Question

Hello! I mainly need part B fully answered but if you are willing to answer the rest that would be awesome! Thank you in advance

iPad? 10:50 PM 12%I My Notes Ask Your Let x be the number of different research programs, and let y be the mean number of patents per program. As in any business, a company can spread itself too thin. For example, too many research programs might lead to decline in overall research productivity. The following data are for a collection of pharmaceutical companies and their research programs Complete parts (a) through (e), given Zr . 90. ?? . 8. Zr2-1420-Xy2 11.5, ??? . 113, and ra-0.917. a) Draw a scatter diagram displaying the data. Submission Data (b) Verify the given sums Ex, Ey, E2, Ey2, Exy, and the value of the sample correlation coefficient r. (Round your value for r to three decimal places) (c) Find x, and y. Then find the equation of the least-squares line y-a+bx. (Round your answers for x and y to two decimal places. Round your answers for a and b to three decimal places) (d) Graph the least-squares line. Be sure to plot the point (x, y) as a point on the line.

Answer 1

(a)

Following table shows the calculations:

	X	Y	X^2	Y^2	XY
	10	1.6	100	2.56	16
	12	1.8	144	3.24	21.6
	14	1.5	196	2.25	21
	16	1.4	256	1.96	22.4
	18	1	324	1	18
	20	0.7	400	0.49	14
Total	90	8	1420	11.5	113

Sample size: n=6

Now,

$S_{yy}=\sum y^{2}-\frac{\left (\sum y \right )^{2}}{n}=0.8333$

$S_{xx}=\sum x^{2}-\frac{\left (\sum x \right )^{2}}{n}=70$

$S_{xy}=\sum xy-\frac{\left (\sum x \right )\left (\sum y \right )}{n}=-7$

The coeffcient of correlation is :

$r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}=\frac{-7}{\sqrt{0.8333\cdot 70}}=-0.917$

(c)

$\bar{x}=\frac{\sum x}{n}=15$
$\bar{y}=\frac{\sum y}{n}=1.3333$

Slope of the regression equation is

$b_{1}=\frac{S_{xy}}{S_{xx}}=-0.1$

and intercept of the equation will be

$b_{0}=\frac{1}{n}(\sum y - b_{1} \sum x)=2.833$

So the regression equation will be

y'=2.833-0.1x

(d)

For graph first option is correct.

(e)

$r^{2}=(-0.917)\cdot (-0.917)=0.84$

explained: 84%

unexplained: 100% -84% = 16%

(f)

The predicted value is:

y'=2.833-0.1*18 = 1.033

Answer:1.03