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Chapter 9: Problem 15

Can a low barometer reading be used to predict maximum wind speed of an approaching tropical cyclone? Data for this problem are based on information taken from Weatherwise (Vol. 46, No. 1\(),\) a publication of the American Meteorological Society. For a random sample of tropical cyclones, let \(x\) be the lowest pressure (in millibars) as a cyclone approaches, and let \(y\) be the maximum wind speed (in miles per hour) of the cyclone. $$ \begin{array}{l|rrrrrr} \hline x & 1004 & 975 & 992 & 935 & 985 & 932 \\ \hline y & 40 & 100 & 65 & 145 & 80 & 150 \\ \hline \end{array} $$ (a) Make a scatter diagram and draw the line you think best fits the data. (b) Would you say the correlation is low, moderate, or strong? positive or negative? (c) Use a calculator to verify that \(\Sigma x=5823, \Sigma x^{2}=5,655,779, \Sigma y=580\), \(\Sigma y^{2}=65,750\), and \(\Sigma x y=556,315\). Compute \(r\). As \(x\) increases, does the value of \(r\) imply that \(y\) should tend to increase or decrease? Explain.

Short Answer

Expert verified
The correlation is strong and negative, as indicated by a negative \(r\) close to -1, suggesting wind speed decreases with higher pressure.

Step by step solution

01

Create a Scatter Diagram

To begin, plot the given data points on a scatter diagram. For each pair \((x, y)\), mark the point where the x-coordinate (pressure) and y-coordinate (wind speed) intersect. This will help visualize any relationship between pressure and wind speed.
02

Draw a Line of Best Fit

Examine the scatter plot and draw a line that appears to best represent the data points. This line should have roughly equal numbers of points above and below it and should minimize the distance from the line to each point.
03

Analyze the Correlation Type

Visually inspect the scatter plot. If the points show an upward trend as \(x\) increases, it's a positive correlation. If the trend is downward, it's a negative correlation. Determine whether this trend appears strong, moderate, or weak based on how closely the points cluster around the line.
04

Calculate the Correlation Coefficient (r)

Use the formula for the correlation coefficient: \[ r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n(\Sigma x^2) - (\Sigma x)^2][n(\Sigma y^2) - (\Sigma y)^2]}} \]Substitute the values: \(n = 6\), \(\Sigma x = 5823\), \(\Sigma x^2 = 5,655,779\), \(\Sigma y = 580\), \(\Sigma y^2 = 65,750\), \(\Sigma xy = 556,315\).
05

Interpret the Correlation Coefficient (r)

Compute the above expression to find \(r\). If \(|r|\) is close to 1, the correlation is strong. If it's close to 0, the correlation is weak. A positive \(r\) indicates a positive correlation, and a negative \(r\) indicates a negative correlation. This value of \(r\) indicates whether, as \(x\) increases, \(y\) tends to either increase or decrease.

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Key Concepts

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

Scatter Diagram
A scatter diagram, also known as a scatter plot, is a graphical representation where two variables are plotted along two axes. This allows us to visually assess the relationship between the two variables. In this case, we have barometer readings as our x-values and wind speed as our y-values. By plotting these pairs of values on a graph, each set of mounting marks a point where the x-coordinate (pressure) and y-coordinate (wind speed) intersect.

Through observing the arrangement of these points on the graph: we can start to detect patterns or trends. Primarily, it helps to determine if there is a relationship between the low barometer readings and the approaching cyclone's maximum wind speed.

The beauty of a scatter diagram is its simplicity: with a quick glance, one can understand whether there is a potential correlation between the variables. Whether it's strong, weak, or nonexistent. By visualizing with scatter diagrams, hypotheses about relationships can be formed and pursued further with mathematical calculations, such as calculating the correlation coefficient.
Line of Best Fit
The line of best fit, or regression line, is a straight line that best represents the data points on a scatter plot. This line attempts to capture the main trend among the data, showing the direction and strength of a potential relationship between variables.

