Question

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

Point C on this Normal curve corresponds to

a. $84$ grams.

b. $82$ grams.

c. $78$ grams.

d. $76$ grams.

e. $74$ grams.

Short answer

The correct option is (c) $78grams$

Step-by-step solution

Given information

Mean, $\mu =80grams$

Standard deviation, $\sigma =2grams$

Concept

Use of $68-95-99.7$ rule

Explanation

According to $68-95-99.7$ rule:

In a normal distribution, $68$ percent of the data lies within $1$ standard deviation of the mean.

A normal distribution has $95$ percent of its data within two standard deviations of the mean.

A normal distribution has $99.7\%$ of its data inside $1$ standard deviation of the mean.

Then

The general Normal density graph is represented as:

Now,

We can see that Point $C$ is located at $1$ standard deviation below the mean on the Normal curve.

Then

The weight on corresponding Point $C$

$$\begin{gathered} x_C=\mu -\sigma \\ =80grams-2grams \\ =78grams \end{gathered}$$