Question

The table below gives the age and bone density for five randomly selected women. Using this data, consider the equation of the regression line, yˆ=b0+b1x', for predicting a woman's bone density based on her age. 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.

Age	40	41	42	43	63
Bone Density	353	344	328	326	322

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:

Determine if the statement "Not all points predicted by the linear model fall on the same line" is true or false.

Step 4 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 yˆ.

Step 5 of 6:

Find the estimated value of y when x=42. 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

	X	Y	X * Y	X2	Y2
	40	353	14120	1600	124609
	41	344	14104	1681	118336
	42	328	13776	1764	107584
	43	326	14018	1849	106276
	63	322	20286	3969	103684
Total	229	1673	76304	10863	560489

Equation of regression line is Ŷ = a + bX

b = ( 5 * 76304 - 229 * 1673 ) / ( 5 * 10863 - ( 229 )²)
b = -0.852
a =( Σ Y - ( b * Σ X) ) / n
a =( 1673 - ( -0.8522 * 229 ) ) / 5
a = 373.63
Equation of regression line becomes Ŷ = 373.6302 - 0.8522 X

Step 1

Slope of regression line is  b = -0.852

Step 2

Y intercept of regression line is   a = 373.63

Step 3

Looking at the scatter plot statement Not all points predicted by the linear model fall on the same line is true

Step 4

Ŷ = 373.6302 - 0.8522 X

Ŷ = 373.6302 - 0.8522 (1)

Ŷ = 372.78

Step 5

When X = 42
Ŷ = 373.63 + -0.852 X
Ŷ = 373.63 + ( -0.852 * 42 )
Ŷ = 337.85

Step 6

r = -0.622

Coefficient of Determination
R² = r² = 0.387