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	79	32	94	201
y km/100	14.8	19.1	5.7	11.8	35.2

(a) Verify that

Σx = 478,

Σy = 86.6,

Σx² = 61,686,

Σy² = 1994.62,

Σxy = 10941.3,

and

r ≈ 0.94662.

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

ρ > 0.

(Use 2 decimal places.)

t
critical t

d) Find the predicted distance (km/100) when a drift bottle has been floating for 70 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	14 km/100
upper limit	15 km/100

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

β > 0.

(Use 2 decimal places.)

t
critical t

Conclusion

Find a 95% confidence interval for β and interpret its meaning in terms of drift rate

lower limit
upper limit

For every day of drift, the distance drifted increases by an amount that falls outside the confidence interval. For every day of drift, the distance drifted decreases by an amount that falls within the confidence interval.     For every day of drift, the distance drifted increases by an amount that falls within the confidence interval. For every day of drift, the distance drifted decreases by an amount that falls outside the confidence interval.
(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.)
22 km/100

Answer 1

x <- c(72,79,32,94,201)
y <- c(14.8,19.1,5.7,11.8,35.2)
model <- lm (y ~x)
summary (model)
cor(x,y)

a)

> cor(x,y)
[1] 0.94662

yes, correlation is 0.94662

b)

summary (model)

Call:
lm(formula = y ~ x)

Residuals:
     1      2      3      4      5 
 1.410  4.544 -1.030 -5.254  0.330 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept)  1.40177    3.63612   0.386   0.7256  
x            0.16651    0.03274   5.086   0.0147 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.139 on 3 degrees of freedom
Multiple R-squared:  0.8961,    Adjusted R-squared:  0.8615 
F-statistic: 25.87 on 1 and 3 DF,  p-value: 0.01469

t = 5.086

critical t = =T.INV(0.99,3) = 4.5407

t > critical t

hence we reject the null hypothesis

d)

 predict(model,data.frame(x=70))
       1 
13.05738 

e)

predict(model,data.frame(x=70),level = 0.90 ,interval = "confidence")

  fit           lwr       upr
1 13.05738   8.275153   17.8396

90% confidence limit

lower = 8.2752

upper = 17.8396

f)

t = 5.086

critical t = =T.INV(0.99,3) = 4.5407

t > critical t

hence we reject the null hypothesis

95% confidence interval for b =

0.062327

0.27069