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Ch. 10 - Correlation and Regression
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 10 - Correlation and RegressionProblem 10.1.9b
Chapter 10, Problem 10.1.9b

Outlier Refer to the accompanying Minitab-generated scatterplot.
Scatterplot showing 10 data points with one outlier above the main cluster of points. Axes labeled X and Y.
b. After identifying the 10 pairs of coordinates corresponding to the 10 points, find the value of the correlation coefficient r and determine whether there is a linear correlation.

Verified step by step guidance
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Step 1: Identify the 10 pairs of coordinates from the scatterplot. Each point represents a pair (x, y). For example, the points in the lower cluster might have coordinates like (1, 1), (1, 2), (1, 3), etc., while the outlier point is at (10, 10). Carefully list all 10 pairs.
Step 2: Calculate the mean of the x-values and the mean of the y-values. These will be used in the formula for the correlation coefficient r.
Step 3: Use the formula for the correlation coefficient r: r = (Σ((x_i - x̄)(y_i - ȳ))) / sqrt(Σ((x_i - x̄)^2) * Σ((y_i - ȳ)^2)), where x̄ and ȳ are the means of x and y, respectively. Compute the numerator and denominator separately.
Step 4: Analyze the value of r. If r is close to 1 or -1, it indicates a strong linear correlation. If r is close to 0, it indicates little to no linear correlation. Consider the impact of the outlier on the correlation coefficient.
Step 5: Determine whether there is a linear correlation by comparing the calculated r value to a critical value from a correlation table (based on the sample size and significance level). If |r| exceeds the critical value, there is significant linear correlation; otherwise, there is not.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Outliers

Outliers are data points that differ significantly from other observations in a dataset. They can occur due to variability in the data or may indicate experimental errors. In scatterplots, outliers can skew the results of statistical analyses, such as correlation, and may need to be addressed to ensure accurate interpretations.
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Correlation Coefficient (r)

The 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 indicates no correlation. Understanding 'r' is crucial for assessing how closely related the variables are in the scatterplot.
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Linear Correlation

Linear correlation refers to the relationship between two variables that can be represented by a straight line. It is assessed using the correlation coefficient 'r'. A strong linear correlation suggests that changes in one variable are associated with changes in another, while a weak correlation indicates little to no relationship. Identifying linear correlation is essential for making predictions based on the data.
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Related Practice
Textbook Question

Notation The author conducted an experiment in which the height of each student was measured in centimeters and those heights were matched with the same students’ scores on the first statistics test.

c. Does r change if the heights are converted from centimeters to inches?

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Textbook Question

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.

b. Find the sum of the squares of the residuals.

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Textbook Question

Clusters Refer to the Minitab-generated scatterplot. The four points in the lower left corner are measurements from women, and the four points in the upper right corner are from men.

a. Examine the pattern of the four points in the lower left corner (from women) only, and subjectively determine whether there appears to be a correlation between x and y for women.

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Textbook Question

Variation and Prediction Intervals

In Exercises 17–20, find the (a) explained variation, (b) unexplained variation, and (c) indicated prediction interval. In each case, there is sufficient evidence to support a claim of a linear correlation, so it is reasonable to use the regression equation when making predictions.

Altitude and Temperature Listed below are altitudes (thousands of feet) and outside air temperatures (°F) recorded by the author during Delta Flight 1053 from New Orleans to Atlanta. For the prediction interval, use a 95% confidence level with the altitude of 6327 ft (or 6.327 thousand feet).

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Textbook Question

Sum of Squares Criterion In addition to the value of another measurement used to assess the quality of a model is the sum of squares of the residuals. Recall from Section 10-2 that a residual is (the difference between an observed y value and the value predicted from the model). Better models have smaller sums of squares. Refer to the U.S. population data in Table 10-7.

c. Verify that according to the sum of squares criterion, the quadratic model is better than the linear model.

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Textbook Question

Notation The author conducted an experiment in which the height of each student was measured in centimeters and those heights were matched with the same students’ scores on the first statistics test.

b. Without doing any research or calculations, estimate the value of r.

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