Question

Find an equation of the least squares linear regression line of y on x for the given data, and sketch both the line and the data. Predict the value of y corresponding to x = 3.5.

X 1 2 3 4 5 6 7
y 1 1.8 2 . 2 4 4.5 7 9

Answer 1

To find the equation of the least squares linear regression line, calculate the slope and y-intercept. The equation is y = -0.057 + 1.0857x. The predicted value of y for x = 3.5 is 3.75.

Step-by-step explanation:

To find the equation of the least squares linear regression line, we need to calculate the slope and y-intercept. Here are the steps:

1. Calculate the mean of x and y. In this case, the mean of x is 4 and the mean of y is 3.8.
2. Calculate the deviation of each x value from the mean, and the deviation of each y value from the mean. Subtract the mean from each x and y value. The deviations for x are: -3 -2 -1 0 1 2 3, and for y are: -2.8 -1.8 -1.8 0.2 1.2 1.7 3.2.
3. Calculate the sum of the product of x deviations and y deviations. Multiply each x deviation with the corresponding y deviation, and then add up all the products. The sum is 15.2.
4. Calculate the sum of the squares of x deviations. Square each x deviation and add up all the squared values. The sum is 14.
5. Calculate the slope of the regression line. The slope (b) is given by the formula: b = sum of product of x deviations and y deviations / sum of squares of x deviations. In this case, b = 15.2 / 14 = 1.0857 (rounded to four decimal places).
6. Calculate the y-intercept of the regression line. The y-intercept (a) is given by the formula: a = mean of y - b * mean of x. In this case, a = 3.8 - 1.0857 * 4 = -0.057 (rounded to three decimal places).

So, the equation of the least squares linear regression line is: y = -0.057 + 1.0857x. To predict the value of y corresponding to x = 3.5, substitute x = 3.5 into the equation and solve for y:

y = -0.057 + 1.0857(3.5) = 3.75 (rounded to two decimal places).