Question

The table shows the amount Andre, as a waiter, earned. The money, in dollars, is the total he earned.

Time(hours): 0, 1, 2, 3, 4, 5, 6, 7
Money(dollars): 0, 16, 35, 47, 60, 69, 90, 103

Part A:
Let c represent the number of hours. Write a regression for the data. Round the slope and y-intercept to the nearest whole number.

Part B:
Find the correlation coefficient, or r-value, of the line. Explain what this indicates.

Answer 1

The regression equation for the given data is y = 14x, where y represents the money earned and x represents the number of hours. The correlation coefficient is approximately 0.817, indicating a strong positive linear relationship between the number of hours worked and the money earned.

Step-by-step explanation:

Part A: To write a regression equation, we need to find the slope and y-intercept. We can use any two points from the table to calculate the slope using the formula: slope = Δy / Δx, where Δy is the change in money and Δx is the change in time. Let's choose the points (0, 0) and (7, 103) to calculate the slope:

slope = (103 - 0) / (7 - 0) = 103 / 7 ≈ 14.71

Next, we can use the slope-intercept form of a linear equation, y = mx + b, where m is the slope and b is the y-intercept. Let's plug in the slope value and choose one of the points, (0, 0), to solve for the y-intercept:

0 = 14.71(0) + b

b = 0

Therefore, the regression equation is y = 14x, where y represents the money earned and x represents the number of hours.

Part B: To find the correlation coefficient, we need to calculate the coefficient of determination (r-squared). The formula for r-squared is: r^2 = SSreg / SStotal, where SSreg is the sum of squares regression and SStotal is the sum of squares total. Since we only have one variable, the correlation coefficient is the square root of the coefficient of determination, r = sqrt(r^2). To calculate r-squared, we need to find the sums of squares:

SSreg = Σ(y_pred - y_mean)^2 = 26488.5

SStotal = Σ(y - y_mean)^2 = 39708

Now, we can calculate r-squared:

r^2 = SSreg / SStotal = 26488.5 / 39708 ≈ 0.668

Finally, we can calculate the correlation coefficient:

r = sqrt(r^2) = sqrt(0.668) ≈ 0.817

The correlation coefficient indicates a strong positive linear relationship between the number of hours worked and the money earned. A value of 0.817 suggests that there is a strong positive association between the two variables, meaning that as the number of hours worked increases, the amount of money earned also increases. However, please note that correlation does not imply causation.