Question

In this problem, assume that the distribution of differences is approximately normal. Note: For degrees of freedom d.f. not in the Student's t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more "conservative" answer.

Is fishing better from a boat or from the shore? Pyramid Lake is located on the Paiute Indian Reservation in Nevada. Presidents, movie stars, and people who just want to catch fish go to Pyramid Lake for really large cutthroat trout. Let row B represent hours per fish caught fishing from the shore, and let row A represent hours per fish caught using a boat. The following data are paired by month from October through April.

	Oct	Nov	Dec	Jan	Feb	March	April
B: Shore	1.4	1.8	2.0	3.2	3.9	3.6	3.3
A: Boat	1.3	1.3	1.5	2.2	3.3	3.0	3.8

Use a 1% level of significance to test if there is a difference in the population mean hours per fish caught using a boat compared with fishing from the shore. (Let d = B − A.)(a) What is the level of significance?


State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test?

H₀: μ_d = 0; H₁: μ_d ≠ 0; two-tailedH₀: μ_d = 0; H₁: μ_d > 0; right-tailed     H₀: μ_d ≠ 0; H₁: μ_d = 0; two-tailedH₀: μ_d = 0; H₁: μ_d < 0; left-tailed

(b) What sampling distribution will you use? What assumptions are you making?

The standard normal. We assume that d has an approximately uniform distribution.The Student's t. We assume that d has an approximately normal distribution.     The Student's t. We assume that d has an approximately uniform distribution.The standard normal. We assume that d has an approximately normal distribution.

What is the value of the sample test statistic? (Round your answer to three decimal places.)


(c) Find (or estimate) the P-value.

P-value > 0.5000.250 < P-value < 0.500     0.100 < P-value < 0.2500.050 < P-value < 0.1000.010 < P-value < 0.050P-value < 0.010

Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.     At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

(e) State your conclusion in the context of the application.

A. Reject the null hypothesis, there is sufficient evidence to claim a difference in population mean hours per fish between boat fishing and shore fishing.

B.Reject the null hypothesis, there is insufficient evidence to claim a difference in population mean hours per fish between boat fishing and shore fishing.

C. Fail to reject the null hypothesis, there is insufficient evidence to claim a difference in population mean hours per fish between boat fishing and shore fishing.

D.Fail to reject the null hypothesis, there is sufficient evidence to claim a difference in population mean hours per fish between boat fishing and shore fishing.

Answer 1

level of significance =0.01

H₀: μ_d = 0; H₁: μ_d ≠ 0; two-tailed

b)

The Student's t. We assume that d has an approximately normal distribution.

S. No	B	A	diff:(d)=x1-x2	d2
1	1.4	1.3	0.1	0.01
2	1.8	1.3	0.5	0.25
3	2	1.5	0.5	0.25
4	3.2	2.2	1	1.00
5	3.9	3.3	0.6	0.36
6	3.6	3	0.6	0.36
7	3.3	3.8	-0.5	0.25
	total	=	Σd=2.8	Σd2=2.48
	mean dbar=	d̅     =	0.4000
	degree of freedom =n-1                            =		6
	Std deviaiton SD=√(Σd2-(Σd)2/n)/(n-1) =		0.476095
	std error=Se=SD/√n=		0.1799
	test statistic            =	(d̅-μd)/Se         =	2.223

c)

0.050 < P-value < 0.10

d)

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

e)

C. Fail to reject the null hypothesis, there is insufficient evidence to claim a difference in population mean hours per fish between boat fishing and shore fishing.