91Ó°ÊÓ

Please do the following. (a) Draw a scatter diagram displaying the data. (b) Verify the given sums \(\Sigma x, \Sigma y, \Sigma x^{2}, \Sigma y^{2},\) and \(\Sigma x y\) and the value of the sample correlation coefficient \(r\) (c) Find \(\bar{x}, \bar{y}, a,\) and \(b .\) Then find the equation of the least- squares line \(\hat{y}=a+b x\) (d) Graph the least-squares line on your scatter diagram. Be sure to use the point \((\bar{x}, \bar{y})\) as one of the points on the line. (e) Interpretation Find the value of the coefficient of determination \(r^{2} .\) What percentage of the variation in \(y\) can be explained by the corresponding variation in \(x\) and the least-squares line? What percentage is unexplained? Answers may vary slightly due to rounding. Cricket Chirps: Temperature Anyone who has been outdoors on a summer evening has probably heard crickets. Did you know that it is possible to use the cricket as a thermometer? Crickets tend to chirp more frequently as temperatures increase. This phenomenon was studied in detail by George W. Pierce, a physics professor at Harvard. In the following data, \(x\) is a random variable representing chirps per second and \(y\) is a random variable representing temperature ('F). These data are also available for download at the Online Study Center.Complete parts (a) through (e), given \(\Sigma x=249.8, \Sigma y=1200.6\) \(\Sigma x^{2}=4200.56, \Sigma y^{2}=96,725.86, \Sigma x y=20,127.47,\) and \(r \approx 0.835\) (f) What is the predicted temperature when \(x=19\) chirps per second?

Step by step solution

01

Draw the Scatter Diagram

First, we plot the pairs of data points \(x, y\) on a graph where \(x\) (chirps per second) is on the horizontal axis and \(y\) (temperature in 'F) is on the vertical axis. Use a consistent scale for each axis to accurately reflect the relationship.

02

Verify Given Sums and Correlation Coefficient

Calculate each of the following to verify the given sums: 1. \(\Sigma x = 249.8\)2. \(\Sigma y = 1200.6\)3. \(\Sigma x^2 = 4200.56\)4. \(\Sigma y^2 = 96,725.86\)5. \(\Sigma xy = 20,127.47\)Using these values, verify the sample correlation coefficient \(r\) using the formula \[ r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n(\Sigma x^2) - (\Sigma x)^2][n(\Sigma y^2) - (\Sigma y)^2]}} \ \approx 0.835\] where \(n\) is the number of data points.

03

Calculate Averages and Constants for Line Equation

Calculate the averages of \(x\) and \(y\): \[\bar{x} = \frac{\Sigma x}{n}\]\[\bar{y} = \frac{\Sigma y}{n}\]Next, calculate the slope \(b\) and intercept \(a\) for the least-squares line: \[b = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{n(\Sigma x^2) - (\Sigma x)^2}\]\[a = \bar{y} - b\bar{x}\]Then, construct the equation \(\hat{y} = a + bx\).

04

Graph the Least-Squares Line

Using the equation from Step 3, draw the least-squares line on the same graph as the scatter plot. Ensure the point \(\bar{x}, \bar{y}\) lies on this line. This visual shows the line of best fit for the data.

05

Find Coefficient of Determination and Interpretation

Calculate the coefficient of determination, \(r^2\), which is \(r\) squared:\[r^2 = (0.835)^2 = 0.6972\]This indicates that approximately 69.72% of the variation in \(y\) is explained by the variation in \(x\) and the least-squares line. Therefore, 30.28% of the variation remains unexplained.

06

Predict Temperature for Given Chirps

Substitute \(x = 19\) into the least-squares line equation \(\hat{y} = a + bx\) to predict the temperature. Solve for \(\hat{y}\) to get the predicted temperature.

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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, often known as a scatter plot, is a graphical representation of the relationship between two variables. In our case, these variables are cricket chirps per second on the x-axis and temperature in degrees Fahrenheit on the y-axis. To create a scatter diagram, plot each pair of data points on a coordinate grid. Each point represents an observation of the two variables.

This visual tool helps to identify patterns or correlations between the variables easily. By observing the scatter diagram, you can see if there is a linear relationship—as the chirping rate increases, the temperature also seems to rise, suggesting a potential positive correlation. The scatter plot serves as the foundation for further statistical analysis, like calculating the least-squares line and the correlation coefficient.

Sample Correlation Coefficient

The sample correlation coefficient, denoted as \( r \), measures the strength and direction of a linear relationship between two variables on a scatter plot. Its value ranges from -1 to 1, where +1 indicates a perfect positive linear relationship, 0 indicates no linear relationship, and -1 indicates a perfect negative linear relationship.

To calculate \( r \), use the formula:

• \( r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n(\Sigma x^2) - (\Sigma x)^2][n(\Sigma y^2) - (\Sigma y)^2]}} \)

Here, \( n \) is the number of observations, \( \Sigma xy \), \( \Sigma x^2 \), and \( \Sigma y^2 \) are the sums of products and squares of the variables, respectively. In our scenario, \( r \) is approximately 0.835, indicating a strong positive linear relationship between chirping rate and temperature. Hence, as chirp frequency increases, so does the temperature.

Least-Squares Line

The least-squares line, or line of best fit, is a straight line that best represents the data on a scatter plot. It minimizes the sum of the squares of the vertical distances of the points from the line. The line is described by the linear equation:

\( \hat{y} = a + bx \)

where \( b \) is the slope and \( a \) is the y-intercept. To determine \( a \) and \( b \), use the formulas:

• Slope: \( b = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{n(\Sigma x^2) - (\Sigma x)^2} \)
• Intercept: \( a = \bar{y} - b\bar{x} \)

You plug in the mean values of x and y, as well as the calculated slope, into these formulas.

This line not only aids in visualizing the extent of the relationship but can also be used to forecast unknown values within the data's range, as shown when predicting temperatures given a certain chirp rate.

Coefficient of Determination

The coefficient of determination, represented by \( r^2 \), quantifies how much of the variability in one variable can be explained by its relationship with another variable. It is the square of the sample correlation coefficient \( r \).

For our example, the calculation is:

• \( r^2 = (0.835)^2 = 0.6972 \)

This means that approximately 69.72% of the variation in temperature can be explained by the frequency of cricket chirps and the linear relationship defined by the least-squares line. Consequently, 30.28% of the variation is due to other factors not explained by this model.

This metric is crucial for understanding the effectiveness of your model: in predicting outcomes or demonstrating the reliability of the observed relationships within your data.