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3. For a random sample of 10 observations of pairs of values

(x, y)

, the equa...

Aug 3, 2024

Solution by Steps

step 1

Given the regression line of

y

x

y = 19.804215 + 0.783997x

step 2

To find the regression line of

x

y

, we need to find the slope of

x

y

. The slope of

x

y

is the reciprocal of the slope of

y

x

step 3

The slope of

y

x

0.783997

. Therefore, the slope of

x

y

\frac{1}{0.783997} \approx 1.275

step 4

The mean value of

y

is given as

83.7

. To find the intercept of

x

y

, we use the formula:

\text{intercept} = \bar{x} - \text{slope} \times \bar{y}

step 5

We need the mean value of

x

, which can be found using the regression line

y = 19.804215 + 0.783997x

and the mean value of

y

step 6

Substituting

\bar{y} = 83.7

into the regression line equation:

83.7 = 19.804215 + 0.783997 \bar{x}

step 7

Solving for

\bar{x}

\bar{x} = \frac{83.7 - 19.804215}{0.783997} \approx 81.5

step 8

Now, using the mean values

\bar{x} = 81.5

and

\bar{y} = 83.7

, the intercept of

x

y

is:

\text{intercept} = 81.5 - 1.275 \times 83.7 \approx -25.2

step 9

Therefore, the equation of the regression line of

x

y

x = -25.2 + 1.275y

Part (b)

step 1

The coefficient of determination,

R^2

, is given as

0.998793

step 2

The coefficient of determination indicates the proportion of the variance in the dependent variable that is predictable from the independent variable

step 3

R^2

value of

0.998793

means that approximately

99.88\%

of the variance in

y

can be explained by the variance in

x

step 4

This high

R^2

value signifies a very strong linear relationship between

x

and

y

Answer

(a) The equation of the regression line of

x

y

x = -25.2 + 1.275y

(b) The coefficient of determination is

0.998793

, indicating a very strong linear relationship between

x

and

y

Key Concept

Regression Line and Coefficient of Determination

Explanation

The regression line of

x

y

is found by taking the reciprocal of the slope of

y

x

and adjusting for the means of

x

and

y

. The coefficient of determination,

R^2

, measures the proportion of variance in the dependent variable that is predictable from the independent variable.