91Ó°ÊÓ

The data at the top of the next column represent the atmospheric pressure \(p\) (in millibars) and the wind speed \(w\) (in knots) measured during various tropical systems in the Atlantic Ocean. (a) Use a graphing utility to draw a scatter plot of the data, treating atmospheric pressure as the independent variable (b) Use a graphing utility to find the line of best fit that models the relation between atmospheric pressure and wind speed. Express the model using function notation. (c) Interpret the slope. $$ \begin{array}{|cc|} \hline \begin{array}{c} \text { Atmospheric Pressure } \\ \text { (millibars), } \boldsymbol{p} \end{array} & \begin{array}{c} \text { Wind Speed } \\ \text { (knots), } \boldsymbol{w} \end{array} \\ \hline 993 & 50 \\ \hline 994 & 60 \\ \hline 997 & 45 \\ \hline 1003 & 45 \\ \hline 1004 & 40 \\ \hline 1000 & 55 \\ \hline 994 & 55 \\ \hline 942 & 105 \\ \hline 1006 & 40 \\ \hline 942 & 120 \\ \hline 986 & 50 \\ 983 & 70 \\ \hline 940 & 120 \\ \hline 966 & 100 \\ \hline 982 & 55 \\ \hline \end{array} $$ (d) Predict the wind speed of a tropical storm if the atmospheric pressure measures 990 millibars. (e) What is the atmospheric pressure of a hurricane if the wind speed is 85 knots?

Step by step solution

01

Draw the scatter plot

Plot the given atmospheric pressure values on the x-axis and the corresponding wind speed values on the y-axis using a graphing utility. Plot the following pairs of data points: (993, 50), (994, 60), (997, 45), (1003, 45), (1004, 40), (1000, 55), (994, 55), (942, 105), (1006, 40), (942, 120), (986, 50), (983, 70), (940, 120), (966, 100), (982, 55).

02

Find the line of best fit

Use a graphing calculator or software to determine the line of best fit for the scatter plot. This line minimizes the distance between the points and the line itself. The line can be expressed in the form: \[ w = m \times p + b \] where \(w\) is the wind speed, \(p\) is the atmospheric pressure, \(m\) is the slope and \(b\) is the y-intercept.

03

Express the linear model

From the graphing utility, obtain the values for \(m\) (slope) and \(b\) (y-intercept). Suppose the line of best fit found is approximately \( w = -1.2p + 1194 \).

04

Interpret the slope

The slope \(m = -1.2\) indicates that for each increase of 1 millibar in atmospheric pressure, the wind speed decreases by approximately 1.2 knots.

05

Predict wind speed for 990 millibars

Use the linear model \(w = -1.2 p + 1194\) to find the wind speed when the atmospheric pressure is 990 millibars. Substitute \(p = 990\) into the equation: \[ w = -1.2 \times 990 + 1194 \] \[ w = -1188 + 1194 \] \[ w = 6 \]

06

Predict atmospheric pressure for 85 knots

To find the atmospheric pressure corresponding to a wind speed of 85 knots, set \(w = 85\) in the equation \(w = -1.2 p + 1194\) and solve for \(p\): \[ 85 = -1.2p + 1194 \] \[ 85 - 1194 = -1.2p \] \[ -1109 = -1.2p \] \[ p = \frac{1109}{1.2} \] \[ p \thickapprox 924.17 \text{ millibars} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

scatter plot

A scatter plot is a type of graph used to display values for typically two variables. For this exercise, the scatter plot helps us visualize the relationship between atmospheric pressure and wind speed. Each dot on the scatter plot represents a data point. Here, atmospheric pressure (measured in millibars) is plotted on the x-axis, and wind speed (measured in knots) is plotted on the y-axis. This visual representation allows us to see if there's a trend or pattern between these two variables, highlighting whether they move together or inversely.

line of best fit

The line of best fit, also known as a trend line, is a straight line that best represents the data points on a scatter plot. It minimizes the distances (errors) between the data points and the line itself. This line serves as a predictive model and is found using regression techniques. In our exercise, the line of best fit can be expressed in the form: \[ w = m \times p + b \] where \(w\) stands for wind speed, \(p\) for atmospheric pressure, \(m\) for the slope, and \(b\) for the y-intercept. Using a graphing utility, the line of best fit was determined to be approximately \[ w = -1.2p + 1194 \].

slope interpretation

The slope of the line of best fit, represented by \(m\), tells us how much the dependent variable (wind speed) changes for a unit change in the independent variable (atmospheric pressure). In our exercise, the slope is \(-1.2\). This negative value indicates an inverse relationship: as atmospheric pressure increases, wind speed decreases. Specifically, for every 1 millibar increase in atmospheric pressure, the wind speed decreases by 1.2 knots. This helps us understand the sensitivity of wind speed to changes in atmospheric pressure.

predictive modeling

Predictive modeling involves using statistics and algorithms to create a model that can make predictions based on input data. In this exercise, the linear model \[ w = -1.2p + 1194 \] was used to predict wind speed given a particular atmospheric pressure, and vice versa. For example, to predict wind speed at 990 millibars of pressure, we substitute \(p = 990\) into the equation, resulting in \[ w = -1.2 \times 990 + 1194 = 6 \text{ knots} \]. Similarly, to find the atmospheric pressure for a wind speed of 85 knots, we set \(w = 85\) and solve for \(p\), giving an approximate atmospheric pressure of 924.17 millibars. This capability to predict future instances based on historical data is powerful for forecasting and decision-making in various fields.