The Amazon River basin includes the largest remaining connected area of tropical
forest in the world, and ecologists regularly monitor its health. One aspect of its
health is the density of different indigenous species. The Amazon basin includes both
areas that are periodically flooded, and areas that are not periodically flooded. If the
densities of species differ in the flooded and unflooded regions, researchers will want
to take stratified samples in their studies. Stratified sampling would be more
expensive, since boats must be rented to get to any randomly selected flooded sites.
The data below are measures of density (individuals per square kilometer) of primates
in random samples of flooded and unflooded areas in the Amazon basin. Primate density
$\begin{array} { l c c c c c c c } \text { Unflooded } & 123 & 137 & 106 & 77 & 103 & 79 & 195 \ & 227 & 182 & & & & & \end{array}$
$\begin{array} { l l l l l l } \text { Flooded } & 201 & 126 & 185 & 355 & 344 \end{array}$ a) Do these data provide convincing evidence of a difference in the primate density
in flooded and unflooded forests in Amazonia? Provide appropriate statistical
justification for your conclusion.
b) Based on your results in part (a), would you recommend using a stratified sample?
Justify your recommendation with a statistical argument.

Question

Quizplus · Accepted Answer

The Answer is a)
$a) Both distributions indicate skew, but there are no outliers. The t procedure is appropriate. Let \mu _ { u } represent the population mean species density in unflooded areas \mu _ { f } represent the population mean species density in flooded areas \begin{array} { l } H _ { 0 } : \mu _ { u } - \mu _ { f } = 0 \ H _ { a } : \mu _ { u } - \mu _ { f } eq 0 \end{array} There are no indications of skew and no outliers in either the flooded or unflooded regions; the t -procedure is appropriate. \begin{array} { l } t = \frac { \left( \bar { x } _ { u } - \bar { x } _ { f } ight) - \left( \mu _ { u } - \mu _ { f } ight) } { \sqrt { \frac { s _ { u } ^ { 2 } } { n _ { u } } + \frac { s _ { f } ^ { 2 } } { n _ { f } } } } = \frac { ( 136.56 - 242.20 ) - 0 } { \sqrt { \frac { 53.343 ^ { 2 } } { 9 } + \frac { 101.92 ^ { 2 } } { 5 } } } = - 2.159 \ P - ext { value } \approx 0.081 , d f = 5.249 \end{array} With a P -value of 0.081 the results are not significant at the .05 level, and the null hypothesis is not rejected. We do not have sufficient evidence of a difference in primate density in the flooded and unflooded regions of the Amazon. b) Based on these results, the less expensive simple random sample should be used. Absent a difference in primate density in the two regions, stratified sampling is not necessary.$

Both distributions indicate skew, but there are no outliers. The

t

procedure is appropriate.
Let

\mu _ { u }

represent the population mean species density in unflooded areas

\mu _ { f }

represent the population mean species density in flooded areas

\begin{array} { l } H _ { 0 } : \mu _ { u } - \mu _ { f } = 0 \H _ { a } : \mu _ { u } - \mu _ { f } eq 0\end{array}

There are no indications of skew and no outliers in either the flooded or unflooded regions; the

t

-procedure is appropriate.

\begin{array} { l } t = \frac { \left( \bar { x } _ { u } - \bar { x } _ { f } ight) - \left( \mu _ { u } - \mu _ { f } ight) } { \sqrt { \frac { s _ { u } ^ { 2 } } { n _ { u } } + \frac { s _ { f } ^ { 2 } } { n _ { f } } } } = \frac { ( 136.56 - 242.20 ) - 0 } { \sqrt { \frac { 53.343 ^ { 2 } } { 9 } + \frac { 101.92 ^ { 2 } } { 5 } } } = - 2.159 \P - ext { value } \approx 0.081 , d f = 5.249\end{array}

With a

P

-value of

0.081

the results are not significant at the

.05

level, and the null hypothesis is not rejected. We do not have sufficient evidence of a difference in primate density in the flooded and unflooded regions of the Amazon.
b) Based on these results, the less expensive simple random sample should be used. Absent a difference in primate density in the two regions, stratified sampling is not necessary.

The Amazon River Basin Includes the Largest Remaining Connected Area Unflooded 1231371067710379195227182\begin{array} { l c c c c c c c } \text { Unflooded } & 123 & 137 & 106 & 77 & 103 & 79 & 195 \\ & 227 & 182 & & & & & \end{array} Unflooded ​123227​137182​106​77​103​79​195​

The Amazon River Basin Includes the Largest Remaining Connected Area $\begin{array} { l c c c c c c c } \text { Unflooded } & 123 & 137 & 106 & 77 & 103 & 79 & 195 \\ & 227 & 182 & & & & & \end{array}$