Problem 54 - Algebra 2

Question

Use the table that shows the life expectancy for people born in various years. \(\begin{array}{|c|c|c|c|c|c|}\hline 1950 & {1960} & {1970} & {1980} & {1990} & {2000} \\ \hline 68.2 & {69.7} & {70.8} & {73.7} & {75.4} & {7.0} \\\ \hline\end{array}\) Draw a scatter plot in which \(x\) is the number of years since 1940 and describe the correlation.

Step-by-Step Solution

Answer

There is a positive correlation in life expectancy from 1950 to 1990, but a decrease in 2000.

Step 1: Define the Year Conversion

Convert the given years into the number of years since 1940. For the years 1950, 1960, 1970, 1980, 1990, and 2000, subtract 1940 from each year to get 10, 20, 30, 40, 50, and 60, respectively.

Step 2: Organize the Converted Data

Create a table of the number of years since 1940 against life expectancies from the given data. This will be a series of points: (10, 68.2), (20, 69.7), (30, 70.8), (40, 73.7), (50, 75.4), and (60, 70.0).

Step 3: Plot the Scatter Plot

On a graph, mark the horizontal axis (x-axis) as the number of years since 1940 and the vertical axis (y-axis) as life expectancy. Plot each of the data points from the previous step on this graph.

Step 4: Analyze the Correlation

Examine the plotted points to identify the trend. The pattern of the points suggests an overall upward movement from 1950 to 1990; however, there is a drop in the year 2000. The trend suggests a positive correlation up to 1990, followed by an anomaly in 2000.

Key Concepts

Life Expectancy

Life expectancy is a statistical measure indicating how long an average person is expected to live, based on the year they are born, their current age, and other demographic factors. It is a crucial indicator of the overall health and social conditions of a country or region. Improved sanitation, medical advances, and better living conditions often lead to increased life expectancy. In tracking these changes over time, historians and demographers study patterns and trends that show how societal shifts have impacted longevity.

• In the context of the exercise, our data spans from 1950 to 2000.
• Noticeable variations showcase societal progress over these decades.

Understanding these changes can provide insights into economic and policy developments that have emerged over time. Disruptions, such as wars or pandemics, can also impact life expectancy figures temporarily or over longer periods.

Correlation

Correlation is a statistical term that describes how two variables move together. If they tend to increase or decrease together, they're said to have a positive correlation. If one increases while the other decreases, they have a negative correlation. In data analysis, understanding correlation helps determine if there is a significant relationship between the variables observed.

• In our exercise, we examine the correlation between time (years since 1940) and life expectancy.

By plotting life expectancy against years, we observe how these values move together over time, enabling us to describe the relationship. Initially, from 1950 to 1990, the life expectancy rises consistently, indicating a positive correlation. However, a drop in 2000 suggests changes in this pattern, showcasing that correlations can vary with changing circumstances.

Data Analysis

Data analysis involves processing and interpreting raw data to make it useful and meaningful. It involves organizing data, applying statistical techniques, and forming interpretations that provide insights.

• For the exercise at hand, we first convert the years into numbers representing years since 1940.
• This transformation allows us to handle data consistently on a time-based scale.

Organizing and plotting this data allows us to visually analyze trends, patterns, and anomalies. While most of the data shows an upward trend, the unexpected drop in the year 2000 prompts further investigation. Recognizing these patterns highlights the importance of thorough data analysis in identifying trends and understanding historical contexts.

Graph Interpretation

Graph interpretation is a skill vital for making sense of visual data representations, like scatter plots. When interpreting graphs, it is essential to understand both the story the data tells and any underlying patterns or exceptions.
In the given exercise, we translate raw data into a scatter plot, where each data point represents life expectancy in a specific year. The horizontal axis marks years since 1940, and the vertical axis denotes life expectancy values.

• This visualization helps to quickly assess trends over time.
• For instance, the graph shows a general positive increase in life expectancy until 1990.

The observation of a sharp drop in 2000 may signal a significant event or anomaly worth exploring further. Effective graph interpretation bridges the gap between data and comprehension, assisting in drawing meaningful conclusions from statistical observations.