Question

At a recent county fair, you observed that at one stand people's weight was forecasted, and were surprised by the accuracy (within a range). Thinking about how the person could have predicted your weight fairly accurately (despite the fact that she did not know about your "heavy bones"), you think about how this could have been accomplished. You remember that medical charts for children contain 5%, 25%, 50%, 75% and 95% lines for a weight/height relationship and decide to conduct an experiment with 110 of your peers. You collect the data and calculate the following sums: where the height is measured in inches and weight in pounds.
(a) Calculate the slope and intercept of the regression.
(b) Find the regression R2.

Answer 1

Answer:

a)Slope: $\hat \beta_1 =\frac{7625.9}{1248.9}=6.106$

Intercept: $\hat \beta_o = 157.955 -6.106 (69.686)=-267.548$

b) $r=\frac{7625.9}{\sqrt{[1248.9][94228.8]}}=0.657$  

And the determination coeffecient is $r^2 = 0.657^2 =0.432$  

Step-by-step explanation:

Data given and definitions

The correlation coefficient is a "statistical measure that calculates the strength of the relationship between the relative movements of two variables". It's denoted by r and its always between -1 and 1.  

The sum of squares "is the sum of the square of variation, where variation is defined as the spread between each individual value and the grand mean"    

When we conduct a multiple regression we want to know about the relationship between several independent or predictor variables and a dependent or criterion variable.  

n=110, $\sum x_i y_i = \sum (X-\bar X)(Y-\bar Y) =7625.9,\sum x^2_i=\sum (x-\bar x)^2 =1248.9 , sum y^2_i=\sum(y-\bar y)^2 =94228.8$  

$\sum Y_i =17375 , \sum X_i = 7665.5$

Part a

The slope is given by this formula:

$\hat \beta_1 =\frac{\sum (x-\bar x) (y-\bar y)}{\sum (x-\bar x )^2}$

If we replace we got:

$\hat \beta_1 =\frac{7625.9}{1248.9}=6.106$

We can find the intercept with the following formula

$\hat \beta_o = \bar y -\hat \beta_1 \bar x$

We can find the average for x and y like this:

$\bar X=7665.5/110 =69.686, \bar y= 17375/110=157.955$

And replacing we got:

$\hat \beta_o = 157.955 -6.106 (69.686)=-267.548$

Part b

In order to calculate the correlation coefficient we can use this formula:  

$r=\frac{\sum (x-\bar x)(y-\bar y) }{\sqrt{[\sum (x-\bar x)^2][\sum(y-\bar y)^2]}}$  

For our case we have this:  

n=110, $\sum x_i y_i = \sum (X-\bar X)(Y-\bar Y) =7625.9,\sum x^2_i=\sum (x-\bar x)^2 =1248.9 , sum y^2_i=\sum(y-\bar y)^2 =94228.8$  

So we can find the correlation coefficient replacing like this:

$r=\frac{7625.9}{\sqrt{[1248.9][94228.8]}}=0.657$  

And the determination coeffecient is $r^2 = 0.657^2 =0.432$