The table below shows the life expectancy of a person related to their birth year

Question

Year of birth, x= 1900, 1910, 1920, 1930, 1940, 1950, 1960
Life expectancy in years, why equals 47.3, 50.0, 54.1, 59.7, 62.9, 68.2,69.7

A. Find the equation for the line of best fit
B. Estimate the life expectancy for someone born in the year 1998

Answer 1

To find the line of best fit for the life expectancy data and to make predictions, we will first express the data in a more convenient form.

A. Finding the Equation for the Line of Best Fit

1. Convert the Year of Birth to a Numerical Scale (x):
   We can use the year itself or convert it to a more manageable scale. For simplicity, let's use the years as they are but remember that they are increasing. \[ x = \begin{bmatrix} 1900 \ 1910 \ 1920 \ 1930 \ 1940 \ 1950 \ 1960 \end{bmatrix} \] The corresponding life expectancy values (y): \[ y = \begin{bmatrix} 47.3 \ 50.0 \ 54.1 \ 59.7 \ 62.9 \ 68.2 \ 69.7 \end{bmatrix} \]

2. Calculate the Line of Best Fit: We will use the least-squares method to find the equation of the line \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept.

   Using a calculator or performing the calculations manually, we find:

   • Number of data points (\(n\)): 7
   • \(\sum x = 1900 + 1910 + 1920 + 1930 + 1940 + 1950 + 1960 = 13600\)
   • \(\sum y = 47.3 + 50.0 + 54.1 + 59.7 + 62.9 + 68.2 + 69.7 = 412.9\)
   • \(\sum xy = 1900 \cdot 47.3 + 1910 \cdot 50.0 + 1920 \cdot 54.1 + 1930 \cdot 59.7 + 1940 \cdot 62.9 + 1950 \cdot 68.2 + 1960 \cdot 69.7 = 107955.7\)
   • \(\sum x^2 = 1900^2 + 1910^2 + 1920^2 + 1930^2 + 1940^2 + 1950^2 + 1960^2 = 26176000\)

   Using the formulas for the slope \( m \) and intercept \( b \): \[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \] \[ b = \frac{(\sum y)(\sum x^2) - (\sum x)(\sum xy)}{n(\sum x^2) - (\sum x)^2} \]

   Plugging the values into the formulas: \[ m = \frac{7(107955.7) - (13600)(412.9)}{7(26176000) - (13600)^2} \approx 0.01079 \] \[ b = \frac{(412.9)(26176000) - (13600)(107955.7)}{7(26176000) - (13600)^2} \approx - 19340.27 \]

   Thus, the equation of the line of best fit is: \[ y \approx 0.01079x - 19340.27 \]

B. Estimate the Life Expectancy for Someone Born in the Year 1998

To estimate the life expectancy for someone born in 1998, we can simply substitute \( x = 1998 \) into our line equation:

\[ y = 0.01079(1998) - 19340.27 \]

Calculating this gives us:

\[ y \approx 0.01079(1998) - 19340.27 \approx 67.68 \]

Thus, the estimated life expectancy for someone born in 1998 is approximately 67.68 years.

Conclusion

A) The equation for the line of best fit is \( y \approx 0.01079x - 19340.27 \).
B) The estimated life expectancy for someone born in 1998 is approximately 67.68 years.

Answer 2

Use the point slope formula for a

Answer 3

The point-slope formula for a linear equation is given by:

\[ y - y_1 = m(x - x_1) \]

where:

• \( (x_1, y_1) \) is a known point on the line,
• \( m \) is the slope of the line.

Step 1: Calculate the slope (\( m \))

From the original data provided, we can use the first two points to calculate the slope:

1. Select two points:

   • First point: \( (1900, 47.3) \) → \((x_1, y_1) = (1900, 47.3)\)
   • Second point: \( (1910, 50.0) \) → \((x_2, y_2) = (1910, 50.0)\)

2. Calculate the slope (\( m \)):

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{50.0 - 47.3}{1910 - 1900} = \frac{2.7}{10} = 0.27 \]

Now we have the slope \( m = 0.27 \).