To draw a line of best fit, place it so that approximately half of the points lie above the line and half below. The line should "run" through the center of the data, minimizing the distance from the line to each point.

  • This concept helps in predicting future data points based on existing data.
  • It gives a clear direction 鈥� upwards for a positive correlation or downwards for a negative correlation.
Even without precise calculations, simply drawing this line helps in visualizing whether the parameters are positively or negatively correlated. Think of it as a guide that simplifies a complex set of data, making trends easier to spot.
Positive and Negative Correlation
When analyzing the relationship depicted in a scatter diagram, understanding positive and negative correlations is crucial.

A positive correlation means that as one variable increases, the other tends to also increase. Graphically, this looks like an upward trend on the scatter plot. For example, if a decrease in millibars (x) consistently aligns with an increase in wind speed (y), there is a positive correlation.

Conversely, a negative correlation indicates that as one variable increases, the other decreases. In a graph, this trend appears as a downward slope.

Correlations can also vary in terms of their strength:
  • Strong Correlation: Data points closely follow a clear linear trend.
  • Moderate Correlation: The trend is still visible, but the points are more scattered.
  • Weak Correlation: Little to no pattern is discernible.
By calculating the correlation coefficient, we can quantify this relationship. A correlation coefficient (r) closer to 1 or -1 indicates a strong positive or negative correlation, respectively. If r is closer to 0, the association between the variables is weak or non-existent. Ultimately, recognizing whether a correlation is present and its strength aids greatly in making predictions and informed decisions based on the data.

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Most popular questions from this chapter

It is not obvious from the formulas, but the values of the sample test statistic \(t\) for the correlation coefficient and for the slope of the least- squares line are equal for the same data set. This fact is based on the relation $$ b=r \frac{s_{y}}{s_{x}} $$ where \(s_{y}\) and \(s_{x}\) are the sample standard deviations of the \(x\) and \(y\) values, respectively. (a) Many computer software packages give the \(t\) value and corresponding \(P\) -value for \(b\). If \(\beta\) is significant, is \(\rho\) significant? (b) When doing statistical tests "by hand," it is easier to compute the sample test statistic \(t\) for the sample correlation coefficient \(r\) than it is to compute the sample test statistic \(t\) for the slope \(b\) of the sample least- squares line. Compare the results of parts (b) and (f) for Problems \(7-12\) of this problem set. Is the sample test statistic \(t\) for \(r\) the same as the corresponding test statistic for \(b\) ? If you conclude that \(\rho\) is positive, can you conclude that \(\beta\) is positive at the same level of significance? If you conclude that \(\rho\) is not significant, is \(\beta\) also not significant at the same level of significance?

Prisons Does prison really deter violent crime? Let \(x\) represent percent change in the rate of violent crime and \(y\) represent percent change in the rate of imprisonment in the general U.S. population. For 7 recent years, the following data have been obtained (Source: The Crime Drop in America, edited by Blumstein and Wallman, Cambridge University Press). $$ \begin{array}{l|rrrrrrr} \hline x & 6.1 & 5.7 & 3.9 & 5.2 & 6.2 & 6.5 & 11.1 \\ \hline y & -1.4 & -4.1 & -7.0 & -4.0 & 3.6 & -0.1 & -4.4 \\ \hline \end{array} $$ Complete parts (a) through (e), given \(\Sigma x=44.7, \Sigma y=-17.4, \Sigma x^{2}=315.85\), \(\Sigma y^{2}=116.1, \Sigma x y=-107.18\), and \(r \approx 0.084 .\) (f) Critical Thinking Considering the values of \(r\) and \(r^{2}\), does it make sense to use the least-squares line for prediction? Explain.

