Problem 74 - Brief Applied Calculus

Question

BIOMEDICAL SCIENCES: Life Expectancy The following tables give the life expectancy for a newborn child born in the indicated year. (Exercise 73 is for males, Exercise 74 for females.) $$ \begin{array}{llllll} \hline \text { Birth Year } & 1970 & 1980 & 1990 & 2000 & 2010 \\ \hline \begin{array}{l} \text { Life Expectancy } \\ \text { (female) } \end{array} & 74.7 & 77.4 & 78.8 & 79.3 & 80.8 \\ \hline \end{array} $$ a. Number the data columns with \(x\) -values \(1-5\) and use linear regression to fit a line to the data. State the regression formula. [Hint: See Example 8.] b. Interpret the slope of the line. From your answer, what is the yearly change in life expectancy? c. Use the regression line to predict life expectancy for a child born in 2025 (this might be your child or grandchild).

Step-by-Step Solution

Answer

The regression line is \( y = 4.02x + 66.14 \). Slope: 4.02 years/unit. Predicted life expectancy for 2025 is 88.25 years.

Step 1: Assign x-values to the data

For the given birth years, 1970, 1980, 1990, 2000, and 2010, we assign the values of \(x\) as 1, 2, 3, 4, and 5 respectively. This step helps to convert birth years into a consecutive numerical series that can be used in linear regression.

Step 2: Organize data for regression

Arrange the data in two columns: \(x\) values and life expectancy.\ \(x = [1, 2, 3, 4, 5]\)\ Life Expectancy (female) = [74.7, 77.4, 78.8, 79.3, 80.8]

Step 3: Calculate the regression line parameters

Linear regression formula is \(y = mx + c\). Calculate \(m\) (slope) and \(c\) (y-intercept) using these formulas:\[ m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \]\[ c = \frac{(\sum y) - m(\sum x)}{N} \]Here, \(x = [1, 2, 3, 4, 5]\) and Life Expectancy = [74.7, 77.4, 78.8, 79.3, 80.8].Substitute \(\sum x = 15\), \(\sum y = 391\), \(\sum xy = 1213.2\), \(\sum x^2 = 55\) and \(N = 5\) into the formulas.

Step 4: Perform calculations for m and c

Substitute the values into the formulas:\( m = \frac{5(1213.2) - (15)(391)}{5(55) - (15)^2} = \frac{6066 - 5865}{275 - 225} = \frac{201}{50} = 4.02 \)\( c = \frac{391 - 4.02(15)}{5} = \frac{391 - 60.3}{5} = 66.14 \)This gives us the regression line equation \( y = 4.02x + 66.14 \).

Step 5: Interpret the slope

The slope \(m = 4.02\) tells us that for each unit increase in \(x\) (representing a decade increase in years), the life expectancy for females increases by 4.02 years.

Step 6: Predict life expectancy for 2025

First, determine the \(x\) value for 2025. The base year is 1970, and \(x = 5\) corresponds to 2010, so: \(x_{2025} = 2025 - 1970 = 55\), \(x_{2010} = 40\), meaning 2025 is \(x = 5.5\).Use the regression equation \( y = 4.02x + 66.14 \):\( y_{2025} = 4.02(5.5) + 66.14 = 22.11 + 66.14 = 88.25 \).Therefore, the predicted life expectancy for a female born in 2025 is 88.25 years.

Key Concepts

Life Expectancy

Life expectancy is an important statistic that indicates the average duration a newborn is expected to live based on current mortality rates. It is a crucial measure for understanding the health and longevity of a population.
In the context of this linear regression exercise, we analyzed the life expectancy of females starting from 1970 up until 2010.
This exercise allows us to predict trends and make educated guesses about future life expectancies. The data set showed that life expectancy for females has gradually increased over decades, highlighting improvements in healthcare, nutrition, and general living conditions.

Slope Interpretation

When we talk about the slope in a linear regression model, we're referring to the rate at which one variable changes concerning another. In this exercise, the slope we calculated was 4.02.
This means that for every unit increase in our x-values (which in this case represent each subsequent decade), the life expectancy increases by 4.02 years.
This is a vital insight because it quantifies the average change over time, allowing us to understand the trend's nature. In practical terms, the increasing slope suggests that as the years move forward, female life expectancy has consistently risen.

Predictive Modeling

Predictive modeling involves using statistical techniques like linear regression to project future data points based on current and historical data.
In the exercise, we used the regression line equation derived to predict the life expectancy for females born in 2025.
The calculation led to a predicted life expectancy of 88.25 years for a female newborn in 2025, emphasizing how such models can offer foresight into future demographics.

• Helps in policymaking
• Encourages health sector advancements

Predictive modeling is a powerful tool in scientific research and public health as it helps in understanding potential futures and planning accordingly.