Q. 6.40 - Elementary Statistics

Question

Green Sea Urchins. From the paper "Effects of Chronic Nitrate Exposure on Gonad Growth in Green Sea Urchin Strongylocentrotus droebachiensis" (Aquaculture, Vol.$242$, No.$1-4$, pp.$357-363$) by S.Siikavuopio et al., we found that weights of adult green sea urchins are normally distributed with mean $52.0$g and standard deviation $17.2$g. Let $x$ denote weight of adult green sea urchins.

a. Sketch the distribution of the variable $x$.

b. Obtain the standardized version,$z,$ of $x$.

c. Identify and sketch the distribution of $z$.

d. The percentage of adult green sea urchins with weights between $50$ g and $60$g is equal to the area under he standard normal curve between ____ and ____ .

e. The percentage of adult green sea urchins with weights above $40$g is equal to the area under the standard normal curve that lies to the ____ of _____ .

Step-by-Step Solution

Answer

Part $\left(a\right)$: The distribution of $x$ is as follows :

Part$\left(b\right)$: The standardized version of $x$ interms of $z$ is, $z=\frac{x-52.0}{17.2}$.

Part$\left(c\right)$: The distribution of $z$is as follows :

Part$\left(d\right)$: The percentage of adult green sea urchins with weights between $50$g and $60$g is equal to the area under the standard normal curve between $-0.12$ and $0.47$.

Part$\left(e\right)$: The percentage of adult green sea urchins with weights above $40$g is equal to the area under the standard normal curve that lies to the right of $-0.70$.

Part a Step 1 . Given information

Let $x$ denote the weight of adult green sea urchins.

Part a Step 2 . The distribution of x is as follows:

Part b Step 1 . The standardized version of x interms of z is,

$z=\frac{x-52.0}{17.2}$.

Part c Step 1 . The distribution of z is as follows:

Part d Step 1 . Substitute x = 50 in the standardized version of x .

$$\begin{gathered} z=\frac{50-52.0}{17.2} \\ =-0.12 \end{gathered}$$

Substitute $x=60$ in the standardized version of $x$.

$$\begin{gathered} z=\frac{60-52.0}{17.2} \\ =0.47 \end{gathered}$$

Therefore, the percentage of adult green sea urchins with weights between $50$g and $60$g is equal to the area under the standard normal curve between $-0.12$ and $0.47$.

Part e Step 1 . Substitute x = 40 in the standardized version of x .

$$\begin{gathered} z=\frac{40-52.0}{17.2} \\ =-0.70 \end{gathered}$$

Therefore, the percentage of adult green sea urchins with weights above $40$g is equal to the area under the standard normal curve that lies to the right of $-0.70$.