Final answer:
The correlation coefficient (r) measures the strength of the linear relationship between x and y variables. A value of r = .55 indicates a moderately strong relationship. The validity of using a linear model to make predictions depends on both the value of r and the sample size.
Step-by-step explanation:
The correlation coefficient, r, is a statistical measure that calculates the strength of the relationship between the independent variable (x) and the dependent variable (y) in a dataset. This coefficient can vary between -1 and +1, with -1 indicating a perfect negative linear relationship, +1 indicating a perfect positive linear relationship, and 0 indicating no linear relationship. The formula to calculate r involves several steps including the multiplication and summation of the x and y values, as well as the adjustment for the number of data points (n).
When analyzing a scatter plot, besides computing the r value, it is critical to examine the pattern of the points. If the points suggest a linear trend, then the line of best fit provided by linear regression can be a good predictor for the data. If the pattern indicates that a curve would fit the data better, then a statistician may use different models instead of a straight-line fit. The line of best fit is calculated by minimizing the sum of the squared errors (SSE), also known as the least-squares criterion.
The value of r squared, or the coefficient of determination (r²), when expressed as a percentage, represents the amount of variation in the dependent variable that can be explained by variation in the independent variable using the regression line. For the provided dataset, the correlation coefficient is r = .55, which signifies a moderately strong linear relationship between x and y. Finally, the reliability of a linear model not only depends on the value of the r, but also on the sample size n, and thus both should be considered together.
In conclusion, if the correlation coefficient is significantly different from zero and the scatter plot shows a linear trend, the regression line can be used to make predictions.