(a) Suppose you are given the following \((x, y)\) data pairs: $$ \begin{array}{l|lll} \hline x & 1 & 3 & 4 \\ \hline y & 2 & 1 & 6 \\ \hline \end{array} $$ Show that the least-squares equation for these data is \(y=1.071 x+0.143\) (rounded to three digits after the decimal). (b) Now suppose you are given these \((x, y)\) data pairs: $$ \begin{array}{l|lll} \hline x & 2 & 1 & 6 \\ \hline y & 1 & 3 & 4 \\ \hline \end{array} $$ Show that the least-squares equation for these data is \(y=0.357 x+1.595\) (rounded to three digits after the decimal). (c) In the data for parts (a) and (b), did we simply exchange the \(x\) and \(y\) values of each data pair? (d) Solve \(y=0.143+1.071 x\) for \(x .\) Do you get the least-squares equation of part (b) with the symbols \(x\) and \(y\) exchanged? (e) In general, suppose we have the least-squares equation \(y=a+b x\) for a set of data pairs \((x, y)\). If we solve this equation for \(x\), will we necessarily get the least-squares equation for the set of data pairs \((y, x)\) (with \(x\) and \(y\) exchanged)? Explain using parts (a) through (d).

All Greens is a franchise store that sells house plants and lawn and garden supplies. Although All Greens is a franchise, each store is owned and managed by private individuals. Some friends have asked you to go into business with them to open a new All Greens store in the suburbs of San Diego. The national franchise headquarters sent you the following information at your request. These data are about 27 All Greens stores in California. Each of the 27 stores has been doing very well, and you would like to use the information to help set up your own new store. The variables for which we have data are \(x_{1}=\) annual net sales, in thousands of dollars \(x_{2}=\) number of square feet of floor display in store, in thousands of square feet \(x_{3}=\) value of store inventory, in thousands of dollars \(x_{4}=\) amount spent on local advertising, in thousands of dollars \(x_{5}=\) size of sales district, in thousands of families \(x_{6}=\) number of competing or similar stores in sales district A sales district was defined to be the region within a 5 -mile radius of an All Greens store. $$ \begin{array}{rlrrrr|rrrrrr} \hline x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} \\ \hline 231 & 3 & 294 & 8.2 & 8.2 & 11 & 65 & 1.2 & 168 & 4.7 & 3.3 & 11 \\ 156 & 2.2 & 232 & 6.9 & 4.1 & 12 & 98 & 1.6 & 151 & 4.6 & 2.7 & 10 \\ 10 & 0.5 & 149 & 3 & 4.3 & 15 & 398 & 4.3 & 342 & 5.5 & 16.0 & 4 \\ 519 & 5.5 & 600 & 12 & 16.1 & 1 & 161 & 2.6 & 196 & 7.2 & 6.3 & 13 \\ 437 & 4.4 & 567 & 10.6 & 14.1 & 5 & 397 & 3.8 & 453 & 10.4 & 13.9 & 7 \\ 487 & 4.8 & 571 & 11.8 & 12.7 & 4 & 497 & 5.3 & 518 & 11.5 & 16.3 & 1 \\ 299 & 3.1 & 512 & 8.1 & 10.1 & 10 & 528 & 5.6 & 615 & 12.3 & 16.0 & 0 \\ 195 & 2.5 & 347 & 7.7 & 8.4 & 12 & 99 & 0.8 & 278 & 2.8 & 6.5 & 14 \\ 20 & 1.2 & 212 & 3.3 & 2.1 & 15 & 0.5 & 1.1 & 142 & 3.1 & 1.6 & 12 \\ 68 & 0.6 & 102 & 4.9 & 4.7 & 8 & 347 & 3.6 & 461 & 9.6 & 11.3 & 6 \\ 570 & 5.4 & 788 & 17.4 & 12.3 & 1 & 341 & 3.5 & 382 & 9.8 & 11.5 & 5 \\ 428 & 4.2 & 577 & 10.5 & 14.0 & 7 & 507 & 5.1 & 590 & 12.0 & 15.7 & 0 \\ 464 & 4.7 & 535 & 11.3 & 15.0 & 3 & 400 & 8.6 & 517 & 7.0 & 12.0 & 8 \\ 15 & 0.6 & 163 & 2.5 & 2.5 & 14 & & & & & & \\ \hline \end{array} $$ (a) Generate summary statistics, including the mean and standard deviation of each variable. Compute the coefficient of variation (see Section \(3.2\) ) for each variable. Relative to its mean, which variable has the largest spread of data values? Which variable has the least spread of data values relative to its mean? (b) For each pair of variables, generate the sample correlation coefficient \(r .