Question

The owner of a sporting goods store gathered data about the numbers of men’s, women’s and children’s bicycles sold each month. She used a graphing tool to organize the data in different scatter plots and found these lines of best fit: The relationship between the numbers of men’s, x, and women’s, y, bicycles sold is modeled by the equation y = 0.95x + 0.83, and the correlation coefficient for the data is 0.791. The relationship between the numbers of men’s, x, and children’s, y, bicycles sold is modeled by the equation y = 0.87x + 0.98, and the correlation coefficient for the data is 0.564. The relationship between the numbers of children’s, x, and women’s, y, bicycles sold is modeled by the equation y = 0.73x + 0.91, and the correlation coefficient for the data is 0.945. Which line of best fit can the store owner expect to produce the most reliable predictions? A. the line modeling the numbers of men’s and women’s bicycles sold B. the line modeling the numbers of men’s and children’s bicycles sold C. the line modeling the numbers of children’s and women’s bicycles sold D. All three lines will give equally reliable predictions.

Answer 1

The line modeling children's and women's bicycles sold, with a correlation coefficient of 0.945, is expected to offer the most reliable predictions among the three lines of best fit.

To determine which line of best fit will provide the most reliable predictions, we need to analyze the correlation coefficients for each relationship. The correlation coefficient, ranging from -1 to 1, indicates the strength and direction of the linear relationship between variables. A higher absolute value of the correlation coefficient suggests a stronger correlation.

In this case, the correlation coefficients are as follows:

Men’s and Women’s Bicycles: 0.791

Men’s and Children’s Bicycles: 0.564

Children’s and Women’s Bicycles: 0.945

The correlation coefficient for the relationship between children’s and women’s bicycles (0.945) is the highest among the three. A correlation close to 1 indicates a strong positive linear relationship. Therefore, the line modeling the numbers of children’s and women’s bicycles sold is likely to produce the most reliable predictions.

In summary, the store owner can expect the line modeling the numbers of children’s and women’s bicycles sold to provide the most reliable predictions due to its high correlation coefficient.