The least-squares regression line for the given data is y = 15.99 + 0.267x.
Step 1: Calculate the mean of x and y
Mean of x (birth weight): (6+10+5+10+7+10+12+3+9+5)/10 = 7.3 pounds
Mean of y (length): (16+20+16+18+19+21+22+16+17+17)/10 = 18.1 inches
Step 2: Calculate the deviations from the mean
For each data point, calculate the deviation from the mean for both x and y. For example, for the first data point (x=6, y=16), the deviation from the mean is x - mean(x) = 6 - 7.3 = -1.3 pounds and y - mean(y) = 16 - 18.1 = -2.1 inches.
Step 3: Calculate the product of the deviations from the mean
For each data point, multiply the deviation from the mean for x by the deviation from the mean for y. For example, for the first data point, the product of the deviations from the mean is (-1.3) * (-2.1) = 2.73.
Step 4: Sum the products of deviations from the mean
Sum the products of deviations from the mean for all data points. In this case, the sum is 2.73 + 2.73 + 0.09 + 2.73 + 0.49 + 2.73 + 5.13 + 0.09 + 0.81 + 0.81 = 17.99.
Step 5: Calculate the sum of squared deviations from the mean for x
For each data point, calculate the squared deviation from the mean for x. For example, for the first data point, the squared deviation from the mean is (-1.3)^2 = 1.69 pounds^2.
Step 6: Sum the squared deviations from the mean for x
Sum the squared deviations from the mean for x for all data points. In this case, the sum is 1.69 + 7.29 + 5.29 + 7.29 + 0.09 + 7.29 + 16.81 + 18.49 + 2.89 + 5.29 = 67.24 pounds^2.
Step 7: Calculate the slope (b1)
Divide the sum of the products of deviations from the mean by the sum of the squared deviations from the mean for x.
b1 = 17.99 / 67.24 = 0.267
Step 8: Calculate the intercept (b0)
Calculate the average of the y values minus the product of the average of the x values and the slope.
b0 = mean(y) - b1 * mean(x)
b0 = 18.1 - 0.267 * 7.3 = 15.99
Step 9: Write the equation of the least-squares regression line**
The equation of the least-squares regression line is:
y = b0 + b1 * x
y = 15.99 + 0.267 * x