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.)
Taxis Using the data from Exercise 15, is there sufficient evidence to support the claim that there is a linear correlation between the distance of the ride and the tip amount? Does it appear that riders base their tips on the distance of the ride?

Verified step by step guidance

1. Step 1: Begin by constructing a scatterplot using the given data. Plot the distance of the ride on the x-axis and the tip amount on the y-axis. This visual representation will help identify any apparent relationship between the two variables.
2. Step 2: Calculate the linear correlation coefficient (r) using the formula: r = (Σ((x - x̄)(y - ȳ))) / √(Σ(x - x̄)² * Σ(y - ȳ)²). Here, x̄ and ȳ represent the means of the x and y variables, respectively. This coefficient measures the strength and direction of the linear relationship.
3. Step 3: Determine the P-value or the critical values of r from Table A-6, corresponding to the sample size (n) and the significance level α = 0.05. The P-value indicates the probability of observing the data if there is no linear correlation, while the critical values define the threshold for significance.
4. Step 4: Compare the calculated r value to the critical values or use the P-value to assess significance. If |r| exceeds the critical value or if the P-value is less than α, there is sufficient evidence to support the claim of a linear correlation.
5. Step 5: Interpret the results in the context of the problem. If a significant linear correlation is found, discuss whether the data suggests that riders base their tips on the distance of the ride. If no significant correlation is found, explain that the data does not support the claim.

Key Concepts

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 suggests no correlation. Understanding r is crucial for assessing how closely two variables are related, which is essential for interpreting scatterplots.

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. A P-value less than the significance level (α = 0.05 in this case) indicates strong evidence against the null hypothesis, suggesting that a linear correlation may exist between the variables being studied.

Scatterplot

A scatterplot is a graphical representation of two quantitative variables, where each point represents an observation. It helps visualize the relationship between the variables, allowing for the identification of patterns, trends, or correlations. Constructing a scatterplot is a fundamental step in analyzing data, as it provides an intuitive understanding of how one variable may affect another, which is key to assessing linear correlation.