Question

Do beavers benefit beetles? Researchers laid out 23 circular plots, each 4 meters in diameter, in an area where beavers were cutting down cottonwood trees. In each plot, they counted the number of stumps from trees cut by beavers and the number of clusters of beetle larvae. Ecologists think that the new sprouts from stumps are more tender than other cottonwood growth, so that beetles prefer them. If so, more stumps should produce more beetle larvae. The table contains the data 2 213 3 4 3125 10 30 12 24 36 40 43112756 18 40 2 1 2 2 11 4 12 1 25 8 2114166 5491314 50 13 Stumps Beetle larvae 4 Stumps Beetle larvae To access the complete data set, click the link for your preferred software format Excel Minitab JMP SPSS TI R Mac-TXT PC-TXT CSV CrunchIt! Analyze these data to see if they support the "beavers benefit beetles" idea. Follow the four-step process in reporting your work STATE: Choose the statement that best describes the goal of the research. The number of tree stumps is the explanatory variable, and we expect an increase in the number of tree stumps to induce a decrease in the response variable, the number of beetle larvae. The number of tree stumps is the explanatory variable, and we expect an increase in the number of tree stumps to induce an increase in the response variable, the number of beetle larvae. Beetle larvae are the explanatory variable, and we expect an increase in beetle larvae to induce a decrease in the response variable, the number of tree stumps. Beetle larvae are the explanatory variable, and we expect an increase in beetle larvae to induce an increase in the response variable, the number of tree stumps.

Answer 1

the number of tree stumps is the explanatory variable, and we expect an increase in the number of tree stumps to induce an increase in the response variable,the number of beetle larvae

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Scatterplot 60 50 40 30 20 10 0 0 1 2 3 4 5 X st

there is a positive linear relationship

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x	y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
2	10	0.05	227.62	3.28
2	30	0.05	24.14	-1.07
1	12	1.48	171.27	15.93
3	24	0.61	1.18	-0.85
3	36	0.61	119.09	8.54
4	40	3.18	222.40	26.58
3	43	0.61	320.88	14.02
1	11	1.48	198.44	17.15
2	27	0.05	3.66	-0.42
5	56	7.74	955.62	86.02
1	18	1.48	50.22	8.63
3	40	0.61	222.40	11.67
2	25	0.05	0.01	0.02
1	8	1.48	291.96	20.80
2	21	0.05	16.70	0.89
2	14	0.05	122.92	2.41
1	16	1.48	82.57	11.06
1	6	1.48	364.31	23.24
4	54	3.18	835.96	51.54
1	9	1.48	258.79	19.58
2	13	0.05	146.09	2.63
1	14	1.48	122.92	13.50
4	50	3.18	620.66	44.41

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	51	577	31.91304348	5379.8	379.57
mean	2.22	25.09	SSxx	SSyy	SSxy

sample size ,   n =   23          
here, x̅ = Σx / n=   2.22   ,     ȳ = Σy/n =   25.09  
                  
SSxx =    Σ(x-x̅)² =    31.9130          
SSxy=   Σ(x-x̅)(y-ȳ) =   379.6          
                  
estimated slope , ß1 = SSxy/SSxx =   379.6   /   31.913   =   11.89
                  
intercept,   ß0 = y̅-ß1* x̄ =   -1.286
                  

correlation coefficient ,    r = Sxy/√(Sx.Sy) =   0.916

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R² =    (Sxy)²/(Sx.Sy) =    0.839

yes, the strong positive association supports the idea that beavers benefit beetles.