The data at the top of the next column represent the...

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?

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?

Key Concepts

Slope Interpretation

The slope in a linear regression model quantifies the rate of change of the dependent variable with respect to changes in the independent variable. It indicates the amount by which the dependent variable is expected to change for each one-unit increase in the independent variable, providing insight into the strength and direction of the relationship.

Function Notation in Modeling

Function notation expresses the relationship between the independent and dependent variables in a formal mathematical way, typically represented as f(x) or y = f(x). This notation clarifies how changes in one variable will affect the other according to the model.

Prediction Using Regression Models

Prediction involves substituting a given value of the independent variable into the regression model to estimate the corresponding value of the dependent variable. This process allows for forecasting or estimating outcomes based on the established linear relationship.

Independent and Dependent Variables

Identifying the independent and dependent variables is crucial in modeling as it defines which variable is presumed to cause or influence the other. The independent variable is manipulated or observed to see its effect on the dependent variable, which is the outcome of interest.

Scatter Plot

A scatter plot is a graphical representation of data where each point represents an observation with its coordinates determined by two variables. It helps visualize the relationship or correlation between the variables, indicating patterns, trends, and potential outliers.

Linear Regression (Line of Best Fit)

Linear regression is a statistical method used to model the relationship between a dependent and an independent variable by fitting a straight line (the line of best fit) through the data points. The line minimizes the squared differences between observed values and predicted values, serving as a compact mathematical representation of the trend in the data.

Transcript

00:02 Hi, today we're going to focus on creating a scatter plot using a graph and calculator, finding a line of best fit, as well as being able to interpret the slope given our linear regression.

00:19 And so in this question, you see you have your pressure and your wind speed.

00:24 So your pressure is going to be your independent variable, which is going to be your x.

00:28 Your wind speed depends on your atmosphere pressure.

00:33 And so that's going to be your y value.

00:36 And so in order for this to work, we're going to take our graphing calculator, and we're going to go to our stat.

00:44 We're going to go to edit.

00:46 And we're going to fill in this information.

00:50 And so you're going to put in all of your atmosphere pressure under l1, which is your x, so 993, 994, 9907.

01:00 You're going to put in all of that x values under l1, and then you're going to put all of your wind speed under l2 so that it matches.

01:11 So that's 50 and then 60 and then 45 and then 45 all the way down until you get to 55.

01:21 And so it should say 982 for l1 and 55 for l2.

01:26 And so you have all of the information for x and you have all the information for y so that we are able to take a look at our scatter plot.

01:37 Make sure your scatter plot is turned on.

01:40 So you're going to do second y.

01:43 Make sure it says plot one on.

01:45 If it does not, hit enter and make sure it's hit on.

01:49 And you can see your scatter plot highlighted.

01:52 You're then going to go to zoom and you're going to go down to number nine to show.

01:58 Our statistics and our stat plot.

02:02 And so when you do that, you'll see your stat plot, which is all of the data we just put in, all in this data here.

02:10 And you see how the slope starts to decrease from left to right.

02:15 It starts to trend downward.

02:16 Well, that's because as pressure starts to go up, your wind speed starts to go up, but then at times it starts to go down and your wind speed doesn't follow.

02:30 And so therefore, we have a statistical plot here to figure out what's actually happening with our slope based off of this model.

02:42 And so what we're going to do is find the line of best fit that falls in this stat plot.

02:48 And so we take that data that we just put in, and instead of going to edit this time, because we already have the data in, we're going to go to calc.

02:56 When we go to calc, we're going to go down to linear regression.

03:00 When we go to linear regression, we're going to put in our l1, comma, l2, to show your values for your x independent variable and your dependent variable.

03:17 And so if you look at r, r states that it is a strong correlation.

03:21 So that means it's a strong linear correlation.

03:23 But we're really just going to focus on our equation, y equals x plus b.

03:27 Or sometimes you know it as y equals mx plus b.

03:30 And we're going to write out our linear regression rounded to the nearest hundredth.

03:36 And so our linear regression is going to be y equals, and we're going to fill in our a and our b, which in this case is negative 1 .19x, and then our y intercept, which is our b value, is going to be 1 ,231 .131...

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