Question

The table below gives the number of hours spent unsupervised each day as well as the overall grade averages for seven randomly selected middle school students. Using this data, consider the equation of the regression line, yˆ=b0+b1x, for predicting the overall grade average for a middle school student based on the number of hours spent unsupervised each day. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant. Hours Unsupervised 2.5 3 3.5 4 4.5 5.5 6 Overall Grades 99 97 87 83 78 69 63 Table

Step 1 of 6 : Find the estimated slope. Round your answer to three decimal places

. Step 2 of 6: Find the estimated y-intercept. Round your answer to three decimal places.

Step 3 of 6: Determine the value of the dependent variable yˆ at x=0. (bo/b1/x/y)

Step 4 of 6: According to the estimated linear model, if the value of the independent variable is increased by one unit, then the change in the dependent variable yˆ is given by? (bo/b1/x/y) Step 5 of 6: Determine if the statement "All points predicted by the linear model fall on the same line" is true or false.

Step 6 of 6: Find the value of the coefficient of determination. Round your answer to three decimal places

Answer 1

Step 1:

Sum of X = 29
Sum of Y = 576
Mean X = 4.1429
Mean Y = 82.2857
Sum of squares (SS_X) = 9.8571
Sum of products (SP) = -102.7857

Regression Equation = ŷ = bX + a

b1 = SP/SS_X = -102.79/9.86 = -10.428

Step 2: b0 = M_Y - b1M_X = 82.29 - (-10.43*4.14) = 125.486

ŷ = -10.428X + 125.486

Step 3: For x=0, ŷ = 125.486

And this is intercept value b0

Step 4: If the value of the independent variable is increased by one unit, then the change in the dependent variable yˆ is given by b1=-10.428 and it is slope value.

Step 5:

All points predicted by the linear model fall on the same line" is true.

Step 6:

X Values
∑ = 29
Mean = 4.143
∑(X - M_x)² = SS_x = 9.857

Y Values
∑ = 576
Mean = 82.286
∑(Y - M_y)² = SS_y = 1085.429

X and Y Combined
N = 7
∑(X - M_x)(Y - M_y) = -102.786

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = -102.786 / √((9.857)(1085.429)) = -0.994

So R^2=-0.994^2=0.988

So 98.8% of variation in y is explained by x