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+b1xy^=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	0.50.5	22	2.52.5	33	3.53.5	44	55
Overall Grades	9393	8888	8080	7777	7272	6868	6161

Table

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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:

Substitute the values you found in steps 1 and 2 into the equation for the regression line to find the estimated linear model. According to this model, if the value of the independent variable is increased by one unit, then find the change in the dependent variable yˆy^.

Step 4 of 6:

Determine if the statement "All points predicted by the linear model fall on the same line" is true or false.

Step 5 of 6:

Find the estimated value of y when x=2.5x=2.5. Round your answer to three decimal places.

Step 6 of 6:

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

Answer 1

Step 1

Hours Unsupervised (x) = 0.5, 2, 2.5, 3, 3.5, 4, 5

Overall Grades (y) = 93, 88, 80, 77, 72, 68, 61

$\sum x = 20.5$

$\bar{x} =$$\sum x/n = 20.5/7 = 2.928571$

$\sum y = 539$

$\bar{y} =$$\sum y/n = 539/7 = 77$

$\sum x^2 = 72.75$

$\sum y^2 = 42251$

$\sum xy = 1482.5$

$S_{xx} = \sum x^2 - \left ( \sum x \right )^2 / n = 72.75 - 20.5^2 / 7 = 12.71429$

$S_{yy} = \sum y^2 - \left ( \sum y \right )^2 / n = 42251 - 539^2 / 7 = 748$

$S_{xy} = \sum xy - \left ( \sum x \sum y \right ) / n = 1482.5 - (20.5 * 539) / 7 = -96$

Slope = Sxy / Sxx = -96 / 12.71429 = -7.551

Step 2 -

y-intercept = $\bar{y}$ - Slope * $\bar{x}$ = 77 - (-7.551) * 2.928571 = 99.114

Step 3 -

Estimated linear model is,

$\hat{y} = 99.114 - 7.551 x$

According to this model, if the value of the independent variable is increased by one unit, the change in the dependent variable is -7.551

Step 4 -

The estimated model is the equation of a linear straight line. Thus, all points predicted by the linear model fall on the same line. It is a true statement.

Step 5 -

For x = 2.5, the estimated value of y is,

$\hat{y} = 99.114 - 7.551 * 2.5$ = 80.237

Step 6.

The coefficient of determination is,

$R^2 = S_{xy}^2 / S_{xx}S_{yy} = (-96)^2 / (12.71429*748) = 0.969$

As, the linear correlation coefficient is not $\pm 1$,