\) For all pairs involving \(x_{1}\), compute the corresponding coefficient of determination \(r^{2}\). Which variable has the greatest influence on annual net sales? Which variable has the least influence on annual net sales? (c) Perform a regression analysis with \(x_{1}\) as the response variable. Use \(x_{2}, x_{3}\), \(x_{4}, x_{5}\), and \(x_{6}\) as explanatory variables. Look at the coefficient of multiple determination. What percentage of the variation in \(x_{1}\) can be explained by the corresponding variations in \(x_{2}, x_{3}, x_{4}, x_{5}\), and \(x_{6}\) taken together? (d) Write out the regression equation. If two new competing stores moved into the sales district but the other explanatory variables did not change, what would you expect for the corresponding change in annual net sales? Explain your answer. If you increased the local advertising by a thousand dollars but the other explanatory variables did not change, what would you expect for the corresponding change in annual net sales? Explain. (e) Test each coefficient to determine if it is or is not zero. Use level of significance \(5 \%\). (f) Suppose you and your business associates rent a store, get a bank loan to start up your business, and do a little research on the size of your sales district and the number of competing stores in the district. If \(x_{2}=2.8\), \(x_{3}=250, x_{4}=3.1, x_{5}=7.3\), and \(x_{6}=2\), use a computer to forecast \(x_{1}=\) annual net sales and find an \(80 \%\) confidence interval for your forecast (if your software produces prediction intervals). (g) Construct a new regression model with \(x_{4}\) as the response variable and \(x_{1}\), \(x_{2}, x_{3}, x_{5}\), and \(x_{6}\) as explanatory variables. Suppose an All Greens store in Sonoma, California, wants to estimate a range of advertising costs appropriate to its store. If it spends too little on advertising, it will not reach enough customers. However, it does not want to overspend on advertising for this type and size of store. At this store, \(x_{1}=163, x_{2}=2.4, x_{3}=188\), \(x_{5}=6.6\), and \(x_{6}=10\). Use these data to predict \(x_{4}\) (advertising costs) and find an \(80 \%\) confidence interval for your prediction. At the \(80 \%\) confidence level, what range of advertising costs do you think is appropriate for this store?

Let \(x=\) day of observation and \(y=\) number of locusts per square meter during a locust infestation in a region of North Africa. $$ \begin{array}{l|llrrr} \hline x & 2 & 3 & 5 & 8 & 10 \\ \hline y & 2 & 3 & 12 & 125 & 630 \\ \hline \end{array} $$ (a) Draw a scatter diagram of the \((x, y)\) data pairs. Do you think a straight line will be a good fit to these data? Do the \(y\) values almost seem to explode as time goes on? (b) Now consider a transformation \(y^{\prime}=\log y .\) We are using common logarithms of base \(10 .\) Draw a scatter diagram of the \(\left(x, y^{\prime}\right)\) data pairs and compare this diagram with the diagram of part (a). Which graph appears to better fit a straight line? (c) Use a calculator with regression keys to find the linear regression equation for the data pairs \(\left(x, y^{\prime}\right) .\) What is the correlation coefficient? (d) The exponential growth model is \(y=\alpha \beta^{x}\). Estimate \(\alpha\) and \(\beta\) and write the exponential growth equation. Hint: See Problem 22 .
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