Question

Application Exercise:
An ecologist hypothesizes that birds with longer wing spans use wider tree branches. The ecologist captured male birds, measured their wing lengths and other characteristics in millimeters, and then marked and released them. During the ensuing winter, the ecologist repeatedly observed the marked birds as they foraged for food on tree branches. He noted the branch diameter on each occasion, and calculated the average branch diameter for each bird in centimeters. The measurement data are below. What can the ecologist conclude with an α of 0.01?

Wing Length	Branch Diameter
79.3 80.1 80.7 81.5 79.5 80.7 81.1 80.5 79.7	1.02 1.04 1.21 1.53 1.21 1.56 1.39 1.31 1.39

a) What is the appropriate statistic?
---Select--- na Correlation Slope Chi-Square
Compute the statistic selected above:  

c) Compute the appropriate test statistic(s) to make a decision about H₀.
(Hint: Make sure to write down the null and alternative hypotheses to help solve the problem.)
critical value =  ; test statistic =  
Decision:  ---Select--- Reject H0 Fail to reject H0

d) Compute the corresponding effect size(s) and indicate magnitude(s).
If not appropriate, input and/or select "na" below.
effect size =  ;   ---Select--- na trivial effect small effect medium effect large effect

e) Make an interpretation based on the results.

There was a significant positive relationship between wing length of the marked birds and branch diameter.

There was a significant negative relationship between wing length of the marked birds and branch diameter.     

There was no significant relationship between wing length of the marked birds and branch diameter.

Answer 1

a)

Here we need to find the relationship between the quantitaive variables so correlation should be used.

Answer: The correlation

b)

Hypotheses are:

Following table shows the calculation for correlation coeffcient:

	Wing Length, X	Branch Diameter, Y	(X-xmean)^2	(Y-ymean)^2	(X-xmean)(Y-ymean)
	79.3	1.02	1.089936	0.076176	0.288144
	80.1	1.04	0.059536	0.065536	0.062464
	80.7	1.21	0.126736	0.007396	-0.030616
	81.5	1.53	1.336336	0.054756	0.270504
	79.5	1.21	0.712336	0.007396	0.072584
	80.7	1.56	0.126736	0.069696	0.093984
	81.1	1.39	0.571536	0.008836	0.071064
	80.5	1.31	0.024336	0.000196	0.002184
	79.7	1.39	0.414736	0.008836	-0.060536
Total	723.1	11.66	4.462224	0.298824	0.769776

Sample size: n=9

Mean:

so r will be:

------------------------

Test statistics:

-------------------------

Here degree of freedom is df=n-2=7 so p-value of the test is 0.0499.

Because the P-value is less than the significance level 0.05, there is sufficient evidence to support the claim that birds with longer wing spans use wider tree branches.

d)

Effect size: r = 0.6666

It is large effect size.

e)

There was a significant positive relationship between wing length of the marked birds and branch diameter.