline 30 & 35490 \\ \hline \end{tabular} A scatter plot for the data with a graph of a linear model in red follows. Use the above scatter plot to decide whether a linear model fits the data. A linear model fits the data A linear model does not fit the data well. Let $I(t)$ be the per capita income in GA in dollars for $t$ years since 1980 . A linear model for the data is $I(t)=$ $\square$ . Use three decimals in your answer. Estimate the per capita income in 2014. \$ $\square$ Round your answer to dollars. Use the model to predict the year in which the income will be $\$ 38800$. $\square$

Solution

Solution Steps

Step 1: Determine if a Linear Model Fits the Data

The scatter plot shows a clear linear trend, indicating that a linear model fits the data well.

Step 2: Formulate the Linear Model

To find the linear model $ I(t) $, we need to determine the slope (m) and y-intercept (b) of the line.
Using two points from the table, for example, (3, 10857) and (30, 35490): \[ m = \frac{35490 - 10857}{30 - 3} = \frac{24633}{27} \approx 912.333 \]
Using the point-slope form $ y - y_1 = m(x - x_1) $ with point (3, 10857): \[ I(t) - 10857 = 912.333(t - 3) \] \[ I(t) = 912.333t + 10857 - 2737 \approx 912.333t + 8120 \]

Step 3: Estimate the Per Capita Income in 2014

2014 is 34 years since 1980, so $ t = 34 $: \[ I(34) = 912.333 \times 34 + 8120 \approx 31019 + 8120 = 39139 \]
The estimated per capita income in 2014 is $39,139.