Question

The weights of laboratory cockroaches can be modeled with a Normal distribution having mean $80$ grams and standard deviation $2$ grams. The following figure is the Normal curve for this distribution of weights.

Point C on this Normal curve corresponds to

$$\begin{gathered} a.84grams. \\ b.82grams. \\ c.78grams. \\ d.76grams. \\ e.74grams. \end{gathered}$$

Short answer

The correct option is (b) $82grams$

Step-by-step solution

Given information

Population mean $\left(\mu \right)=80$

Population standard deviation $\left(s\right)=2$

Calculation

Consider $X$ to be a random variable that represents cockroach weight in grams.

The percentage of cockroaches weighing between $76$ and $84$ grams is computed as follows:

$$\begin{gathered} P(76<X<84)=P(\frac{76-\mu }{\sigma }<Z<\frac{84-\mu }{\sigma }) \\ =P(\frac{76-80}{2}<Z<\frac{84-80}{2}) \\ =P(-2<Z<2) \\ =0.95 \end{gathered}$$

Thus, the required proportion is approximately $95\%$

Hence, the correct option is$\left(b\right)$