Question

11-13 please

11) (3) Answer the three questions below about the correlation coefficient, r. a. (1) What is the range of values that can take on? b. (1) For what value(s) of r will be predictions be best? c(1) For what value(s) of r will be predictions be worst? 12) (7) Researchers are interested in the relationship between children's IQ and the size of a family. The following hypothetical data represent eight families. number of children (X) 4 2 2 1 3 5 1 2 children's average IQ (Y) 104 111 101 131 103 96 123 115 a. (3) Given the following summary statistics: SSx = 14, SSY = 996, SP -97, find a regression equation for using the numbe of children in a family to predict the average 1 of those children. b. (2) Use the regression equation you found in part a to predict the average IQ for the children in a family with 3 childre с. (2) You should find in part b that the predicted average IQ for the children in a family with 3 children is different than the actual average IQ for the family with 3 children in the hypothetical data above (i.e., that family's children have an average IQ of 103). Why isn't the prediction the same as the actual value? 13) (2) What is a source of variance? (Provide a definition as best you can.)

Answer 1

11.

a. The range of values that r can take on? -1 to +1

b. For values of , the predictions will be best.

r> 0.9

c. For values of $\left | r \right |>0.3$ , the predictions will e worst.

12.

We are given Sx=14,Sy=996,sp=-97. We need the mean of x and y.

	X	Y
	4	104
	2	111
	2	101
	1	131
	3	103
	5	96
	1	123
	2	115
Total	20	884

n=8 here.

$\bar{x}=\frac{\sum_{i=1} ^{n}x_{i}}{n}$

$\bar{x}=\frac{20}{8}=2.5$

Σ=1 Ji 3 η

$\bar{y}=\frac{884}{8}=110.5$

The slope of the regression line $\hat{\beta }_{1}=\frac{CP}{S_{x}}$

  $\hat{\beta }_{1}=\frac{-97}{14}$

$\hat{\beta }_{1}=-9.9286$

The intercept of the equation $\hat{\beta }_{0}=\bar{y}-\hat{\beta }_{1}*\bar{x}$

  $\hat{\beta }_{0}=110.5-(-9.9286)*2.5$

$\hat{\beta }_{0}=110.5+17.3214$

  $\hat{\beta }_{0}=127.8214$

The equation to predict IQ scores is

$\hat{IQ}=127.8214-9.9286*children$

(b). The average IQ for children with number of children=3 is

$\hat{IQ}=127.8214-9.9286*3=98.0356$

(c). The actual IQ for families with 3 children is 103 and the predicted value is 98 which is not the same. This difference is observed since the regression uses least squares based on the concept of average and the line is the one that best fits teh data. Or this is the line which is closer to the points and that is why we observe the difference.

13. Source of variance;

In the estimation of regression, we are fitting a line which mimics the data. In doing so, we are trying to explain the variation in the data by a line which is called a regression. Therefore, we are splitting the total variation due to various sources called regression. Ideally, we expect all points to lie on the line so that the entire variation in the data is being explained by the regression. Whatever part is not being explained is called error. Naturally, the regression and the error are the sources of variation.