Problems 17–22 use the results from Problems 27–32 in Section 4.1.
[DATA] Hurricanes (Refer to Problem 32, Section 4.1.) The following data represent the maximum wind speed (in knots) and atmospheric pressure (in millibars) for a random sample of hurricanes that originated in the Atlantic Ocean.
b. Explain what the slope represents.

Verified step by step guidance

1. Step 1: Understand the variables involved. Here, the explanatory variable (independent variable) is the atmospheric pressure (in millibars), and the response variable (dependent variable) is the wind speed (in knots).
2. Step 2: Recall that the slope in a linear regression equation represents the rate of change of the response variable with respect to the explanatory variable. In this context, the slope tells us how much the wind speed changes for each one-unit change in atmospheric pressure.
3. Step 3: Interpret the sign of the slope. If the slope is negative, it means that as atmospheric pressure decreases, wind speed increases, and vice versa. If the slope is positive, it means wind speed increases as atmospheric pressure increases.
4. Step 4: Formally, if the regression equation is \(\hat{y} = b_0 + b_1 x\), where \(\hat{y}\) is the predicted wind speed and \(x\) is the atmospheric pressure, then \(b_1\) (the slope) represents the change in predicted wind speed for each 1 millibar increase in atmospheric pressure.
5. Step 5: Summarize the interpretation: The slope quantifies the relationship between atmospheric pressure and wind speed, indicating how wind speed is expected to change as atmospheric pressure changes in the sample of hurricanes.

Key Concepts

Slope in Linear Regression

The slope in a linear regression model represents the rate of change in the response variable for each one-unit increase in the explanatory variable. In this context, it shows how much the wind speed changes for each unit change in atmospheric pressure. A positive slope indicates a direct relationship, while a negative slope indicates an inverse relationship.

Relationship Between Atmospheric Pressure and Wind Speed

Atmospheric pressure and wind speed in hurricanes are often inversely related; as pressure decreases, wind speed tends to increase. Understanding this relationship helps interpret the slope and predict wind speeds based on pressure readings, which is crucial for meteorological analysis and hurricane intensity estimation.

Interpreting Data from a Table for Statistical Analysis

Data tables provide raw values for variables under study, which are essential for calculating statistical measures like slope. Proper interpretation involves recognizing paired data points, understanding units, and preparing the data for analysis such as plotting or regression to uncover relationships between variables.