Problem 67 - Prealgebra

Question

The table shows the heat index and relative humidity for an air temperature of \(75^{\circ} \mathrm{F}\). $$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|} \hline \text { Eacture ty uster } & 0 \% & 5 \% & 10 \% & 15 \% & 20 \% & 25 \% & 30 \% & 35 \% & 40 \% & 45 \% & 50 \% \\ \hline \text { Lesures }(x) & 69 & 69 & 70 & \pi & 72 & 72 & 73 & 73 & 74 & 74 & 75 \\ \hline \end{array}$$ Make a scatter plot and draw a line of fit.

Step-by-Step Solution

Answer

Plot the points on a graph and draw a slightly increasing linear line that fits the data trend, balancing points above and below it.

Step 1: Construct the Data Table

The given table provides heat index and relative humidity data for an air temperature of \(75^{\circ} \mathrm{F}\). The independent variable \(x\) represents the relative humidity percentage, and the dependent variable \(y\) represents the heat index. The data points are: \((0, 69), (5, 69), (10, 70), (15, 71), (20, 72), (25, 72), (30, 73), (35, 73), (40, 74), (45, 74), (50, 75)\).

Step 2: Create the Scatter Plot

Plot each data point on a coordinate graph where the x-axis represents the relative humidity as a percentage and the y-axis represents the heat index. Mark each point clearly: (0, 69), (5, 69), (10, 70), (15, 71), (20, 72), (25, 72), (30, 73), (35, 73), (40, 74), (45, 74), (50, 75).

Step 3: Estimate and Draw the Line of Fit

Observe the plotted points to determine the line that best represents the trend of the data. In this case, the general trend appears to be slowly increasing. Choose two points that align closely and estimate a linear line of fit by visually positioning a straight line that balances the points above and below it.

Key Concepts

Data Table

A data table is an organized way to display information that allows you to view relationships and trends in the data easily. The data table from our exercise shows how the heat index varies with changes in relative humidity at a constant air temperature of \(75^{\circ} \mathrm{F}\).

In this table:

• The first row lists the relative humidity percentages.
• The second row shows the corresponding heat index values for each relative humidity point.

By presenting data this way, it becomes simple to refer to specific points, such as how a 25% humidity corresponds to a heat index of 72. Data tables are crucial for displaying numerical information clearly.

Creating a data table involves organizing data points, usually with columns and rows, to present information about relationships between different variables. In our example, one row represents a specific condition (humidity), while another provides the response (heat index). This structured format makes comparing individual data points more intuitive and assists in further analysis, such as creating scatter plots or finding trends.

Line of Best Fit

A line of best fit, or trend line, is a straight line drawn through the center of a group of data points on a scatter plot. This line aims to represent the data's overall trend. It provides a visual representation of the relationship between the independent and dependent variables. In our example, as you plot each data point, you will observe an upward trend in the data.

When estimating a line of best fit:

• The line should balance the data points, ensuring that there is an equal number of points on either side of the line, as much as possible.
• It is an estimation, so it might not touch all points, but it should be close to the central tendency of the data.

Creating a line of best fit helps in making predictions and understanding the data's behavior. For example, it could predict the heat index for an unlisted humidity percentage.

Independent and Dependent Variables

Understanding independent and dependent variables is crucial for analyzing data and identifying cause-effect relationships. In our heat index example, relative humidity is the independent variable, while the heat index is the dependent variable.

• Independent Variable: This is the variable you change directly, or it is the factor that influences changes in another variable. Here, it's the relative humidity percentage, controlled or altered to observe its impact on another aspect of the data.
• Dependent Variable: This variable changes in response to the independent variable. The heat index is dependent because its changes are measured based on variations in humidity.

Differentiating these variables helps in setting up equations and models to predict outcomes. Recognizing the roles of these variables when analyzing data is fundamental to understanding how changes in one aspect can affect another.