Question

Bardi Trucking Co., located in Cleveland, Ohio, makes deliveries in the Great Lakes region, the Southeast, and the Northeast. Jim Bardi, the president, is studying the relationship between the distance a shipment must travel and the length of time, in days, it takes the shipment to arrive at its destination. To investigate, Mr. Bardi selected a random sample of 20 shipments made last month. Shipping distance is the independent variable and shipping time is the dependent variable. The results are as follows:

#	Shipment Distance Miles		Shipment time (days)
1	626		5
2	801		11
3	733		4
4	640		3
5	756		13
6	790		8
7	718		8
8	652		3
9	792		12
10	629		4
11	696		9
12	718		10
13	754		3
14	648		4
15	666		3
16	831		10
17	691		13
18	822		9
19	793		3
20	784		11

QUESTION: State the decision rule for 0.01 significance level: H0: ρ ≤ 0; H1: ρ > 0. (Round your answer to 3 decimal places.)

Reject H0 if t > _______ I understand the answer is 2.552

Explain how to get the decision rule. In this case 2.552 ?

Answer 1

	X	Y	X * Y	X2	Y2
	626	5	3130	391876	25
	801	11	8811	641601	121
	733	4	2932	537289	16
	640	3	1920	409600	9
	756	13	9828	571536	169
	790	8	6320	624100	64
	718	8	5744	515524	64
	652	3	1956	425104	9
	792	12	9504	627264	144
	629	4	2516	395641	16
	696	9	6264	484416	81
	718	10	7180	515524	100
	754	3	2262	568516	9
	648	4	2592	419904	16
	666	3	1998	443556	9
	831	10	8310	690561	100
	691	13	8983	477481	169
	822	9	7398	675684	81
	793	3	2379	628849	9
	784	11	8624	614656	121
Total	14540	146	108651	10658682	1332

r = 0.5181

To Test :-
H0 :- ρ = 0
H1 :- ρ > 0

Test Statistic :-
t = (r * √(n - 2) / (√(1 - r²))
t = ( 0.5181 * √(20 - 2) ) / (√(1 - 0.2684) )
t = 2.5699

Test Criteria :-
Reject null hypothesis if t > t(α)
Critical value t(α,n-2) = t(0.01 , 20 - 2 ) = 2.552 ( from t table )
t > t(α, n-2) = 2.5699 > 2.552
Result :- Reject null hypothesis

Decision based on P value
P - value = P ( t > 2.5699 ) = 0.0096
Reject null hypothesis if P value < α = 0.01 level of significance
P - value = 0.0096 < 0.01 ,hence we reject null hypothesis
Conclusion :- We reject H0

There is sufficient evidence to support the claim that H1: ρ > 0.