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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 13
Chapter 10, Problem 13

Testing for a Linear Correlation
In Exercises 13–28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of α = 0.05. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.)
Powerball Jackpots and Tickets Sold Listed below are the same data from Table 10-1 in the Chapter Problem, but an additional pair of values has been added in the last column. Is there sufficient evidence to conclude that there is a linear correlation between lottery jackpot amounts and numbers of tickets sold? Comment on the effect of the added pair of values in the last column. Compare the results to those obtained in Example 4.


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Step 1: Construct a scatterplot. Plot the given data points on a graph with the x-axis representing the lottery jackpot amounts and the y-axis representing the number of tickets sold. This visual representation will help identify any potential linear relationship between the two variables.
Step 2: Calculate the linear correlation coefficient (r). Use the formula for r: r = (Σ((x - x̄)(y - ȳ))) / (sqrt(Σ(x - x̄)²) * sqrt(Σ(y - ȳ)²)), where x̄ and ȳ are the means of the x and y values, respectively. This measures the strength and direction of the linear relationship.
Step 3: Determine the critical values of r or the P-value. Refer to Table A-6 for the critical values of r at a significance level of α = 0.05, based on the sample size (n). Alternatively, calculate the P-value using statistical software or a calculator.
Step 4: Compare the calculated r value to the critical values or interpret the P-value. If |r| is greater than the critical value or if the P-value is less than α = 0.05, there is sufficient evidence to support a claim of a linear correlation.
Step 5: Analyze the effect of the added pair of values. Recalculate the correlation coefficient and P-value with the additional data point included. Compare the results to the original analysis to determine how the added pair influences the strength and significance of the correlation.

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Key Concepts

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

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 indicates no correlation. Understanding r is crucial for assessing how closely the data points cluster around a straight line in a scatterplot.
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Correlation Coefficient

P-value

The P-value is a statistical measure that helps determine the significance of results obtained in hypothesis testing. It represents the probability of observing the data, or something more extreme, assuming the null hypothesis is true. In the context of correlation, a low P-value (typically less than 0.05) suggests that there is sufficient evidence to reject the null hypothesis, indicating a significant linear correlation between the variables.
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Step 3: Get P-Value

Scatterplot

A scatterplot is a graphical representation of two quantitative variables, where each point represents an observation. It is used to visualize the relationship between the variables, helping to identify patterns, trends, or correlations. Constructing a scatterplot is a fundamental step in analyzing data for linear correlation, as it provides a visual context for interpreting the correlation coefficient and the significance of the relationship.
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Related Practice
Textbook Question

Comparing Two Means Treating the data as samples from larger populations, test the claim that there is a significant difference between the mean of presidents and the mean of popes.

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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.

Find the value of the linear correlation coefficient using all eight points. What does that value suggest about the relationship between x and y?

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Textbook Question

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 Repeat the preceding exercise assuming that the linear correlation coefficient is r = 0.132.

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