Let's break this problem down step by step.

Given Information:

We have a least-squares regression equation provided as: $y = 83.30 - 0.49x$ Where:

• $x$ represents the birthrate (number of births per 1000 people),
• $y$ represents the female life expectancy in years.

Part (a) - Observed Female Life Expectancy at Birthrate of 31.1:

You are asked to find the observed life expectancy at a birthrate of 31.1 births per 1000 people, from the given data table.

From the data provided: $x = 31.1, \quad y = 64.6$ So, the observed female life expectancy when the birthrate was 31.1 is 64.6 years.

Part (b) - Predicted Female Life Expectancy Using the Regression Equation:

Now, we will use the regression equation to predict the female life expectancy when the birthrate is $x = 31.1$.

The regression equation is: $y = 83.30 - 0.49x$

Substitute $x = 31.1$ into the equation: $y = 83.30 - 0.49(31.1)$ $y = 83.30 - 15.239$ $y \approx 68.06 \, \text{years}$

Thus, the predicted female life expectancy when the birthrate is 31.1 is approximately 68.1 years.

Summary of Answers:

(a) 64.6 years (observed from the data). (b) 68.1 years (predicted using the regression equation).

Would you like more details or have any further questions?

Related Questions:

1. How do we interpret the slope and intercept of the regression equation in real-life terms?
2. How do outliers affect the least-squares regression line?
3. What would be the predicted life expectancy for a birthrate of 40 births per 1000 people?
4. How accurate is the regression model in this context?
5. What are some common methods for testing the goodness-of-fit for a regression model?

Tip:

When interpreting regression equations, the slope tells you how much the dependent variable (life expectancy) changes for each unit increase in the independent variable (birthrate).