Let's break down the problem into the three main parts:

Part 1: Regression Line

You are given two sets of data points:

• Year (x): 0 (1900), 2 (1920), 4 (1940), 6 (1960), 8 (1980)
• Life Expectancy (y): 47.3, 49.6, 51.2, 52.4, 53.4

The correct regression line is calculated as:

$y = 0.750x + 47.78$

Thus, the correct option is D.

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Part 2: Coefficient of Correlation (r)

The coefficient of correlation measures how well the regression line fits the data. It is given as:

$r = 0.985$

This indicates a very strong positive correlation between the number of decades since 1900 and life expectancy.

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Part 3: Predict Life Expectancy in 2010

The year 2010 corresponds to $x = 11$ (since it is 11 decades after 1900). Using the regression line equation:

$y = 0.750x + 47.78$

Substituting $x = 11$:

$y = 0.750(11) + 47.78 = 8.25 + 47.78 = 56.03$

Rounding to one decimal place, the predicted life expectancy in 2010 is approximately 56.0 years.

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Summary:

1. The correct regression line is: $y = 0.750x + 47.78$.
2. The coefficient of correlation $r$ is 0.985.
3. The predicted life expectancy for the year 2010 is 56.0 years.

Would you like to go into further details on how these calculations are performed?

Here are 5 related questions to extend your understanding:

1. How do you calculate the slope (m) of a regression line?
2. What does the coefficient of correlation tell us about data trends?
3. How can the equation of the regression line be used for future predictions?
4. What happens if the correlation is negative?
5. How is the coefficient of determination (r²) related to the correlation?

Tip: The closer the correlation coefficient $r$ is to 1 or -1, the better the data fits a straight line.