Chapter 23, Problem 23.1AYK

a)

To determine

To construct scatter plot and conclude about the correlation.

Explanation of Solution

Given:

Number of D.magna grazers	Net growth rate of G. semen
1	-1.9
2	-2.5
3	-2.2
4	-3.9
5	-4.1
6	-4.3

The scatter plot:

Yes, there is strong relationship between two variables.

b)

To determine

To find slope and y-intercept.

Explanation of Solution

Given:

Number of D.magna grazers	Net growth rate of G. semen
1	-1.9
2	-2.5
3	-2.2
4	-3.9
5	-4.1
6	-4.3

Regression analysis:

Regression Analysis
	r²	0.862	n	6
	r	-0.928	k	1
	Std. Error	0.443	Dep. Var.	Y
ANOVA table
Source	SS	df	MS	F	p-value
Regression	4.8893	1	4.8893	24.89	.0075
Residual	0.7857	4	0.1964
Total	5.6750	5
Regression output
variables	coefficients	std. error	t (df=4)	p-value
Intercept	-1.30	0.41	-3.151	.0345
X	-0.53	0.11	-4.989	.0075

Slope = $\beta$ = -0.53

Y-intercept = $\alpha$ = -1.30

Regression equation is,

  $\widehat{y}=-1.30+(-0.53)x$

c)

To determine

To find the residuals and standard error.

Explanation of Solution

Given:

Number of D.magna grazers	Net growth rate of G. semen
1	-1.9
2	-2.5
3	-2.2
4	-3.9
5	-4.1
6	-4.3

Slope = $\beta$ = -0.53

Y-intercept = $\alpha$ = -1.30

Regression equation is,

  $\widehat{y}=-1.30+(-0.53)x$

The residuals are:

X	Y	Predicted	Residual	(Residual)2
1	-1.90	-1.83	-0.07	0.005
2	-2.50	-2.36	-0.14	0.020
3	-2.20	-2.89	0.69	0.470
4	-3.90	-3.41	-0.49	0.236
5	-4.10	-3.94	-0.16	0.025
6	-4.30	-4.47	0.17	0.029
Total			0.00	0.786

So, SSE = 0.786

Therefore, standard error is,

  $\sigma =\sqrt{\frac{SSE}{n-2}}=\sqrt{\frac{0.786}{6-2}}=0.443$