Question

1. the population of austin, texas has been growing exponentially since 1950. the scatterplot graphs years since 1950 versus the log(population of austin, tx). the equation for the least squares regression line shown is given by $hat{y}=5.0941 + 0.0172x$ where $x$ is years since 1950 and $y$ is log(population of austin, tx). a. interpret the meaning of 5.0941 in the least squares regression line equation b. interpret the meaning of 0.0172 in the least squares regression line equation. c. use the least squares regression line to predict the population of austin, texas for the year 2010.

Explanation:

Step1: Identify the regression - line form

The regression line is in the form $\hat{y}=a + bx$, where $\hat{y}=\log(\text{Population of Austin, TX})$, $a = 5.0941$, $b=0.0172$, and $x$ is years since 1950.

Step2: Interpret the y - intercept (5.0941)

When $x = 0$ (i.e., in the year 1950), $\hat{y}=5.0941$. Since $\hat{y}=\log(\text{Population of Austin, TX})$, the value 5.0941 is the predicted value of the logarithm (base - 10) of the population of Austin, TX in 1950.

Step3: Interpret the slope (0.0172)

The slope $b = 0.0172$ means that for each additional year (increase in $x$ by 1), the predicted value of $\log(\text{Population of Austin, TX})$ increases by 0.0172.

Step4: Calculate $x$ for 2010

For the year 2010, $x=2010 - 1950=60$.

Step5: Predict $\log(\text{Population})$

Substitute $x = 60$ into the regression equation: $\hat{y}=5.0941+0.0172\times60$.
$\hat{y}=5.0941 + 1.032=6.1261$.

Step6: Find the population

Since $\hat{y}=\log(\text{Population})$, then $\text{Population}=10^{\hat{y}}$. So, $\text{Population}=10^{6.1261}\approx1336338.9$.

Answer:

a. The value 5.0941 is the predicted value of the logarithm (base - 10) of the population of Austin, TX in 1950.
b. The value 0.0172 means that for each additional year since 1950, the predicted value of the logarithm (base - 10) of the population of Austin, TX increases by 0.0172.
c. The predicted population of Austin, TX in 2010 is approximately 1336339.