Step 2: Use the Point-Slope Formula

Now, we can use the point-slope formula with one of our points (let's use \( (1900, 47.3) \)):

Using the point \( (1900, 47.3) \):

\[ y - 47.3 = 0.27(x - 1900) \]

Step 3: Rearranging the equation to the slope-intercept form (optional)

To convert it to the slope-intercept form \( y = mx + b \):

1. Distributing \( 0.27 \):

\[ y - 47.3 = 0.27x - 0.27 \cdot 1900 \] \[ y - 47.3 = 0.27x - 513 \]

2. Adding \( 47.3 \) to both sides:

\[ y = 0.27x - 513 + 47.3 \] \[ y = 0.27x - 465.7 \]

Conclusion

Thus, the equation of the line using the point-slope formula starting from the point \( (1900, 47.3) \) is:

\[ y - 47.3 = 0.27(x - 1900) \]

In slope-intercept form, it can also be written as:

\[ y = 0.27x - 465.7 \]

This equation can now also be used to estimate the life expectancy for a person born in 1998 by substituting \( x = 1998 \).

Estimating Life Expectancy for 1998

Substituting \( x = 1998 \):

\[ y = 0.27(1998) - 465.7 \] \[ y = 539.46 - 465.7 \approx 73.76 \]

So using this method, the estimated life expectancy for someone born in 1998 is approximately 73.76 years.

Answer 4

What was the life expectancy for someone born in 1987 be

Answer 5

To estimate the life expectancy for someone born in 1987 using the derived equation from the point-slope form, we can use the equation we found earlier:

\[ y = 0.27x - 465.7 \]

Substitute \( x = 1987 \):

\[ y = 0.27(1987) - 465.7 \]

Calculating:

1. Calculate \( 0.27 \times 1987 \):

\[ 0.27 \times 1987 = 534.51 \]

2. Substitute back into the equation:

\[ y = 534.51 - 465.7 \] \[ y = 68.81 \]

Conclusion

The estimated life expectancy for someone born in 1987 is approximately 68.81 years.

Answer 6

What about 1984

Answer 7

To estimate the life expectancy for someone born in 1984 using the derived equation from earlier, we will use:

\[ y = 0.27x - 465.7 \]

Substitute \( x = 1984 \):

\[ y = 0.27(1984) - 465.7 \]

Calculating step-by-step:

1. Calculate \( 0.27 \times 1984 \):

\[ 0.27 \times 1984 = 535.68 \]

2. Substitute back into the equation:

\[ y = 535.68 - 465.7 \] \[ y = 69.98 \]

Conclusion

The estimated life expectancy for someone born in 1984 is approximately 69.98 years.

Answer 8

What about 2009

Answer 9

To estimate the life expectancy for someone born in 2009 using the equation:

\[ y = 0.27x - 465.7 \]

Substitute \( x = 2009 \):

\[ y = 0.27(2009) - 465.7 \]

Calculating step-by-step:

1. Calculate \( 0.27 \times 2009 \):

\[ 0.27 \times 2009 = 542.43 \]

2. Substitute back into the equation:

\[ y = 542.43 - 465.7 \] \[ y = 76.73 \]

Conclusion

The estimated life expectancy for someone born in 2009 is approximately 76.73 years.

Answer 10

What about 2020

Answer 11

To estimate the life expectancy for someone born in 2020 using the equation:

\[ y = 0.27x - 465.7 \]

Substitute \( x = 2020 \):

\[ y = 0.27(2020) - 465.7 \]

Calculating step-by-step:

1. Calculate \( 0.27 \times 2020 \):

\[ 0.27 \times 2020 = 545.4 \]

2. Substitute back into the equation:

\[ y = 545.4 - 465.7 \] \[ y = 79.7 \]

Conclusion

The estimated life expectancy for someone born in 2020 is approximately 79.7 years.