Chapter 23, Problem 23.5AYK

a)

To determine

To find t statistic and p-value.

Answer to Problem 23.5AYK

Test statistic is -4.989 and p-value is 0.0038

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
	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

Null and alternative hypotheses:

  $H_0:\beta =0$

  $H_0:\beta \ne 0$

Test statistic is,

t = -4.989

Therefore, P-value of one tailed test is half of the two-tailed test.

  $P-value=\frac{0.0075}{2}=0.0038$

Decision: P-value< 0.05, reject H0.

Conclusion: There is sufficient evidence to conclude that there is linear relationship between two variables.

b)

To determine

To find correlation coefficient and find p-value for this one-tailed test.

Answer to Problem 23.5AYK

Test statistic is -4.981 and p-value is 0.0038

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
	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

Formula:

  $t=r\sqrt{\frac{n-2}{1-r^2}}$

Correlation coefficient is r = -0.928

Null and alternative hypotheses:

  $H_0:\rho =0$

  $H_0:\rho <0$

Test statistic is

  $\begin{array}{l}t=-0.928\sqrt{\frac{6-2}{1-{(-0.928)}^2}}\\ t=-4.981\end{array}$

Therefore, P-value of one tailed test is.

  $P-value=0.0038$

Decision: P-value< 0.05, reject H0.

Conclusion: There is sufficient evidence to conclude that there is linear relationship between two variables.