Question

Ocean currents are important in studies of climate change, as well as ecology studies of dispersal of plankton. Drift bottles are used to study ocean currents in the Pacific near Hawaii, the Solomon Islands, New Guinea, and other islands. Let x represent the number of days to recovery of a drift bottle after release and y represent the distance from point of release to point of recovery in km/100. The following data are representative of one study using drift bottles to study ocean currents.

x days	72	76	32	93	207
y km/100	14.8	19.1	5.3	11.8	35.7

(a) Verify that

Σx = 480,

Σy = 86.7,

Σx² = 63,482,

Σy² = 2025.67,

Σxy = 11174.1,

and

r ≈ 0.94564.

Σx
Σy
Σx2
Σy2
Σxy
r

(b) Use a 1% level of significance to test the claim

ρ > 0.

(Use 2 decimal places.)

t
critical t

Conclusion

(c) Verify that

S_e ≈ 4.2911,

a ≈ 1.6127,

and

b ≈ 0.1638.

Se
a
b

(d) Find the predicted distance (km/100) when a drift bottle has been floating for 60 days. (Use 2 decimal places.)

= km/100

(e) Find a 90% confidence interval for your prediction of part (d). (Use 1 decimal place.)

lower limit	km/100
upper limit	km/100

(f) Use a 1% level of significance to test the claim that

β > 0.

(Use 2 decimal places.)

t
critical t

(g) Find a 95% confidence interval for β and interpret its meaning in terms of drift rate. (Use 2 decimal places.)

Lower limit=

Upper limit=

(h) Consider the following scenario. A sailboat had an accident and radioed a Mayday alert with a given latitude and longitude just before it sank. The survivors are in a small (but well provisioned) life raft drifting in the part of the Pacific Ocean under study. After 30 days, how far from the accident site should a rescue plane expect to look? (Use 2 decimal places.)
= km/100

Answer 1

Answer a)

Σx	480
Σy	86.7
Σx2	63482
Σy2	2025.67
Σxy	11174.1
r	0.9456

Answer b)

Critical t value corresponding to df = 3 and α = 0.01 for a right tailed test is 4.541 (Obtained using t distribution table. Screenshot attached)

t	5.034
critical t	4.541

Conclusion

Since t = 5.034 > critical t = 4.541, we reject the null hypothesis. There is enough evidence t to claim that the population correlation ρ is greater than 0, at the 0.01 significance level.

Answer c)

a = 1.6127

b = 0.1638

s_e = 4.2911

Answer d)

Based on Part c) regression equation can be written as:

y = 1.61327 + 0.16438*x

At x = 60

y =  1.61327 + 0.16438*60

Predicted distance y (km/100) = 11.48

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