Scatter plots and least square lines.
The number of home runs hit by a player.
1990 1991 92 93 94 95 96 97 98 99
39. 22. 42 9. 9. 39 52 58 70 65
Find the equation of the least square line? Y=4.9x+18.5
Find the correlation, r, to the nearest hundredth? 0.68
Use the least square line to predict the number of home runs the player hits in 2000. Can't figure this one out. Thanks
To find the equation of the least square line, we use the method of linear regression. This method allows us to find a line that best fits the data points in a scatter plot.
Step 1: Calculate the mean of the x-values (years) and the mean of the y-values (number of home runs).
In this case, the mean of the x-values is (1990 + 1991 + 1992 + ... + 1999) / 10 = 1994.5, and the mean of the y-values is (39 + 22 + 42 + ... + 65) / 10 = 46.1.
Step 2: Calculate the differences between each x-value and the mean of the x-values (x - x̄), and the differences between each y-value and the mean of the y-values (y - ȳ).
For example, for the first data point (1990, 39), the difference between the x-value and the mean of x-values is 1990 - 1994.5 = -4.5, and the difference between the y-value and the mean of y-values is 39 - 46.1 = -7.1.
Step 3: Calculate the product of the differences obtained in Step 2 for each data point.
For the first data point, the product of the differences is (-4.5 * -7.1) = 31.95.
Step 4: Square each difference obtained in Step 2 for each data point.
For the first data point, the square of the difference is (-4.5)^2 = 20.25.
Step 5: Calculate the sum of the squared differences obtained in Step 4.
For all the data points, the sum of the squared differences is 20.25 + ... + (last data point's squared difference).
Step 6: Calculate the sum of the product of the differences obtained in Step 3 for each data point.
For all the data points, the sum of the product of the differences is 31.95 + ... + (last data point's product of differences).
Step 7: Calculate the slope of the least square line.
The slope (b) is given by: b = sum of the product of differences divided by the sum of the squared differences.
Step 8: Calculate the y-intercept of the least square line (a).
The y-intercept (a) is given by: a = mean of y-values - (slope * mean of x-values).
Step 9: Write the equation of the least square line.
The equation of the least square line is: Y = b * X + a, where Y is the dependent variable (in this case, number of home runs) and X is the independent variable (in this case, years).
By following this step-by-step process, we can find the equation of the least square line.