Q 3.215 - Elementary Statistics

Question

Frog Thumb Length. W. Duellman and J. Kohler explore a new species of frog in the article "New Species of Marsupial Frog (Hylidae: Hemiphractinae: Gastrotheca) from the Yungas of Bolivia" (Journal of Herpetology, Vol. $39$, No. $1$. pp. $91-100$). These two museum researchers collected information on the lengths and widths of different body parts for the male and female Gastrotheca piper ata. Thumb length for the female Gastrotheca piperata has a mean of $6.71$ mm and a standard deviation of $0.67$ mm. Let x denote thumb length for a female specimen.

a. Find the standardized version of x.

b. Determine and interpret the $2$-scores for thumb lengths of $5.2$ mm and $8.1$ mm. Round your answers to two decimal places.

Step-by-Step Solution

Answer

Part (a):

Standardized variable of x is $z=\frac{x-266}{16}$

Part (b):

Interpretation for $5.2$ mm: Thumb length is $2.25$ standard deviations below the mean

Interpretation for $8.1$ mm: Thumb length is $2.07$ standard deviations below the mean

Step 1. Given information is:

Thumb length for the female Gastrotheca piperata has a mean of  $6.71$ mm and a standard deviation of $0.67$ mm.  

Part (a) Step 1. Standardized Version of x

We have to find the standardized version of x.

For a variable x, the variable $z=\frac{x-\mu }{\sigma }$is called the  standardized version of x or the standardized variable corresponding to the variable x.

Standardized variable of x is

$z=\frac{x-266}{16}$

Part (b) Step 1. Finding z-scores

The z-scores for thumb length of $5.2$ mm

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ z=\frac{5.2-6.71}{0.67} \\ z=-2.2537 \\ z=-2.25 \end{gathered}$$

Interpretation: Thumb length is $2.25$ standard deviations below the mean

The z-scores for thumb length of $8.1$ mm

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ z=\frac{8.1-6.71}{0.67} \\ z=2.0746 \\ z=2.07 \end{gathered}$$

Interpretation: Thumb length is $2.07$ standard deviations above the mean