Question

For Problems $7-18$, 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 $\sum x y$ 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 $\left({ }^{\circ} \mathrm{F}\right)$. These data are also available for download at the Online Study Center. $$ \begin{aligned} &\begin{array}{l|llllllll} \hline x & 20.0 & 16.0 & 19.8 & 18.4 & 17.1 & 15.5 & 14.7 & 17.1 \\ \hline y & 88.6 & 71.6 & 93.3 & 84.3 & 80.6 & 75.2 & 69.7 & 82.0 \\ \hline \end{array}\\ &6\\ &\begin{array}{l|lllllll} \hline x & 15.4 & 16.2 & 15.0 & 17.2 & 16.0 & 17.0 & 14.4 \\ \hline y & 69.4 & 83.3 & 79.6 & 82.6 & 80.6 & 83.5 & 76.3 \\ \hline \end{array} \end{aligned} $$ 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?

Name: Problem 18, Chapter 9: Understandable Statistics Concepts and Methods
Uploaded: 2021-04-17T00:03:25-08:00
Duration: 2 min 5 s
Channel: Tyler Moulton

   For Problems $7-18$, 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 $\sum x y$ 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 $\left({ }^{\circ} \mathrm{F}\right)$. These data are also available for download at the Online Study Center.
$$
\begin{aligned}
&\begin{array}{l|llllllll}
\hline x & 20.0 & 16.0 & 19.8 & 18.4 & 17.1 & 15.5 & 14.7 & 17.1 \\
\hline y & 88.6 & 71.6 & 93.3 & 84.3 & 80.6 & 75.2 & 69.7 & 82.0 \\
\hline
\end{array}\\
&6\\
&\begin{array}{l|lllllll}
\hline x & 15.4 & 16.2 & 15.0 & 17.2 & 16.0 & 17.0 & 14.4 \\
\hline y & 69.4 & 83.3 & 79.6 & 82.6 & 80.6 & 83.5 & 76.3 \\
\hline
\end{array}
\end{aligned}
$$
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?

Understandable Statistics Concepts and Methods

Charles Brase,… 12th Edition

Chapter 9, Problem 18

Instant Answer

Solved by Expert Tyler Moulton

04/17/2021

Step 1

The sample mean $\bar{x}$ is the sum of the data divided by $n$, which in this case is $15.1$. The sample standard deviation $S$ is the sum of deviations about the mean squared divided by $n - 1$, which is $2.51$. Show more…

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For Problems $7-18$, 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 $\sum x y$ 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 $\left({ }^{\circ} \mathrm{F}\right)$. These data are also available for download at the Online Study Center. $$ \begin{aligned} &\begin{array}{l|llllllll} \hline x & 20.0 & 16.0 & 19.8 & 18.4 & 17.1 & 15.5 & 14.7 & 17.1 \\ \hline y & 88.6 & 71.6 & 93.3 & 84.3 & 80.6 & 75.2 & 69.7 & 82.0 \\ \hline \end{array}\\ &6\\ &\begin{array}{l|lllllll} \hline x & 15.4 & 16.2 & 15.0 & 17.2 & 16.0 & 17.0 & 14.4 \\ \hline y & 69.4 & 83.3 & 79.6 & 82.6 & 80.6 & 83.5 & 76.3 \\ \hline \end{array} \end{aligned} $$ 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?

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Key Concepts

Coefficient of Determination

The coefficient of determination, denoted as r², represents the proportion of the variance in the dependent variable that is predictable from the independent variable using the linear regression model. It provides an indication of the goodness of fit, quantifying how well the regression line explains the variability of the data.

Scatter Diagram

A scatter diagram, or scatter plot, is a graphical representation of data where each point represents a pair of values. It is used to visualize the possible relationship or association between two random variables and helps to identify patterns, trends, clusters, or outliers in the data.

Verification of Data Summations

Verifying given sums, such as the sum of x values, y values, squares of x and y, and the products of x and y, is an essential step in confirming the accuracy of computed statistics. This process ensures that any further calculations based on these summaries, like determining regression parameters or correlations, are based on correct data totals.

Correlation Coefficient

The sample correlation coefficient, denoted by r, quantifies the degree and direction of the linear relationship between two variables. Its value ranges from -1 to 1, where values close to 1 indicate a strong positive linear relationship, values close to -1 indicate a strong negative relationship, and values near 0 suggest little to no linear relation.

Least-Squares Regression

Least-squares regression is a method used to determine the best fitting line through a set of data points by minimizing the sum of the squared differences between observed and predicted values. The resulting equation, y? = a + bx, where a is the y-intercept and b is the slope, describes the linear relationship between the independent variable x and the dependent variable y.

Point of Averages (Mean Point)

The point (x?, y?), where x? and y? are the means of the x and y data respectively, is always located on the least-squares regression line. This property ensures that the line accurately reflects the central tendency of the data as it balances the cumulative deviations of the observations.

Prediction Using the Regression Line

Once the regression equation is established, it can be used to make predictions by substituting a specified value of the independent variable x to estimate the corresponding value of y. This application is fundamental in practical scenarios where forecasting based on observed trends is required.

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