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Ch. 10 - Correlation and Regression
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoTriola 14th EditionCh. 10 - Correlation and RegressionProblema 10.RE.3a
Capitolo 10, Problema 10.RE.3a

Time and Motion In a physics experiment at Doane College, a soccer ball was thrown upward from the bed of a moving truck. The table below lists the time (sec) that has lapsed from the throw and the corresponding height (m) of the soccer ball.
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a. Find the value of the linear correlation coefficient r.

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Step 1: Understand the problem. The goal is to calculate the linear correlation coefficient (r), which measures the strength and direction of the linear relationship between two variables: time (x) and height (y).
Step 2: Use the formula for the linear correlation coefficient: r = (nΣ(xy) - ΣxΣy) / sqrt([(nΣ(x^2) - (Σx)^2)][(nΣ(y^2) - (Σy)^2)]). Here, n is the number of data points, Σ(xy) is the sum of the product of x and y, Σx and Σy are the sums of x and y respectively, Σ(x^2) is the sum of the squares of x, and Σ(y^2) is the sum of the squares of y.
Step 3: Organize the data into a table with columns for x (time), y (height), x^2, y^2, and xy. Compute the values for each column and sum them up to find Σx, Σy, Σ(x^2), Σ(y^2), and Σ(xy).
Step 4: Substitute the computed sums and the number of data points (n) into the formula for r. Simplify the numerator and denominator separately before dividing.
Step 5: Interpret the value of r. If r is close to 1 or -1, it indicates a strong linear relationship. If r is close to 0, it indicates a weak or no linear relationship.

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Linear Correlation Coefficient (r)

The linear correlation coefficient, denoted as r, quantifies the strength and direction of a linear relationship between two variables. Its value ranges from -1 to 1, where 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 signifies no correlation. Understanding how to calculate and interpret r is essential for analyzing the relationship between the time elapsed and the height of the soccer ball in the given experiment.
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Correlation Coefficient

Data Representation

Data representation involves organizing and displaying data in a way that makes it easier to analyze and interpret. In this context, the height of the soccer ball over time can be represented in a table or graph, which helps visualize the relationship between the two variables. Proper data representation is crucial for identifying trends and patterns that inform the calculation of the correlation coefficient.
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Statistical Significance

Statistical significance refers to the likelihood that a relationship observed in data is not due to random chance. In the context of calculating the correlation coefficient, it is important to assess whether the correlation found is statistically significant, which can be determined through hypothesis testing. This concept helps in making informed conclusions about the relationship between the time and height of the soccer ball in the experiment.
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Exercises 1–10 are based on the following sample data consisting of costs of dinner (dollars) and the amounts of tips (dollars) left by diners. The data were collected by students of the author.

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Scatterplot Construct a scatterplot and comment on the pattern of points.

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Exercises 1–10 are based on the following sample data consisting of costs of dinner (dollars) and the amounts of tips (dollars) left by diners. The data were collected by students of the author.

Predictions The sample data result in a linear correlation coefficient of r = 0.846 and the regression equation y^ = -0.00777 + 0.145x. What is the best predicted amount of tip, given that the cost of dinner was \$84.62? How was the predicted value found?

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Domanda del libro di testo

Time and Motion In a physics experiment at Doane College, a soccer ball was thrown upward from the bed of a moving truck. The table below lists the time (sec) that has lapsed from the throw and the corresponding height (m) of the soccer ball.

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c. What horrible mistake would be easy to make if the analysis is conducted without a scatterplot?

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Least-Squares Property According to the least-squares property, the regression line minimizes the sum of the squares of the residuals. Refer to the jackpot/tickets data in Table 10-1 and use the regression equation y^ = -10.9 + 0.174x that was found in Examples 1 and 2 of this section.

a. Identify the nine residuals.

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Casino Size and Revenue Use the same paired data from the preceding exercise.

b. What is the best predicted amount of revenue for a casino with a size of 200 thousand square feet? Is it likely that the best predicted amount of revenue will be accurate?



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Time and Motion In a physics experiment at Doane College, a soccer ball was thrown upward from the bed of a moving truck. The table below lists the time (sec) that has lapsed from the throw and the corresponding height (m) of the soccer ball.

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b. Based on the result from part (a), what do you conclude about a linear correlation between time and height?

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