Let's analyze the content of the uploaded image step by step:

The image includes:

1. A table showing U.S. life expectancy at birth for selected years.
2. A scatter plot of life expectancy as a function of years after 1950.
3. The task to determine the equation of the regression line and use it to predict life expectancy for specific years.

Step-by-Step Solution

Part (b) - Select the equation for the regression line.

The regression line equation is $y = mx + b$, where:

• $m$ is the slope,
• $b$ is the y-intercept.

From the table:

• The slope, $m$, is calculated as: $m = \frac{\Delta y}{\Delta x} = \frac{77 - 68.2}{50 - 0} = \frac{8.8}{50} = 0.176$

Thus, the regression equation is: $y = 0.176x + 68.2$

This matches the choice $y = 0.18x + 68.2$ (rounded to two decimal places).

Part (c) - Estimate life expectancy for 1985.

Here, $t = \text{years after 1950} = 1985 - 1950 = 35$.

Using the regression equation: $y = 0.176(35) + 68.2$ $y = 6.16 + 68.2 = 74.36$

The estimated life expectancy for 1985 is 74.36 years.

Part (d) - Predict life expectancy for 2010.

Here, $t = \text{years after 1950} = 2010 - 1950 = 60$.

Using the regression equation: $y = 0.176(60) + 68.2$ $y = 10.56 + 68.2 = 78.76$

The predicted life expectancy for 2010 is 78.76 years.

Would you like a detailed breakdown of how the regression line was calculated? Let me know if you'd like assistance with other questions!

Here are 5 related questions to consider:

1. How is the slope of a regression line calculated in general?
2. What does the slope represent in the context of life expectancy over time?
3. How accurate is the prediction for years far beyond the data range (e.g., 2050)?
4. How would predictions differ if the slope were higher or lower?
5. What could cause life expectancy to deviate from the predicted trend?

Tip: Always verify the fit of the regression line by assessing the scatter plot visually to ensure the trend line matches the data closely.